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In the absence of any ready-made machine, an aluminum, ball-bearing pulley about 20 cm in diameter may be used. The necessary distances can be determined fairly accurately by trial. Perhaps the best method is to check the rising weight by means of a stop placed just below the pulley. Release the weights at one click of the clock and adjust the distance so that the weight strikes the stop at the next click, the second, or the third as desired. The distances the weight rises in these times should be measured.
EXPERIMENT NO. 111
VELOCITY OF A BULLET BY BALLISTIC PENDULUM.
References: Stewart, Physics, Sect. 69, 72, 87; Kimball, College Physics, Sect. 68, 69, 107, 109, 110; Duff, College Physics, Sect. 17, 18, 80; Spinney, Text-Book of Physics, Sect. 96-98.
One of the oldest methods of finding the velocity of a bullet is to fire the bullet into a suspended block of wood, determine the velocity of the block, and from it, in turn, calculate the velocity of the bullet. The principles of conservation of energy and conservation of momentum are both made use of. The latter principle is merely Newton's law of action and reaction.
A block of wood, hung by four strings from the ceiling, serves as a ballistic pendulum. A meter stick is fixed in a horizontal position by a clamp, so that the pendulum swings alongside the stick without quite touching it. A small slider of celluloid, having a projecting wing, is set on the meter stick so that the rear end of the suspended block just touches the wing when the pendulum is at rest. Read the position of one edge of the slider.
When ready, discharge a BB shot from an air gun into a hole in the end of the block where there is some wax to catch and hold the shot. The pendulum swings and the slider is pushed along the meter stick. Read the new position of the edge noted before and compute the distance the pendulum has swung. Make at least ten trials of this part and take the average. Weigh the pendulum block and before each trial be sure to remove any
shot imbedded in the wax.
Find the mass of a single shot by
weighing a quantity of them and counting the number weighed. Also determine the length of the pendulum, measuring to the center of the block.
The velocity of the shot is to be computed as follows:
Let m mass of a shot,
M=mass of the pendulum block,
v velocity of shot before impact,
V-velocity of shot and pendulum immediately after impact.
Then by the principle of conservation of momentum we have,
If V were known we could calculate v. To get V we make use of the principle of conservation of energy as expressed by the equation,
where h is the height to which the pendulum is raised by the impact of the shot. This distance is easily obtained from a knowledge of the length, 1, of the pendulum, and the distance, d, through which the slider on the meter stick was pushed.
As h is only a small fraction of a centimeter, its square is a very small quantity, and h2 may therefore be neglected in the equation obtained by applying the Pythagorean theorem to the right triangle in Fig. 4. This gives the expression:
from which his easily determined. Substitute this value of h in equation (2) and compute V. This value, substituted in equation (1), enables one to find the velocity of the bullet.
A 22-caliber rifle may be used in this experiment, but a heavy block of 4 x 4 timber is needed, while for the air gun, a 100 gm block will be heavy enough. A rifle, shooting powder
cartridges, should be clamped in a vise and always aimed toward a thick wall, so as to avoid accidents. Inconsistent results will be obtained unless the gun is cleaned after every shot.
It is interesting to compute the kinetic energy of the shot before impact and that of the block and shot after impact and note what a large percentage of the original energy is changed into heat at the impact, more than 99% as a rule.
EXPERIMENT NO. 112
THE WHEEL AND AXLE.
References: Stewart, Physics, Sect. 96-99; Kimball, College Physics, Sect. 73-83; Duff, College Physics, Sect. 15, 18, 34-40; Spinney, Text-Book of Physics, Sect. 57-59, 74, 76.
The mechanical advantage of a machine is defined as the ratio between the weight lifted (or force overcome) and the force applied to the machine. This is the actual mechanical advantage. Assuming the machine to be perfect, another ratio between machine dimensions can usually be worked out, which is equal to the ratio between weight and applied force. This ratio is the theoretical mechanical advantage. In any real machine there is some friction, so the work done against the weight will prove to be less than the energy put into the machine by the amount used in overcoming friction. The ratio between the output work and the input energy is the efficiency of the machine. It is also equal to the ratio between the actual and theoretical mechanical advantages of the machine.
The purpose of this experiment is to study the relations between the weight lifted and the force applied, using a wheel and axle as the machine.
Place the wheel and axle on a support so that the plane of the wheel is vertical. Attach one end of a piece of cord to the axle, wind it around the axle several times, and attach the free end to a scale-pan. Fasten another piece of cord to the edge of the wheel, wrap it around the wheel a few times and fix another scale-pan to this free end. The load consists of a kilogram weight plus the weight of the scale-pan, suspended from the axle.
Place enough weights in the other scale-pan to make the load just move upward with uniform velocity. Call this lifting force f'. Now place just enough weights in the pan to make the load move slowly downward without acceleration. Call this force f". The average of f' and f" is the force, f, necessary to produce equilibrium. The weight of the scale-pan is included in f' and f”.
Measure the radius, R, of the wheel and the radius, r, of the axle, and see whether your data satisfy the second condition of equilibrium of a rigid body. (See Exp. No. 104).
Measure the distance the load is raised when the second scale-pan moves down a distance of 30 cm, and calculate the amount of work done on the load. Also, calculate the work done by the average force, f, in moving 30 cm. These two quantities of work should be equal.
Repeat the experiment, using each axle in turn. Repeat with a different load.
What relation exists between the load, W, the force, f, the radius of the wheel, and the radius of the axle? What advantage does the wheel and axle have over the ordinary lever? Tabulate data under the headings specified in Experiment No. 113, substiR
tuting a column,, for the last column of that experiment.
What is the mechanical advantage with each axle used? What is the efficiency? Why is the latter not 100% ?
EXPERIMENT NO. 113
References: Stewart, Physics, Sect. 98, 99, 104; Kimball, College Physics, Sect. 79-81; Spinney, Text-Book of Physics, Sect. 57-59, 74, 77.
Read the introduction to Experiment No. 112 and the portions of your text-book relating to pulley systems.
In this experiment, one may test a set of small pulleys with weights of a few hundred grams or a commercial block and tackle using fairly large ropes and lifting large weights. The small pulleys will usually show very high efficiency, but the use of the larger pulleys is recommended, wherever feasible, as the student will learn what may be expected of such apparatus in
commercial use. With the large pulleys it is more convenient to measure the applied forces with spring balances of 30 or 50 lb. capacity. The distances are then measured in feet and work calculated in foot-pounds. The following directions for the use of small pulleys, may be readily adapted to the use of the commercial apparatus.
Arrange the apparatus as shown in Fig. 5. The weight attached to the movable pulley should be about 1 kilogram. Place weights in the scale-pan until the movable pulley will just move upward at a uniform rate. Call the weight of the
Figs. 5 & 6.
scale-pan plus the weights in it, f'. Now load the pan so that the movable pulley just moves downward with uniform speed, and call the corresponding weight of scale-pan and contents, f". The average of f' and f" is the force that would produce equilibrium.
Measure the distance the weight is raised when the scalepan moves down a distance of 20 cm. Calculate the work done by the average force, f, in moving this distance, and also the work done on the weight and the movable pulley. The two amounts of work should be equal. (Why?) Note the number of strings running to the movable pulley. Show from your readings that this number is equal to the ratio between the weight lifted and the force required. Repeat with a load of two kgm. Repeat the observations with the string attached on the movable pulley, as in Fig. 6. Then substitute a double pulley at the lower end of the tackle, and repeat tests with end of string attached first to the upper and then to the lower