Matter in Space and Time

Section 1: Uses high school math, geometry, algebra, and mensuration leading to kinematics, then to physics of mechanics, dynamics, and in some cases, basics of calculus. You have studied these topics. Browse through some cases!

Prove: (A - B) 2 = A 2 - 2AB + B 2

The Greeks understood the totality of anything
to equal the sum of its parts.

An easy way to precede is to begin by with a slightly easier task regarding ( A + B )². The common result was first obtained by Euclid around 300 B.C.

Theorem of Pythagoras

The Pythagorean Theorem, a statement of an algebraic truth regarding the sides and hypotenuse of right triangles, can be proved by use of geometry and the general truth:

"The whole equals the sum of its parts."

Eratosthenes Measures Earth

Knowing geometry and having observed eclipses of the moon, Eratosthanes came to believed Earth was a sphere. To measure its circumference, he devised a plan to used the rays of light form the sun. He assumed all rays from the sun arrived at Earth parallel, each to all others.

Drilling Rig Visibility

drilling rig visibility

The image shows an oil rig at sea which rises 500 feet above sea level and is located 50 "surface" miles from shore. Can the rig be seen from a restaurant atop an 800 foot hotel at the shoreline?

Prove: "The rig can be seen," or tell why it cannot be seen.

To get at it, lets assume some important lady at dinner in the resturant sees the gas flare at the rig and screams, "EGad! Pollution!" By geometry we calculate the distance then compare it to the stated 50 "surface" miles to sea.

Prove: (-1) x (-1) = 1

Most of us use this fact without thinking about it. To prove the formula, we use the results of a previous proof.

Pharaoh's Engineers

The Great Pyramid of Egypt was constructed to precise proportions. A hypothesis is that the pyramid was constructed to fit inside an imaginary hemisphere with each of its corners and its peak touching the hemisphere. Suppose the hypothesis were true. Draw a sketch of a pyramid and calculate the angle each face makes with the horizontal plane of the desert.

Vectors Derive Trig Laws

A typical planar triangle is shown. The triangle vertices, lengths of sides, and angles are identified by notation.

We choose to define a vector space to be the plane of the triangle. We name the two-dimensional space 0XY. We place the origin of the space at the A0B vertex of the triangle and we align its X-axis along the triangle side, 0A...


Crank, Rod and Piston

"Power trains" are the mechanical parts by which the power of combustion is transformed into power of a rotating shaft. A sketch of a simple crank-rod-piston arrangement is shown. Engine designers must know the position of the piston face for every position of the crank. The mathematical tool, vectors, makes this task logical.

Ladder-Boom Rescue

Imagine a retirement home is ablaze and flames have backed two elderly women to the opening of a rear, third story window. A ladder truck has arrived in a back alley but a building obstructs its reach.

To place the ladder at the window, it will be extended nearly vertical to the correct, laser determined, length L, then tipped downward to the angle θ. Lasers will obtain the dimensions (given in the sketch) and a computer will solve the geometry. To provide a test case, use the information of the sketch.

Quick Return

The mechanism O-A-B has two links and ground. The input, Bar-OA, oscillates through small angles,­ α ± Δα. Bar-AB, slot-connected to Bar-OA at A, has a changing, active length, X(t). Figure 1. is the basic configuration with notation. Write a general "specification" for all positions of this planar mechanism. Use a math/physics approach that leads to a solution statement readily expressible in computer code.

High Wire Apparatus

After performing with ponies, the dogs of a traveling show do feats on wire strung on a cubical steel frame. The frame measures ten meters on each edge. Once it is erected cables are strung for specific dogs or tricks. Since the feats of the dogs must move smoothly and promptly, the rigger must have a good knowledge of cable lengths and places of connection. In this example we apply vectors to that task.

Valentino's Wake

When Rudolph Valentino died (1936) his body was displayed at his wake in an open casket. One newspaper report asserted:

"Today 9,000 persons per hour passed
Valentino's casket in solemn tribute."

Is it possible that 9,000 people walked past his casket in an hour?

Dog Greets Owners

Problems in HS physics texts some­times omit detail then request an "answer" when only an "approximated" answer is obtainable. For example!

A dog runs back and forth between its two owners, who are walking toward one another. Initially the owners are 10 meters apart, the dog runs at 3 m/s and the owners each walk at 1.3 m/s. How far will the dog have traveled when the owners meet?"

Yacht and Sea Buoy

A yacht sets sail eastward from a dock at a speed of one mile per hour. A sea buoy is anchored two miles northeast of the dock. Our event starts at t = 0+ with the yacht (position and velocity) and the sea buoy (position) as indicated in the scenario sketch.

When will the yacht pass closest to the Sea Buoy?


This sketch is a physical scenario. It depicts what will happen.
Looking at it closely, we anticipate the lift-rate of the scissor-jack 
will vary during any lift with constant input rotation.

A scissor-type car jack is shown. Member BC is threaded. When it is rotated by the hand-held driver, the distance |BC| shortens which in turn causes the top of the jack (with elevation, h(t)) to move upward. The lead of the threaded bar BC is 0.1 cm/rev, meaning one rotation of the screw shortens the distance |BC| by 0.1 cm. For the position shown,

Calculate the upward velocity of the jack, point D.

Train Passes Two Boys

Two boys, walking beside railroad tracks heard a train approaching from behind. The older boy knew city train speeds were limited to 30 mph and that he and his buddy walked about 3 feet per second. When the nose of the engine was abreast of them, the smaller boy began to count. The count, the instant the caboose passed, was, "... 34 seconds." A moment later, the older boy said, "... only about 1400 feet long."

Civil War Memorial

Many statues have been erected in honor of the soldiers who died in the Civil War. A typical monument (this one in Mulberry, Tennessee) is shown. Suppose you were to observe the original and to notice that it was at rest.

Model the statue as a BODY. What does Newton's Second Law of Motion tell us?

Hand Supports a Mass

The "standard" acceleration of gravity (at sea level and 45° latitude) is 9.80665 m/s². What force is required to hold a mass of 2 kilograms at rest in this gravitational field? What mass can a force of one Newton support?

Lunar "Carry-Off" Luggage

Suppose on Earth an astronaut, using his maximum "lifting force," lifts 30 kilograms. Since unloading on the moon will be easier, let's use a forklift to load on Earth. What should be the mass such that the astronaut can apply the same magnitude of force to unload on the moon?

Scenario shows person holding pota-
toes. To the right is a FBD in which 
the ambient density of air is assumed 
equal to zero.

Ten Pounds of Potatoes

When English Engineering units were specified (~1824) mass was set to be a dimension and defined as a certain, specific quantity of matter. The "founding mass" was a cube of metal kept in a vault in France with replicas for distribution. In those times "force" was an idea more subtle than mass. One immediate idea of force was the "effort" required by someone to hold or support some mass in his hand. Some clever lads supposed force to be a dimension as is mass. Going further, they gave quantification to force. They defined a "unit force" (one pound force) to equal in magnitude, "that force" required to support a unit mass (one pound mass) at sea level.

Blue Ocean Towing

Our company has contracted to protect an off-shore oil rigs. In the Davis Strait, a massive ice slab (~ 800x106kg) has cleaved from the ice-shelf and is drifting with the sea current directly toward the rig.

The map (right) shows the initial location of the the slab relative to the the production rig. We plan that our largest tug, (pulling constantly at 90° to the current), will drag the slab off-course such that it will pass, abreast of the oil rig, at a distance no closer than 4000 meters.

Calculate the towing force the tug must sustain to accomplish the task.

Projectile Arcs Intersect

A body is projected from a point O on horizontal ground with speed of 70 m/s. It passes through a point P, which is 45 m above the ground and 50 m horizontally from O. Calculate the tangents of the two possible angles of elevation of the projected body. Find also the gradients at P of the two paths.

