(b) Is this difference equal to that found in the second problem? Why?
6. A stone dropped from a cliff strikes the foot of it in 5 seconds.
What is the height of the cliff?
7. Why is it that the increased weight of a body when taken to higher lat.i.tudes causes it to fall faster, while at the same place a heavy body falls no faster than a light one?
8. When a train is leaving a station its acceleration gradually decreases to zero, although the engine continues to pull. Explain.
9. Would you expect the motion of equally smooth and perfect spheres of different weight and material to be equally accelerated on the same inclined plane? Give reason for your answer. Try the experiment.
10. A body is thrown upward with the velocity of 64.32 ft. per sec. How many seconds will it rise? How far will it rise? How many seconds will it stay in the air before striking the ground?
11. 32.16 feet = how many centimeters?
12. The acceleration of a freely falling body is constant at any one place. What does this show about the pull which the earth exerts on the body?
(7) THE PENDULUM
=100. The Simple Pendulum.=--Any body suspended so as to swing freely to and fro is a pendulum, as in Fig. 81. A simple pendulum is defined as a single particle of matter suspended by a cord without weight. It is of course impossible to construct such a pendulum. A small metal ball suspended by a thread is approximately a simple pendulum. When allowed to swing its vibrations are made in equal times. This feature of the motion of a pendulum was first noticed by Galileo while watching the slow oscillations of a bronze chandelier suspended in the Cathedral in Pisa.
[Ill.u.s.tration: FIG. 81--A simple pendulum.]
=101. Definition of Terms.= _The center of suspension_ is the point about which the pendulum swings. A _single vibration_ is one swing across the arc. A _complete_ or _double_ vibration is the swing across the arc and back again. The time required for a double vibration is called the _period_. The _length_ of a simple pendulum is approximately the distance from the point of support to the center of the bob.
A _seconds pendulum_ is one making a single vibration per second. Its length at sea-level, at New York is 99.31 cm. or 39.1 in., at the equator 39.01 in., at the poles 39.22 in.
A _compound pendulum_ is one having an appreciable portion of its ma.s.s elsewhere than in the small compact body or sphere called a bob. The ordinary clock pendulum or a meter stick suspended by one end are examples of compound pendulums.
The _amplitude_ of a vibration is one-half the arc through which it swings, for example, the arc _DC_ or the angle _DAC_ in Fig. 81.
=102. Laws of the Pendulum.=--The following laws may be stated:
1. The period of a pendulum is not affected by its ma.s.s or the material of which the pendulum is made.
2. For small amplitudes, the period is not affected by the length of the arc through which it swings.
3. The period is directly proportional to the square root of the length.
Expressed mathematically, _t_/_t"_ = v_l_/v_l"_.
=103. Uses of the Pendulum.=--The chief use of the pendulum is to regulate motion in clocks. The wheels are kept in motion by a spring or a weight and the regulation is effected by an escapement (Fig. 82). At each vibration of the pendulum one tooth of the wheel _D_ slips past the p.r.o.ng at one end of the escapement _C_, at the same time giving a slight push to the escapement. This push transmitted to the pendulum keeps it in motion. In this way, the motion of the wheel work and the hands is controlled. Another use of the pendulum is in finding the acceleration of gravity, by using the formula, _t_ = pv(_l_/_g_), in which _t_ is the time in seconds of a single vibration and _l_ the length of the pendulum. If, for example, the length of the seconds pendulum is 99.31 cm., then 1 = pv(99.31/_g_); squaring both sides of the equation, we have 1 = p(99.31/_g_), or _g_ = p 99.31/1 = 980.1 cm. per sec., per sec. From this it follows that, since the force of gravity depends upon the distance from the center of the earth, the pendulum may be used to determine the elevation of a place above sea level and also the shape of the earth.
Important Topics
1. Simple pendulum.
2. Definitions of terms used.
3. Laws of the pendulum.
4. Uses of the pendulum.
Exercises
1. What is the usual shape of the bob of a clock pendulum? Why is this shape used instead of a sphere?
2. Removing the bob from a clock pendulum has what effect on its motion?
Also on the motion of the hands?
3. How does the expansion of the rod of a pendulum in summer and its contraction in winter affect the keeping of time by a clock? How can this be corrected?
4. Master clocks that control the time of a railway system have a cup of mercury for a bob. This automatically keeps the same rate of vibration through any changes of temperature. How?
5. How will the length of a seconds pendulum at Denver, 1 mile above sea-level, compare with one at New York? Why?
[Ill.u.s.tration: FIG. 82--Escapement and pendulum of a clock.]
6. What is the period of a pendulum 9 in. long? _Note._ In problems involving the use of the third law, use the length of a seconds pendulum for _l_, and call its period 1.
7. A swing is 20 ft. high, find the time required for one swing across the arc.
8. A pendulum is 60 cm. long. What is its period?
9. If in a gymnasium a pupil takes 3 sec. to swing once across while hanging from a ring, how long a pendulum is formed?
10. A clock pendulum makes four vibrations a second, what is its length?
Review Outline: Force and Motion
Force; definition, elements, how measured, units, dyne.
Graphic Representation; typical examples of finding a component, a resultant, or an equilibrant.
Motion; Laws of motion (3), inertia, curvilinear motion, centrifugal force, momentum, (_M = mv_), reaction, stress and strain.
Moment of Force; parallel forces, couple, effective and non-effective component.
Gravitation; law; gravity, center of; weight. Equilibrium 3 forms; stability, how increased.
Falling Bodies; velocity, acceleration, "g," Laws; _V_ = _gt_, _S_ = (1/2)_gt_ - _s_ = (1/2)_g_(2_t_ - 1).
Pendulum; simple, seconds, laws (3), _t_ = pv(_l_/_g_).