Category: Vectors Define Space - Examples, one step to the next lead to competency:
Of the characteristics, position, velocity and acceleration, position is least complicated but requires vectors to specify. Velocity and acceleration are more complicated. A sound capability with vectors is essential. Category: Vectors lists an approach to vectors.
1.3 Position: the First Vector
"Where is something" is a question persons have asked over all of time. The answer tells the "position of that something" in a mutually understood space. The "something" might be your car keys or a location such as a freeway exit or the rest room, etc. For a person to answer, "Where is it?," requires both familiarity of both persons with the surrounding space. The space "it is in" must be mutually known.
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.
"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.
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.
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.
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....
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?
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. Suppose, (in the position shown - dimensions given below), the driver rotates member BC at = 200 revolutions per minute.
For the position shown, Calculate the upward velocity of the top of the jack, point D.
Two boys, walking beside railroad tracks heard a train approaching from behind. Their backs to the train, they continued to walk. 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 his job; to count. The count, at the instant end of the caboose was beside the boys was, "... 34 seconds." In a moment the older boy said, "A short one, only about 1400 feet long."
Typical vectors for a BODY (as system) are its spatial properties: position and velocity (also acceleration). The gravity force and sum of surface (or external) forces, ΣF, are vectors. The figure shows a BODY in 0XZ-space with a notation for its vectors.
1.16 Notations: Position and Velocity
Some say position and velocity are system "characteristics," not properties. It is helpful to use simple but effective vector notations with beginning mechanics and thermodynamic analyses. The table presents some representations of position and velocity (of a BODY or a Point).
Altitude of Geo-stationary 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.
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?