Galileo's Experiments

When Galileo attempted to measure the free fall acceleration of a BODY in Earth's gravity he found its effect in vertical fall too fast to measure. To diminish the effect, to slow the event, (so as to be measurable), Galileo constructed a very smooth, rigid, inclined plane.

Given the measurements of one event, calculate the value of Earth's gravity, go.

Acceleration - Initial Condition

A problem statement, regarding dynamics of a BODY, contains specification of the acceleration of the BODY at the instant of initiation of its event as a "given." Thus the author implies, by some physical means or manner, "at the commencing instant of an event" the acceleration of the BODY is determined. Below is a discussion of how, physically, that "initial condition" might be known.

Parachutist's Drag Force

At the precise instance of leap, the acceleration is Earthward at 9.8m/s². A few moments later, in free fall, the acceleration of a parachutist becomes 6.8 m/s². The mass of the person is 90 kilograms. Calculate the frictional drag force of the surrounding air at that instance of the fall?

GOD Lifted Earth I

This approach takes the Woman and
Earth together as the system.

In Massachusetts (1710) an accused witch was hanged in the public square. At the moment of her death, when that rope snapped taut, an experience occurred to her family and loved ones, they believed an earthquake had happened. At night, in full darkness, friends cut her down and a rumor got started. "When she died, God lifted Earth to receive her," whispered one bystander to the hanging. Within a day minds changed; the entire village came to believed she was innocent.

GOD Lifted Earth II

Often, to understand an event, it is convenient to construct a simpler model to solve. The sketch to the right is a 'physical scenario" similar to the previous example, "GOD Lifted Earth I." Our sketch shows two spheres of equal mass held in a state of constrained equilibrium by a rigid frame and cord. We assume these are the only masses and that the surface gravity of each is represented by go. Think of the cord as a catalyst. When it is cut, an event will occur.

Point Blank

Warships abreast, fewer than 300 yards apart, were in the firing position Lord Nelson's crews called "point-blank." Suppose Ship A fires a cannon, aimed perfectly horizontal, at Ship B. The shot mass 24 pounds. Assume its muzzle speed constant over its flight at 500 ft/s. What does Newton's Second Law tell us about "Point Blank?"


Virginia Shot Tower


From years before the Revolutionary War and through the Civil War over a million spherical lead shot for muskets were manufactured at the shot tower in Wytheville, Virginia. The shot sold in twenty-five pound bags. Of the lead shot produced, 10% failed to qualify and were remelted.

What least mass of lead was melted per bag of shot produced?
Approximately how many shot would a bag contain?
Calculate the speed attained by a shot that falls for three seconds.

Geostationary Orbits

Modern rockets routinely place communications satellites in Earth orbit such that they maintain a constant position above and relative to Earth. Such orbits (which are possible only in the plane of the equator) are accomplished when the rocket "parks" the satellite at the proper altitude and with the proper velocity. The satellite is parked ina Geostationary orbit. How high up is that?


A caution of physics texts about usage of Newton's 2nd Law is: "Be sure to use a non-acccelerating, that is inertial coordinate reference." However, after that state­ment, through numerous examples next to nothing further is said or done regarding inertial coordinate reference selection. The examples, being in an Earth environment, does text disregard of reference imply all places on Earth are inertial?

Upon further reading one encounters a second "caution." If coordinates "not inertial (by error or choice)" are used, special "fictitious" or "pseudo" forces might be needed. Centrifugal force is the simplest case.

Polar Versus Equatorial Weight

Suppose we have a certified mass of 1,000 pounds. Demonstrate analytically that the mass will weigh 1,000 pounds force at the North Pole but only 996 pounds force at the equator? (Ignore the buoyant effect of atmospheric air).

Mass and Sled

The sketch is from a physics text. According to the drawing, there is one horizontal force that acts with magnitude, F, directed to the right. The implication is acceleration of masses m and M occurs "at the very instance" of observation.

Calculate: The greatest force such that NO sliding of the mass on the sled will occur.

Kinematics of Bar-AB

Typical text-book "given-information" sketch.

"Bar-AB and Ground" constitute a kinematic mechanism. Connections at A and B (not shown) maintain sliding contact with the ground. At the instant shown, End-B has a velocity of 0.5m/s and an acceleration of 0.3m/s² (in the directions shown). Calculate the angular acceleration of the Bar-AB and the acceleration of End-A.

Lift-Off Acceleration


Motion char­acteristics of a rocket during its lift-off are observed by a tracking laser housed in a blast-proof bunker at a distance from the launch pad. The figure shows the initial situation.

Ideal Fluids

The ideal fluid is a frictionless liquid or gas. Properties are mass, volume, density, pressure and temperature. Consequences of Newton's 2nd Law are called "hydrostatics." The hydrostatic equation is a first order equation. The Ideal Gas Equation is introduced.

LNG Tanker

The photo of a Norwegian tanker is shown. The tanker capacity (all five of its spherical tanks completely full) is 128,000 cubic meters of liquid natural gas (LNG). A scale profile of the tanker is available.

Estimate the length of the tanker.

Sphere, Tank and Water

The diameter of the sphere suspended above the tank containing water is one foot. When the sphere is lowered into the tank the water contained rises to a final depth of two feet. No water spills over.

Calculate the mass of water in the tank.

Concrete Mixture

At the simplest level a mixture is classified as homogeneous or non-homogeneous. Homogeneous means uniform through out. The text example discussed here (Sonntag et al. Example 2.1) treats its mixture as being homogeneous but the figure provided by the authors shows clearly that the mixture is not uniform throughout. Indeed, since air is included as a system component, the mixture cannot possibly be uniform throughout.

Hurricane Wilma

This very large hurricane set some records in October, 2005. At times its winds reached 185 miles per hour. Measurements showed the barometric pressure in her eye to be 882 mbars of mercury. These pressure measurements were "static."

Torricelli's Barometer

Torricelli, a hydraulics engineer, established a number for the pressure of the atmosphere. The idea of pressure was known to many but Torricelli made that property quantitative.

He and fellow engineers (1645) were puzzled that their best water pumps, used in mining, failed completely to draw water columns any higher that 34 feet. At that height, "vacuum was created" and the flow abruptly failed. Thinking about water and liquids, Torricelli rationalized...

Pascal's Experiment

In 1648, the brother-in-law of Blaise Pascal carried a mercury barometer from the base to the top of the French volcanic formation, Puy de Dome. The peak of Puy de Dome is 4803 feet above its base.

Approximately what difference in heights of mercury in the barometer were observed?

Density of Fluid "A"

The glass U-shaped tube contains Fluid "A" and water. Since the fluids are static, the hydrostatic equation applies.

Calculate the density of fluid A.

Oil, Cable Support a Rod

A cylindrical steel bar of length "L" and a cross-sectional area, "A" is suspended by a cable. The upper third of the bar is surrounded by atmospheric air; the lower two-thirds of the bar is submerged in oil. Determine an equation for the tension of the cable?

Water Barometer

A sketch is shown being the configuration of a barometer but the fluid is water not mercury. The "pressure of the void" for water is the vapor pressure of water at 80°F. This pressure is about 0.5 psi. Use the schematic.

Calculate the height of the column of water, H.


The "piezometer" is a pressure measuring device of antiquity (and with a few applications now). How it works is our interest.

The image depicts water flowing through a long horizontal pipe. The pipe elevation is above sea level; the water flows "to the sea." The hydrostatic principle can be used to determine the pressure of the "flowing" water in the pipe.

Manometer, Bourdon Gage

The ele­vation of the horizontal pipe is slightly above sea level. Water flows through a pipe that has been fitted with a manometer and a Bourdon gage. Use the readings of the manometer and Bourdon gage:

Calculate the pressure of water flowing through the pipe.

