An outline of how
we will proceed.
By previous studies we know Newton's Laws of Motion provide a valuable means of understanding motion. In the 60-90 years after his death, scholars extended the utility of the 2nd Law. Though this extension can be told in a casual way, physics or engineering applications require a quantitative development.
Mathematical extension of the 2nd Law statement begins regarding the rate of change of momentum of a BODY under action of forces. Newton's 2nd Law is "extended" simply. Consider that the BODY is displaced. Multiply Newton's 2nd Law by that displacement.
This writing needs revision!
Newton's 2'nd Law of Motion (a mathematical expression) that tells the change of momentum in time provided the initial state of the BODY and the forces acting on that BODY for the duration are known. Since his death, scientists have tinkered with his Law of Motion. Two new ideas useful in understanding physical behaviors of a BODY have been determined. These new ideas are the constructs: Energy and Work.
The idea Potential Energy was constructed to be a new and useful perspective for the understanding (and study) of physical reality. One point needs to be made clear. When the idea "potential energy" is NOT used, the perspective of behavior of a BODY is the Understanding of by which (will abiding the full scope of Newton's Laws) the work of an event of a BODY associated with "gravity force" is recast. This subtle idea requires. Work is the displacement of a force through a distance.
This development addresses the system model, BODY. For all events of the BODY, Newton's Second Law applies. Newton categorized forces as acting either "at a distance" or "at the surface" of the system. Forces are categorized as "BODY forces" which act at a distance and "surface forces" which act at the system surface. Our form of "f = mA" is written:
Newton's Momentum Equation:
BODY as system.
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.
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?
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?
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):
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.
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.
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.
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.
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.
Work is defined by physics, by engineering and elsewhere in our language.
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.
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.
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.
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?
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.
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.
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.
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.
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.
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.
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?
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.
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).
Every engineering analysis or experiment has a "system" as its focus. System is a theoretical idea patterned after the work of Isaac Newton. The subject of every consideration of thermodynamics is a specific, initially selected amount of matter called the system . System definition is a matter of both selection and isolation.
The previous derivation of the energy equation for a BODY (sometimes called the "extrinsic energy equation") was accomplished mathematically without reference to anything specifically physical. Below we will derive the same equation but do so in reference to the physical events of a forklift.
Energy principles (though not always in the context of an energy equation) are introduced in beginning physics. Newton's Second Law of Motion described "change of momentum" of a BODY in consequence of action of a sum of forces. Another perspective of such events, a perspective that introduces the construct, energy, can be developed. When a positive change of momentum of a BODY is multiplied by its positive displacement with that change, a change of "energy" of the BODY is said to have occurred as a consequence of an energy "transfer" mechanism; "work."
Newton's models of physical reality were the BODY, two BODIES and the EXTENDED BODY. These system models worked very well for planetary motion and for simple, low-friction, events of solids on Earth. The simplicity of there being but two types of energy, kinetic energy and potential, and the fact that their sum was (nearly) always constant, supported the common idea, "energy is conserved." The trend of study of energy these days is less-so "energy is conserved," and more-so "How does energy change."
Compression work will be introduced as occurring to a gas (many bodies as a collective system) contained in a horizontal piston and cylinder apparatus which is idealized to function without leakage or friction. By action of the assumed frictionless mechanical crank and gear, the piston is advanced a differential distance into the cylinder. With the crank, piston and gas as system The force-displacement of the crank is work delivered to it. With compression of the gas, it is obvious there is work but the apparatus is arranged so the gas experiences zero kinetic energy change and zero potential energy change.
The general ideas of internal energy and compression of a substance arrive early in science education. Our goal is to give form to those ideas and a representation for them suitable for a thermodynamic energy equation. Our extrinsic equation addressed system displacement, kinetic and potential energies. Our immediate task is to identify and explain internal energy and compression work which are associated not with system displacement but with its deformation.
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?
The system of this development is an Ideal Gas (IG). The IG, having its simple Equation of State, permits the math to go smoothly. There are two idealizations of the event - frictionles ans adiabatic.
Our path is to introduce internal energy and enthalpy in a general manner, simply as properties, then to discuss how they apply to physical events or processes.
Entropy is a thermodynamic property of the system (substance) and of the surroundings. "Entropy of Everything" is the subject of a physical law; the Second Law of Thermodynamics. Everything is the Universe has the parts: system and surroundings.
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.
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)?
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.
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.
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.
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
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 new model is SUBSTANCE (or material (metal, solid and so). Also a new energy form is introduced: internal energy.
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.
While every thermodynamicist uses only one sign convention, it might not be the same as another uses. The clearest way to state your sign convention is by a quick sketch.
Which, of the sign conventions shown, if any, is wrong and why?
Previous pages presented definitions and assembled an energy equation relating interactions of system energy, work and heat. The table serves as a way to take stock of the Energy Equation we have and how it was obtained.
Our goal is a complete, competent Energy Equation. So far we have the simplest external (extrinsic) energy equation in which energy changes and work relate directly to displacement of the body (system).
Many physical gases of systems qualify as being "ideal" in their events because their properties pressure, temperature and specific volume conform to the ideal gas equation. In those states, the specific heats of gases depend solely on temperature (are independent of pressure). How to check, using property information, whether (and over what ranges) pv = (R/M)T is valid is an important and easy task but is special topic.
The simplest model of physical matter is the BODY which is an approximation of the selected matter as being localized at a point, that is, to have mass but no extent. For mass without extent, rotation makes no sense. Otherwise stated, a BODY does not rotate.
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.
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.
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?
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.
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.
Frequently in engineering one is required to evaluate an integral (or double integral). Students, having just learned calculus, tend to belabor such tasks. To obtain a quick answer, simply assume you know the constant, average value of the integrand over the range, then apply the MVT.
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.
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.
There are hundreds of varieties of springs. Our development will apply in general but will be made in terms of the simplest, ideal, coiled spring. In elementary design, the kinetic energy and potential energy of a spring are taken to equal zero and since high temperatures destroy a spring, their events are assumed adiabatic or nearly so. The work is intrinsic, the energy equation is simply:
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.
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.
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.
The principles of thermodynamics can be expressed in either of three distinct mathematical formats. The differential format is used to develop equations and for derivation of special results. Time is not a factor with differential considerations.
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.
That work passes through shafts of machines is obvious. Yet a thermodynamic discussion is worthwhile. Consider a machine that has a power shaft but which is off, inoperative (top view). While the shaft is inoperative (not rotating) press a sharp-edged bar of steel against the shaft and draw it from one end to the other thereby leaving a scribed line the length of the shaft (shown as "inoperative").
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?
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.
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?
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).
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.
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?
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.
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."
... 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.
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.