Pressure Calculations

The sketch shows a gage connected to a tank. The gage has "vacuum" written on its face and its reading (GR) is 5.8 psi. Also attached to the tank is a manometer containing oil with a specific gravity (S.G.) of 0.9. The ambient pressure is known to be 14.5 psi. The schematic also shows a mercury barometer and a water barometer beside the tank.

Pressure of the Void

To high precision, standard atmospheric pressure equals 101.32 kPa and will support a barometric column of mercury 760 millimeters in height. These numbers are defined to be (assumed as) the surface-of-Earth average.

Torricelli assumed the void at the top of the column of a mercury barometer to be "vacuum." He also assumed vacuum meant numerically zero pressure. The actual, non-zero pressure, "in the void" of a mercury barometer is called the vapor pressure of mercury at the prevailing temperature. Use the information of the sketch. Calculate the "Pressure of the Void."

Cave Diver

On average, 8 persons a year drown while exploring spring-fed underwater caves in Florida. In one cave a local dive club posted warning signs and installed a mercury barometer. The schematic shows a diver's view of the barometer and its location relative to the spring surface.

Calculate the depth of the diver beneath the surface of the spring.

Wilma's Surge


Many persons who have experienced the eye of a hurricane have witnessed its calmness, its winds with speeds and also its very high "tides." These are not tides. The seemingly "tide" phenomenon is more precisely called "storm surge." High water levels that occur in the eye of a hurricane are not directly related to location of the moon. The water levels are forced by atmospheric air that surrounds the storm at a distance. Estimate the height of water surge within the eye of Hurricane Wilma.

Off-Shore Natural Gas

An image of an off-shore natural gas well is shown. In normal operation natural gas flows from a gas reservoir beneath the ocean floor upward through pipes to the production rig then through pipes leading to shore facilities. Occasionally valves are closed such the gas flow is stopped. The "stopped no flow" flow" condition is our interest. Use the information provided to estimate the well-head pressure of the natural gas.


A glass U-tube is shown in its "Initial Condition," 1). The tube is filled with mercury to the level "0 - - 0", its legs are open at their tops to the atmosphere. The cross-sectional areas, left and right, are AL and AR. Beside the U-tube is a cup containing mw grams of water.

Bubble-Tube Depth Gage

The sketch show a bubble-tube apparatus. An open tube extends to very near the bottom of a tank of oil. In operation, compressed nitrogen is supplied to the tubing at low pressure. The nitrogen bubbles from the bottom of the tube into the oil so slowly that the pressure in the gas at depth equals the oil pressure adjacent to it (at B). Use the information given. Calculate the depth from seal level to the surface of the oil.

Monster Tires

The load carried by the six large tires of a fully-loaded strip-mine hauler is 360 tons. Carrying such a great load, we expect that the air pressure in the tires to be great. It is not possible to calculate the pressure of the air in the tires of a given truck. However knowing the hub dimensions (given below), a calculation is possible.

Estimate the least possible air pressure required within each tire to support the truck and its load.

Volume of a Sculpture

Many replicas of a sculpture are to be made. The completed model is given to a technician to determine the volume of precious metal each replica will require. The material of the model is compact and its density is known. Although the model cannot be immersed in water, two precision scales are available.

Arctic Shelf-Ice

The image shows an edge of the Ross Ice Shelf northeast of Canada. From its top to sea level the ice slab measures about twenty meters. Presently about 5 million square miles of similar shelf-ice float on the seas of Earth. In many places that ice is melting. As floating ice melts into the sea, calculate the height of rise of sea level.

Ideal Gas Calculations

The specific energy of a single phase state of a pure substance is least as solid, greater as liquid and greatest as a gas. In general, the greater the temperature or lower the pressure (or both trends combined) the more likely a gas is to be Ideal. Although we discuss the first, extensive form of the ideal gas equation, the results apply to either form.

Casting Rail Car Wheels

A somewhat elaborate apparatus is used to cast the steel wheels for railroad cars. The process begins when a large "ladle" of molten steel is placed in a chamber which is then covered with a heavy air-tight lid. Ceramic wheel molds are arranged on top of the chamber. A "refractory tube" or pipe extends downward from each mold to the bottom of the pool of molten steel. When the chamber is pressurized with gaseous nitrogen, molten steel flows from the ladle upward into the molds.

Professor Foghorn's Gage

Professor Foghorn constructed a super-sensitive gage for his Arctic Laboratory. He claims the gage can measure in the range of 1 megaPascal to an accuracy of
±1 Pascal.

Foghorn set up the apparatus as shown and asked you what his gage should read. Neglect the density of the ambient air and the air in the tank.

What reading is expected?

Abandoned Car

While cleaning trash from an estuary, students discovered an abandoned car exposed only at low tide. They decided to remove the eyesore.

Their plan was to insert inverted steel washtubs inside the car at low tide. With the bottoms of the tubs against the roof, the incoming tide would trap air in the them and float the vehicle. Once floated, the car body would be dragged to much deeper water then sunk from sight.

First Equestrian Ascent

At the World Fair (Paris ~ 1798) the crowd watched curiously as Louis Testu-Brissy assembled his hot air balloon with its platform-shaped gondola. His aides lighted the "lifting flame," the balloon inflated, became spherical and then, to the astonishment of all, the balloon and horse (with Louis astride) slowly ascended high above the fairground.

Otto Von Guernike

German scientist, inventor and engineer.

At a fair in 1654 Otto amazed the spectators. He showed them two hemispherical brass shells that fit together (to form a hollow sphere) but fell apart when not held. Next, (using the air pump he invented) Otto removed air from the inside of the sphere. Each hemisphere had built-in ring to which a rope was attached. The other ends of the ropes were attached to harnesses of two teams of horses. The horses, urged to pull, strained and snorted but, to the surprise of everyone, they were unable to pull the hemispherical shells apart.

Victor Regnault

This careful and meticulous scientist (1810 - 1878) was one of the first to measure the density of atmospheric air. A master glass blower, Victor crafted two near identical flasks; one being a test flask, the other being a "counterpoise." Both flasks were filled with dry (standard) atmospheric air. Next, an air pump was used to extract air from the test flask until the inside air pressure became 560 mm Hg barometric. Both flasks were placed on a beam balance...

Roof-Top Exit

A building 1200 feet in height will have a doorway for rooftop exit. Air interior to the building will be maintained at 70°F and prevailing local pressure. The worst case of air outside in a light breeze is 40°F and 14.7 psi.

On the roof is a powerful fan which when turned on, will act to force air from the building and into the outside air. At ground level the building has a typical door which we assume to be air-tight such that air cannot leak past the seals along its sides, top and bottom. Furthermore assume the entirety of the rest of the building has no "leaks" or paths for air to enter or leave the building - the single exception is the exhaust duct of the fan on the roof.

Two Tanks

Each tank is square in cross-section and has a valve for filling with water (or venting of air) at its top. The tanks are connected at their bottoms by a pipe with a valve.

Two cases are to be studied.  The sketch shows the setup for both: all valves are closed, the depth of water in tank A is 5 meters and the water depth in tank B is zero. Events initiate when valves are opened. For each event, calculate the final depth of water in each tank.

Ocean of Air - its Average Density

Earth is spherical, and above Earth, our atmosphere is a spherical shell of gases. The Earth diameter is about 6,380 kilometers and the depth of the atmosphere above its surface is about 30 kilometers. The standard atmospheric pressure, is assumed to have an average over the Earth surface of 101.3 kPa (enough precision for now). Estimate the average density of the atmosphere of Earth.

Hand Lotion Event

An air-traveler packed a 1/2 liter, 1/4 full, rigid plastic bottle of hand lotion in a "bottom-up" position in her bag with clothing (shown in the sketch). The little spout of the bottle was in the open position but lotions hardly flow at all. The bottle was loaded into the cargo hold of the aircraft which then climbed to 35,000 feet where pressure and temperature were 30 kPa and -30°C. When the plane landed, approximately what volume of lotion will remain in the bottle?

Fuel Cell Proportions

A fuel-cell is a battery-like chemical device into which flows oxygen and hydrogen. In the fuel cell these species react to produce electrical potential and current. The exhaust of the fuel cell is water. As one comes to understand basic thermodynamics, one is obliged to learn some basic chemistry.

Concreted Nuclear Waste

Fast-Breeder Reactors produce fuel that is used for other reactors or for weapons. Spent fuel (just out of the reactor) is agitated, hot, and lethal. To calm down, it is placed in cooling ponds for several years. From the pond it is solidified and mixed into cubical concrete blocks which are buried...

Work, Energy and Heat

Section 3: Newton created the construct, FORCE, as the cause of motion. The matter in question was modeled to the simplest extent, to be a BODY. Thus mass of a BODY and Force were beginning ideas about events of mass, i.e., about its motion - its basic activity. Today, meaningful understanding of motion requires math at a vectors and calculus level. If you don't like math - not to worry. Someone else will understand this physics and tell you about it. So let's begin... What is work and energy will be told in due time.

Champagne Cork Muzzle Velocity

After the speeches and glad-handing at the Professors Emeritus Banquet," a student bartender, opened the first bottle of champagne with a loud "POP." Everyone laughed as the cork flew upward to pass over a gymnasium rafter.

Perfect Bounce

A son who borrowed his father's Mercedes is driving safely at 50 miles per hour but bit too closely behind a truck loaded with bricks. His date says,

"Honey, is a brick falling off that truck?"

Is there any chance the brick might bounce over Dad's Mercedes?

Kids Clean the Garage

Two boys struggled to move a tractor tire from flat on the floor to a position leaning against a wall. The sketch shows the tire initially 1) and finally 2). What "least" work did the boys do?

Planet Gizmo

Consider a block of mass 6 kilograms. The block was first observed moving down a frictionless plane with a kinetic energy of 19,200 Joules, State (1). We set the time (the moment of first observation) to be, t = 0+. The coordinate "S," is added to denote distance down the plane.

Calculate the speed, velocity, kinetic energy, position as a function of time, potential energy, and position and velocity in 3 seconds (not necessarily in order):

Slap Shot

A hockey player received the puck and checked it. Then with a quick (approximately 25 centimeter) sweep of his stick, he slapped the puck across ice into the net. Our system, is the 0.2 kilogram puck received with low speed. Our event is its "slap" to goal, delivered at 30 meters per second.

Calculate the Work of the shot.

Bilge Pump Work

At sea in a small boat during a storm, a man operates a manual bilge pump. t. Raised by hand, the piston lifts water from the bilge of the boat and dumps it into the bay. Given the information provided by the sketch.

Calculate the least (theoretical) work of the man per stroke of the pump.

Buoy Lift I

The sketch depicts a buoy floating in the sea. A cable is lowered from a dock and attached to the buoy (1). The event is that by the cable, the buoy is hauled to a second state, (2).

Calculate the least work required of the cable that lifts the buoy.

Buoy Lift II

The sketch, (1) shows a buoy floating in the sea. A cable lowered from a dock is attached to the buoy and the buoy is hauled to state (2). Take the system to be the buoy, a short length of the cable and part of the sea. The cable moves the buoy for condition (1) to condition (2) as shown.

Calculate  the least work required of the cable.

Work to Move a Wall

C240_i01 A tank contains a shallow body of water. Its bottom, left wall, and ends are fixed. The right wall can be pushed to the left and will remain vertical should that happen. Seals prevent all leakage of water. Figure 1. shows the wall in its initial and final positions. Derive an algebraic expression for the "Least Work" required of the worker to push the wall a distance, L, to the left.

Harpy's Nest

To observe nesting eagles, naturalists built a blind. With a cross-bow, a line was passed through the canopy of a neighboring tree. Next a block and tackle (a "gun tackle" shown below) was hoisted and manipulated to attach to a secure limb.

First Lift:   The woman got into the harness and the man hauled rope to lift her (mg = 500 N) to the tree top. Calculate the work by integration of hauling force applied to the rope times its displacement. Next calculate the same work by use of the energy equation.

Back Squat

The Bulgarian, Ivan Charakov, is shown in the "squat stage" of a competition lift. Once he adjusts his stance, he will extend his legs to raise the weight to his full height.

Analysis using principles of physics make reality more understandable. Stated briefly, analysis addresses three physical properties: mass, momentum and energy.

Newton's Rock Climb

Consider a rock climber (50 kilograms) who climbs to the summit of a popular practice-level rock formation. The increase of elevation of the climber is 80 meters.

Typically, persons would view this task or event at one that involves work. In the common meaning of the word, it does involve "effort" or work. First apply Newton's 2nd Law, then apply the principles of thermodynamics.

Calculate the work of the climber.

Work to Raise the Castle Bridge

Our system is the simple span of a drawbridge. To "select the bridge as system," is to isolate it. To do this, let your mind trace the entirety of the boundary of bridge. The event for the bridge will be that the "drawing ropes" pulling it from its horizontal to its vertical position.

What least work is required to raise the span?

Elephant Weighs Teak

An elephant and her mahut can weigh a teak log. The mahut shows the elephant a piece of chalk then points to a teak log; the elephant knows what to do. With its trunk and tusks the elephant lifts the log slowly and carefully to a balanced, level position, pauses, then puts the log down. The mahut marks the balance point of the log as his helper climbs onto one end. The elephant is shown the chalk again. It lifts and balances the log a second time with the helper standing on one end, grasping the elephant's ear for balance. The mahut uses this information to estimate the weight of the log.

Polaris Scale Model

Members of a rocketry club constructed a scale model of the submarine-launched ICBM, Polaris. Since the model was very light, the idea occurred that its buoyancy might accomplish the model's first stage of flight (without a rocket). Upon release at a depth in water, buoyancy of the model might project it from the water into air. As a test of this possibility they built a balsa-wood model which they secured to the bottom of a pond planning that its buoyancy would be the first stage. Simple tests are costly and time-consuming. Perform an energy analysis of the proposed event. Assume the event proceeds without friction.

Calculate the greatest possible elevation of launch above water of this first stage.

Aircraft Work Against Drag

An airliner of mass 300 tons is powered by four engines each of which develops a steady power of 15,000 kW. The take-off speed of the aircraft 75 m/s. The aircraft requires 11 minutes to attain its cruising speed of 210 m/s at an altitude of 10,000 meters.

Calculate the work of the aircraft expended to overcome the air resistance, or drag, it experiences for the duration of the climb.

Workers A and B

Two events are analyzed
with approximation.

Basic mechanics can be used with the energy equation to evaluate the efficacies of work tasks. Consider the manners by which each of two workers moves a BOX (m = 12kg) from the storage floor to the tailgate of a truck. Use the data of the figures.

Calculate the energy expended (J), and the average power (W) for the moving events of A and B as described below.

Crate Pushed Uphill

This concerns the simple task of pushing a crate slowly up a plane inclined at 45 degrees. The surface opposes the motion; it has a coefficient of friction, μ. The simple geometry permits both the energy change and work of the event to be calculated then compared. Note: Earth and the Block are taken as system.

Calculate the energy change of the crate.

Optimized Move of a Crate

A force is applied to move a heavy crate across a horizontal surface. The coefficient of friction of the crate and surface is 0.3 (μ = 0.3). The crate is modeled as an extended body. The sketch shows its free-body-diagram and the orientation of the applied force. The event is that the crate is pushed a distance, L, to the right.

a) Obtain a general expression for the least magnitude of the applied force required, as a function of the angle of application of that force, θ.

b) Calculate the required force and work of the event for the angle of application, θ = 0

c) Calculate the least force required to accomplish the event and the least work.

Drunk Driving Awareness

At youth fairs across the State, Law Enforcement Agencies simulate the aftermath a head-on collision for all to see. A crane hoists the hulk of an automobile so high that when it is dropped, the car impacts solid concrete at 70 miles per hour.

Calculate the least height the hulk must be raised such that 70 mph is attained at impact with the ground?

Wreck at Batavia

On Feb 18, 1885, a train engine moving at high speed collided with a second, stationary engine. Their engines were identical, having been manufactured by the same company. The aftermath of the collision was engine A (stationary upon impact) situated near vertically on top of engine B.

Rock Dropped on a Spring

State (1) shows a rock of mass, m, supported by a cord above an upright spring. There is "touching" contact but the initial spring force is zero. The length of the spring is its upright free length, Lo. Upon cut of the cord, the rock falls and compresses the spring to its shortest length, whereupon the spring push the rock upward, and so, oscillating and eventually come to rest - State (2).

Batch Mix Event

The apparatus mixes the ingredients of a polymer that is used to seal and fire-proof the interiors of aircraft fuel tanks. Ingredients are poured in from the top which remains open to the atmosphere. The stirring propeller rotates at 60 rpm, driven by the motor below. The table contains information regarding the event.

Apply the energy equation. What does it tell us?

Blowgun Hunter

Men hunt monkeys and birds in the Amazon forest using long wooden tubes called blowguns. A monkey in sight, the hunter slips a poisoned dart into the tube and quietly lifts it to vertical. Taking a deep breath and a steady aim, he contracts his chest to blow the dart skyward. When successful, the dart hits a monkey and it falls to the ground. The image shows geometry of a hunt.

Calculate the least muzzle velocity of the dart.

Skate Board Skill


A standard feat of skate board competitions is "get air" which means simply to "jump" in which the skater starts by tipping into the ramp (1), gaining momentum with the fall then "getting air" or height on the opposite side (2). Skate boarders know "Air is not for free."

What must a skate boarder "pay" to attain one meter of air (height)?

Newman's Annihilator

In 1860, Alfred E. Newman patented this clever device to annihilate bugs that might hide on the ceiling. Al and I bought one. Be quiet, we are are about to demolish our first fly. It is sitting on our bathroom ceiling.

By forklift, we quietly positioned the annihilator to enclose the fly. I'm a little nervous, we are standing under a very large piston. Al claims this tested just fine on a pesky raccoon.

Kids on a Swing

Two children sit in a porch swing which is initially restrained from motion by a rope, 1). When child B releases the rope, the two swing through air until the swing and boys cease to move. The final, at rest angle (θfin), defines the final condition of the system, 2).

Calculate the final angle the swing will attain.

SCUBA Horsepower

The regulator of a SCUBA apparatus operates in two stages. The first stage lets high pressure air leave the tank to enter a small inter-stage chamber. When the diver attempts to draw a breath through the mouthpiece, the valve exiting the second stage opens to supply the breathing air. Given data, estimate the possible duration a dive.

Scalar Product of Unit Vectors

It is fun to let vectors beat up trigonometry. Vectors and their operations contain trigonometry. Here we write two unit vectors, then scalar multiply them to obtain the formula:

cos(α + β) = cosα cosβ - sinα sinβ

3.16 About Heat and Temperature

Science students are physically aware of temperature and the associated event, heat. When heat and temperature are being considered, the model BODY or point mass is inadequate. The model required is pure substance or material (metal, solid and so). In addition a new energy form is needed: internal energy.

Discharge of an Air Pistol

A great­ly simplified sketch depicts a loaded "air pistol." The chamber of the pistol contains air compressed to 3 MPa. The "trigger" is a pin that constrains the bullet in the barrel. When the pin is pulled, the bullet is released and the compressed air expands to expel it from the barrel into the surrounding atmosphere. Use the data of the sketch. Calculate the theoretially maximum velocity the bullet might attain.

Air/Water Mortar

It is proposed to construct a mortar to shoot water vertically. It is intended that water be the "projectile." The air compressed beneath the piston will be the "propellant." The figure shows the steps to "load" the mortar. We will Analyze the "loading" and subsequent of "firing" of the motar.

Tank, Spring and Heater

Initially the temperature of the everything is uniformly 300 K and the water in the stand-pipe resides at (1). The spring is compressed, it holds the frictionless piston to the left. At time, t = 0+, the heater is turned on. It deliver 6 kilowatts of thermal energy to the air.

The final state occurs when the water level in the stand-pipe is 8 meters higher. At what time is State (2) attained?

Candle Beneath a Piston

The event of this dis­cussion is an "idealized heating" of an ideal gas inside an "idealized piston/cylinder" assembly. You must believe in make-believe! The theoretical, frictionless apparatus assures a constant pressure process of the ideal gas. Conditions of the figure correspond to the instant the candle is lighted. When the candle has burned completely it will have provided 200 Joules of heat to the diatomic ideal gas (M = 83.14) contained.

Average Force to Brake

As a driver in a car traveling at 60 km/hr approaches an intersection, the light suddenly changes. The driver applies the breaks until the car stop. If the vehicle (plus driver) mass is 1075 kilograms and the deceleration is constant over the 5 seconds of stopping, calculate the average force that acts at the points of tire-to-road contact.

Vacuum-Launched Rocket

To place scientific packages into Earth orbit is expensive. A great part of the cost is the fuel consumed with lift-off. The schematic shows a system proposed as the first, lift-off stage.

Setup:  The tall circular launch tube is a shaft that extends downward into a large mine with numerous accesses. The rocket, positioned in the tube snugly, is supported at the bottom of the shaft. A strong, deployable hatch covers the top of the shaft and beneath the rocket, sealing the shaft below, is an explosively destructible shield. Prior to a launch, vacuum machines slowly and economically extract air from the shaft until a pressure of 3 kPa is attained.

Hot Shot

In the earlier Example Point Blank a sailing man-of-war fired a first cannon shot at the enemy. If that shot was on target it is likely the next round would be a "hot shot." These rounds were heated in a coal fire until they were cherry red (about 1200°C). The shot was then loaded, "oh so carefully," and fired.

Bungee Jumper

The design and setup of bungee jumping ropes is a matter of "life or death." When done wrong, we read about it in the newspapers. The rail of the New River Gorge Bridge passes 785 feet above the surface of the slow, trickling stream, the New River.

Our jumper, a 120 pound "living-dangerous" lady, wants the ropes set so at the bottom of her jump she can grab a fistful of water from the river.

Box and Springs

The sketch (left part) shows a box shoved against the strong spring compressing it 4 centimeters and restrained there by a "trigger" mechanism. When the trigger (our catalyst) is pulled the box is projected to slide across the "assumed smooth" plane, ultimately to encounter the weaker spring and compress it.

Washer Water-Level Switch

When a washing machine is started, a valve opens and to allow water to fill the washing tub. Rising slowly, the water level eventually closes the bottom of the water shut-off mechanism. An approximation of the mechanism, a tube open at the bottom and closed at the top by a spring-type bellows is shown. As the water level rises, air trapped in the shut-off tube is compressed, its pressure (acting over the inside area of the bellows), extends the bellows slowly upward until a switch is thrown setting the water supply valve to its "off" position.

♦  Calculate the "shut-off" depth of water in the washer.

Elevator Speed

An elevator operates by cables, pulleys, an electric motor and controls. This elevator is driven by an 8-kW motor. Assume the elevator operates with no power lost to friction.

Calculate the maximum rate of ascent of the elevator.

Gravity Work Rate:

This is being developed!

Ship's Reduction Gear

The marine turbine of a destroyer produces 5000 horsepower while rotating 25 thousand revolutions per minute (rpm). But the ship propellers work best while rotating as 180 rpm.

Were the gear to operate ideally, what would be the output torque?

Horsepower of an Ox

In many countries, oxen, guided by children, plow straight, level furrows through soil. The ox of the figure pulls its plow a length of 200 meters in five minutes.

Calculate the least average horsepower expended by the ox.

Truck Least Horsepower

The power, or energy rate, of a truck varies considerably with its motion through the gears, speeds and such. Suppose in normal operation a truck loaded to 80,000 pounds moved horizontally from stop to a speed of 40 miles per hour in a distance of about 1/4 mile. Use the definition that “truck horsepower” is the time-average change of kinetic and potential energy of the truck in its event.

What average horsepower is required by the truck to accomplish this event?

Tank Locomotion

The system of this discussion is a typical military tank that moves only in the horizontal. The perspective of discussion is the energy equation, rate form, (potential energy omitted ~ horizontal motion).

Cyclist Power

The events of a bicycle race are explained simplistically as "the cyclist expends physical (metabolic) energy to move himself/herself and the cycle along a road and through the surrounding air. In places the road is flat and further along there are hills that must be climbed. Descents from hilltops are the easiest events of a cyclist's race.

Racers describe their racing as having three types: time trial, climbing, and downhill. In this example we compare the three phases at those times, at instances for which speed of the cyclist/cycle system is constant.

Car Drives Uphill

A car that weighs of 3000 N has been tested thoroughly. Under full power its engine produces 160 kW of power (in excess of internal losses, rolling friction and other losses). Traveling at top speed on a level road its maximum speed is 35 m/s. Suppose the car were to travel up a hill with a rise of 1 meter for every 20 meters traveled. What constant speed would be expected?

18 Wheeler Safe-Braking Speeds

The event of "braking" of an 18-Wheeler falls in the category "energy is conserved" (at least temporarily). When the fast-moving truck brakes, its kinetic energy transforms into increased internal energy of the truck. This energy resides principally and momentarily, in the brake shoes and drums. By basic physics we know increased internal energy of a solid is exhibited by its increase of temperature. In this example, we stop a big truck safely so the brakes get hot but are not ruined.

Atwood's Machine

In the time of Newton accurate determination of the surface acceleration of gravity was difficult. Some forty years later, the Reverend George Atwood devised (and built) a quite clever apparatus to measure that acceleration. So successful was his device that thereafter all devices developed to measure Earth's acceleration of gravity were named, in his honor, "Atwood Machines."

Mao Zedong

... ordered massive tracts of land along the banks of the Yellow River to be cleared of trees and turned to agriculture. But snow and rain flowed from the mountains across the treeless banks and stripped fertile soil into Yellow River, whereupon it was carried all the way to the Yantze to contribute to severe flooding. Next, to control the flooding, Mao had his people build large dikes along the Yantze - by hand.

Pulling Sheet Piling

Sheet piling are used to construct temporary barriers in waterways to put in bridge footers, lay fuel lines and such. Once placed, the longer those piles remain (especially in salt water), the greater the headache they are to remove. Worth salvage, the embedded piles must come out, one at a time with their connecting edges intact. By watching a load gage mounted on the drag-line boom, experienced operators can decide when salvage is worth the wear that removing them will put on the lifting machine. Sometimes the piles are pulled and scrapped.

Thermodynamic Properties

Section 4: Some thermodynamic events involve change of phase of the system substance. This is a complication which Section 4 addresses.

Chef Thickens the Soup.

A pot on a range contains four liters of soup and is at a gentle boil. The chef intends that the soup be thickened slowly by boiling away 30% of its water. The rating of the stove element is 1800 watts.

What is the least time required to thicken the soup?

When will the Teapot Whistle?

When you turn it on, a thermally insulated, electrically heated teapot pour contains 1 liter of water at @25°C. The metal of the teapot is equivalent to 200 cubic centimeters of water and the label of the teapot says it is rated at 1250 W. How long will you have to wait for the water to boil and for the pot to whistle?

WATER at 1 Atmosphere

The table presents selected thermodynamic properties of water. Water at 0° can be either solid (with its properties) or liquid (with its properties). To list both sets of properties, 0° is notated as 0-°C and 0+°C. The temperatures, 0-°C, means 0°C but solid; 0+°C means 0°C but liquid.

Citrus Concentrate

Great orange juice is made from fresh oranges sliced and squeezed. The alternative is to prepare the juice from a concentrated product. Water and valuable oils are removed from fresh citrus at a processing plant then "concentrate" is shipped to consumers with the instructions,

"Mix concentrate with 3 cans of tap water."


Microwave Coffee

A cup of day-old coffee (25°C, 500 cc) was put into a microwave oven and "heated." After a minute and fifteen seconds, the oven bell rang and the coffee was removed. Its temperature was measured to be 90°C.

Calculate the least electric work the water received from the microwave?

Grease Fire Experiment

When a deep-frying cooker catches fire, to douse the flaming grease with water, is a bad idea. The water sinks beneath the grease surface and is heated immediately to steam. The explosive change of volume, as liquid water becomes gas, will blast flaming grease throughout the kitchen.

One morning around 2 AM two bored new-hires at CHICKEN OUTLET decided it would be "cool" to explode some ice in the deep-fryer. "No flame, no problem," they assumed...

Yardley's Extractor (1856)

This introductory writing about thermodynamics is restricted.
To present the full use of water in industrial applications 
(at its varying and much higher pressures) would require pages 
beyond our scope. However, one high-pressure example is presented 
for those students who boil bones to make glue.

Yardley used a spherical vat with hot high-pressure steam to render animal bones into glue. The vat was filled with bones, fat and cartilage. Steam was admitted to purge the air, then the vat port cover was pulled snug. The vat was charged with steam to 250 kPa and 500°C then rotated slowly. It usually took about an hour for the steam to cool until the pressure in the vat became one atmosphere whereupon the port cover would pop open. Then the glue was drained into the sump and the process was repeated.

Stone Boiling

The Indians of the early southwest used stone boiling to cook tender parts of bison, particularly the liver. After a successful hunt the paunch and liver were cut from a bison. Fire wood was gathered and stacked in criss-crossed layers then stones were placed on top and the wood was set afire. Nearby, the paunch was inverted, filled with water and hung on a tripod of branches.

closed system at t = 0-

Pressure Cooker

Pressure cooking is largely an industrial process but many domestic kitchens also pressure cook. Thorough analysis of a pressure cooker can be tedious because they are transient. Usually an open-system is used.

Also, at the beginning of operation the cooker contains a mixture: water and air. "Texts" might avoid these difficulties by "starting the problem" at a special time after the cooker has been heated on the stove, has attained its pressure and expelled all air.

Emergency Power

An example in the thermo text by Moran and Shapiro provides the schematic (right) then proceeds to calculate the work that the apparatus might produce when activated. The authors provide the information of the sketch and the final pressure and temperature of the steam; 15 bar and 400°C.

Neon Signage

As capitalism grow, vastly greater amounts of signage will appear. Neon is cheaper for smaller signs. Glass tube is cut, heated and bent to shape. Into each tube end, an electrode fitting is welded to bring power to the bulb. A last step, prior to filling the bulb with gas and sealing, is to heat treat the assembly to burn out impurities and water and to stress-relieve the glass welds. Burn-out is accomplished by passing direct current electricity at very high voltage (but low current) into the lead at one end of the tube through the glass and out the other.

Leaded Pipe Joint

The joint or connection of two sections of cast iron pipe is accomplished by pouring and caulking lead. Preparing to "lead" a pipe joint, a plumber puts 15 pounds of lead into a "hot pot." The lead is initially 25°C Lead melts at 327°C. What least heat is required to melt all of the lead?

Specific Heat of Liquid "X"

To raise the temperature of a container by 1 K requires 50 Joules. 250 grams of a fluid is placed in the container (assume both at 1 atm and 300 K). The temperature is raised with a heater, P = 15 Watts. After a while the temperature is stabilized and remains constant until the heater is turned off. The temperature then falls by 1.2K/min. Calculate the specific heat of the fluid.

Space Shuttle Re-entry

The shuttle has a mass 2.7 million kilograms. When it returns from mission, its re-entry commences with a speed near 7000 meters per second and an altitude of about 125 kilometers. The shuttle must descend and arrive at a runway with a landing speed of 160 m/s. This happens quickly - in about 30 minutes. Thus the immense kinetic energy and potential energy of orbit must be dissipated to friction with the atmosphere.

Copper Block Slides on Ice

A block of copper (m = 1.6 kg) rests upon a smooth plane of ice. Initially the temperatures of the copper, the surrounding atmosphere and the ice are 0°C. In an instant motion is imparted to the block causing it to slide across the ice slab at 2.5 m/s. Immediately sliding-friction causes the speed of the block to diminished; ultimately the block attains the speed, 0 m/s.

Calculate  the greatest mass of ice (solid water @0°C) that is changed water (liquid water @0°C) by the event.

Sausage Preparation

Once prepared, sausages must be refrigerated promptly. The sketch shows ingredients for a "batch" of sausage (at 15°C) just prior their last pass through the grinder. To hasten chilling, rushed dry ice (solid carbon dioxide) is added this time. Assume the grinder requires four minutes of its 3/4 horsepower (HP) motor for the final grind. The final condition required is that the meats and carbon dioxide exit the grinder at a temperature of 1°C. Calculate the least amount of Dry Ice that must be added to the mix.

Ice vrs Dry Ice: A Competition

Water as a solid (frozen water) is commonly called "ice." On picnics, we put ice into the cooler so it will cool our beer and soft drinks to refreshing temperatures. The ice is cold, as it melts it cools its surroundings. "Short Term" means for that ice or dry-ice is used to accomplish some cooling task it exhausts itself.

Rifle Musket (1861)

These rifles fired the
famous “Minie” ball.

Lead bullets recovered from Civil War battlefields exhibit severe fragmentation and distortion from their original shapes. Many of these were fired by Springfield rifle muskets. A possible explanation is that impacts of the lead bullets caused them to melt partially, smear, then freeze back to solid in a contorted form. Is it possible the lead bullets actually became liquid upon impact?

Laser Retina Surgery

A blow to the head can cause the retina, the sight-giving lining of one's eye, to tear or detach from the inner surface of the orb, to float in its aqueous. For a minor tear sight might be restored by "welding" the retina back to the wall of the eye with carefully applied blasts of a laser powered at 10,000 W/(mm sq). Calculate the correct time, the duration, of applied laser power?

Thermodynamic Analysis

Section 5: Previous sections have addressed the tools of thermodynamic analysis. The attempt here is to treat every thermodynamic consideration with a consistent method. THERMO Spoken Here! was written to teach the method of thermodynamic analysis.

trans-Alaska Pipeline

Initially some 800,000 barrels of oil per day were pumped 800 miles from Prudehome Bay to Valdez where it is loaded into tankers. At a location in the pipeline, a gage measures the average velocity of the flow of oil passing through that measurement section of the four-foot diameter pipe. What reading of "average speed of flow" is expected?

Dig Suez

The Suez Canal was dug across 40 miles of desert by a team of dredges. The canal initiated in a lake at its north end. It was "dug" by pumping a water-saturated sand mix (slurry) from the canal to a bank some half mile away. The average rate of sand removed by the digging machines was six million cubic feet per month. The sketch shows an average profile of the canal.
Estimate the years required to complete the digging of the canal.

Linear Valve

The supply of water to an empty tank is controlled by a valve which is initially closed. While opening the valve, the flow of water into the tank increases linearly. To open the valve completely requires two minutes and when the valve is "full-open" one cubic foot per second of water flows into the tank. The time required to close the valve completely is also two minutes.

Calculate the least time required to fill the tank and the overflow that will happen.

Pipe Pig

When oil refinery operations change it is common that the previous product of a pipe is cleared of the previous product so the new product can pass through.

To clean a pipe a tight-fitting rubber plug, called a "pig" is placed in one end of the pipe then using compressed nitrogen it is forced through the pipe and into the holding tank. The plug pushes the previous product out, leaving the pipe clean.

Boost Pump

The schematic shows a piston/cylinder component of a hydraulic circuit. Normally the circuit operates at low-pressure. The piston is motionless and the cylinder contains its maximum volume of fluid. In this condition, hydraulic fluid pumped to the task bypasses the cylinder. Periodically, when extra "oomph" is needed, an electrical signal shuts the one-way valve and activates the motor to drive the piston to the right.

Freeze–Dried Rattlesnakes

The Bible says "snakes are evil." Conse­quently some people kill every snake they possibly can. One business sells dead snakes.

Rattlesnakes are suf­focated then, while flexible, a stiff wire is shoved through the corpse, length-wise from rattles to fangs. The wire is bent supporting the dead snake in a threatening "poised-to-strike," pose. Next the snake is frozen solid (-20°C) then placed inside a sealed chamber equipped with a vacuum pump. With prolonged pumping of first air, then air annd ater vapor, (a great while later) all of the water in the snake's body is pumped out, leaving a preserved rattlesnake trophy.

Civil Rights Memorial

This mem­orial in Montgomery, Alabama was designed by Maya Lin to honor the Civil Rights movement and some 40 persons who were murdered because they believed in equal rights for all. The names of those killed are engraved, aligned radially, around the circular edge of a massive, flat, granite stone. Water (pumped from below) rises steadily near the center of the stone, wells upward with speed, then spreads and calms as it flows outward to pass over the names of the dead. The water then wraps, without a ripple, over the rounded edge in a circular waterfall.

Expansion Flow

The sketch depicts water flowing from the left to the right without friction (we assume) through a horizontal pipe. A fixed "origin" is notated. Left of the origin the pipe diameter is d. An expansion is located at X = 0. The pipe dimeter thereafter is D. Suppose at the instant, t = t*, a point in the flow at X = XL(t*) is marked and observed to have the velocity vL(t*).

Prove the velocity of the water at X = XR(t*) is equal to (AL/AR)vL(t*).

Depth of Wine

The sketch shows an open-top cask that contains wine initially having a depth of four feet. At the start of the party (at time, t = 0+) wine is steadily released steadily from the overhead cask by abruptly removing the bung. Assume the speed of the flow of wine to be predicted by Torricelli's in Theorem:

(1) 1

As wine flowed into their flagons, some guests realized the decreasing depth of wine in the cask, H(t), was. in a way, a measure or indication of how long the party would last. Use calculus to determine if the moment half the wine is consumed corresponds to the time being one half the duration of the party.

Extruded Rod

Aluminum exits from an extrusion die at constant speed of 2 centimeters per second and a cross-sectional area of 3 square centimeters. Once the extrusion is established, a cutter actuates regularly to chop the exiting aluminum in lengths that have a mass equal to 100 grams. The sketch shows the process at the instant (t = 0+) a section was cut. That is:

Calculate the time lapse between each cut.

Reaction of a Jet

The sketch shows a jet of water that issues from a nozzle. The stream, in an 0XY plane arrives tangent to the beginning edge of a blade that has a 90-degree blade angle. In this beginning level analysis we assume the water does not splatter and that it leaves the blade having the same speed as when it entered.

Calculate the force exerted on the block by the jet.

Jet-ski Static Pull

Two boys bought a high performance carburetor kit for their jet ski. They devised a test to evaluate the improvement. The sketch shows the machine running full-power but restrained by a cable from it to a dock. They made part of the cable include a spring scale. Use the data provided in the sketch.

Refill the Pool

Suppose you noticed the water level of your neighbor's swimming pool to be about 6 inches low. You inform him of the fact whereupon he directs a garden hose into the pool and turns it on. The pool has vertical sides and measures 15 by 36 feet.

The sketch shows the geometry of this "pool filling" event. For a moment the two of you watch the water flowing into the pool. Then, knowing you are an engineer, the neighbor asks: "When should I return and turn the water off?"


Pitot-Static Gage

Air moves with uniform velocity (section [B]) through a pipe. Some of that air headed directly for the projecting, tube-end of the gage, is decelerated ahead of the tube then comes nearly to rest, momentarily, just ahead of the tube opening, [A]. The event just described is called "stagnation of the flow." The kinetic energy of the flow changes into an increased pressure of the air at the tip of the gage. This pressure difference created is important information.

Where Water Jets Collide

The sketch shows a reservoir of water restrained by a wall. The wall has two openings, one above the other. For the instant shown, jets of water that issue from the holes intersect at a distance, L, from the tank wall. Assume the flow frictionless, the speed of each jet of water issuing from the wall outlet is given by Torricelli's Theorem. Show intersection of streams occurs at distance: L = 2 (H1 H2)1/2.

P-51 Mustang

On strafing runs WWII P-51 Mustangs often fired bursts of 1000 rounds from its six - 50 caliber machine guns. These streams of "fired lead" changed the air speed of the Mustang. This calculation will include some guess-work and approximation.
Calculate the P-51's change of air speed after it fires a burst of 1000 rounds.

There are two water tanks connected by
a pipe with a closed valve. One tank is full,
the other is empty. The valve constrains the water.

Water Seeks its Level

The waters of Earth exist predominantly as liquid. Where liquid water is found it resides with its surface flat and tangential to Earth. Furthermore the surface is as physically low, or close to Earth as possible - subject to constraints that contain it. This example considers an amount of water initially constrained within a tank. The "event" initiates when the constraint is removed. The water, freed from the constraint flows toward Earth until a new "lesser" constraint arises. At the least constraint, water becomes part of the sea.

Gear Pump

The gear pump of a running engine receives oil at atmospheric pressure from the oil pan. Power passes from the engine through a belt-driven pulley to the shaft of the "driver gear." As the driver gear rotates, it forces the "driven gear" to rotate. Supply oil fills the spaces or "pockets" between the gear teeth of both gears. As the gears turn these "pockets" of oil are forced around the outer perimeter of the pump housing to exit at high pressure.

What ideal, steady mass rate of oil does the pump deliver?

Aquarium Turbines

Some background about aquarium pumps is needed. An aquarium pump and filter are shown in typical operation. Water, originally at the surface in the filter box (1) passes upward through the pump to jet into the aquarium with properties; (2). Water of the aquarium returns to the filter box by action of a siphon (3).

Calculate the least wattage of the pump.

NOT a "Cool Idea"

In sweltering summer heat, outdoor pools feel just like hot tubs. It's silly, but some pool owners try to "beat the heat" by dropping hundreds of pounds of ice into their pools. On the day of his daughter's wedding, an architect had 800 pounds of ice dumped into his pool which was completely full of water. In about an hour all of the ice had melted and the water in the pool was still unpleasantly warm. The architect wondered "What went wrong?"

Most of all, the Orca needs to be calm (sedated).
Next, near freezing water is a must.
Shipping Maeku

A killer whale captured off Japan is to be shipped air-freight to Miami in a container. To travel, Maeku will float in water, sedated, and harnessed to the sides of his aquarium. During the 48 hour flight,dry ice (solid carbon dioxide), will be dumped steadily into the aquarium to maintain its water at 5°C.

Instant Hot Water

The sketch shows a wall-mounted, electric-powered water heater that operated "on demand." There is no hot water tank. When the temperature control is switched, the device delivers water as shown in the promotional sketch.

Calculate the least electrical power required?

Pipe Line Flow

The sketch shows a section of a pipeline. Oil flows through the pipe steadily. Two sections (planes that cut through the pipe) are identified. Take the system to be the oil contained within the pipe between the "in" and "out" planes. Masses of oil flow across these planes.

What does the energy equation say about the flow of this physical situation?

Torricelli's Theorem

A reservoir of water extends a great distance to the left. The waters are constrained by banks, one close to us, one at a distance into the page. To the right the water is constrained by the left face of a dam. Near the base of the dam is a pipe with a closed valve. Question: When the valve is opened, what maximum exiting velocity of water would happen?

Scrapped-Surface Heat Exchanger

This device is used to "slush-freeze" concentrated orange juice. The juice is pumped through the inner annulus. A shaft passes the length of that tube with a spiral scraper attached. As ammonia passes through the outer annulus it causes the juice to freeze to the inside wall of its path. The scraper, rotated by a motor, forces the liquid/solid concentrate to exit upper right. Calculate the least steady mass rate of ammonia required by the device.

Cardiac Horsepower

At rest, your heart causes about 5 liters of blood per minute to circulate to your toes and back and elsewhere. The flow returning to the heart enters the vena cava at low velocity, with an average pressure of 10 millimeters of mercury. Your heart pumps your blood twice. First it pushes blood into your lungs. That heart-exiting flow enters the pulmonary artery at about 30 mm Hg. Returning from the lungs, oxygenated blood flows into the heart at a pressure of about 10 mm Hg. Then the heart routes the blood into the aorta (100 mm Hg) to flow back to your toes and other important (at rest) places. Estimate the average horsepower of a human heart at rest.

Niagara Power Inc.

Water that flows down the Niagara River passes over the spectacular Niagara Falls (partly in Canada, partly in the United States). A great volume of water (15 million cubic feet per minute) fall 167 feet from the escapement headed to Lake Erie.

Suppose "upstream" were dammed so the entire elevation change from the surface of Lake Erie to the surface of Lake Ontario could be used as massive hydroelectric power facility.

Tomato Juice

This juice is easily damaged by high temperature. Consequently its concentration is accomplished with modest heating by hot water in a heat exchanger followed by extraction of water from the juice by action of a vane pump that maintains a low pressure over the juice and ejects water vapor from it in a separator.

The process has two flow streams. The hot water simply enters (is cooled as the juice is heated) and exits. The other stream, juice, enters, is warmed then exits partly as concentrate and partly as vapor exhausted by the vane pump. Use the information of the drawing.

Calculate the mass rate of water that exits through the vane pump as water vapor.

ZULIA ~ Side-Casting Dredge

The shipping channel from the Caribbean Sea to the port of Caracas, Venezuela, passes through a long pond-like expanse of water called Lake Maricaibo. Since the water is too shallow for commercial shipping, the side-casting dredge, ZULIA, must dredge constantly and arduously to maintain the shipping channel.

In operation, its diesel powered pumps dredge a slurry of water and mud from the bottom of the channel and side-cast it the length of its 400 foot boom. Day by day, years pass as ZULIA toils to ever-restore the ever-filling channel.

Calculate the least pumping horsepower of the side-casting dredge.

Water Pumped Vertically

Displays of water-and-light at hotels in Las Vegas require complex piping, pumping and lighting systems. The photo shows the Bellagio water display in operation. Learn a little about such cosmetic water displays.

Light Bulb Efficiency

As a science project, students decided to evaluate the efficiency of 100-watt light bulbs. They defined "bulb efficiency" to be the visible energy produced by the bulb divided by the power expended. When a light bulb is powered, it attains steady usage of power and steady temperatures of operation nearly immediately. The tungsten filament becomes incandescent and the glass globe of the bulb becomes hot.