120. The physical, or compound, pendulum. In the theory of the simple pendulum it is assumed that the weight and mass of the cord or wire may be neglected and that the ball is so small that its mass may be regarded as concentrated at its center of gravity. Most pendulums depart greatly from these simple conditions. For example, a long cylindrical bar is often used in our laboratories. One which does not satisfy these simple conditions is called a physical, or compound, pendulum.

It can be proved that a physical pendulum has the same period as a simple pendulum of length I if 1

1=

mR

where m is the mass of the physical pendulum, R the distance of its center of gravity from the point of support, and I the moment of inertia of the pendulum (see section 132). If this value of I is substituted in equation (66), for the period of the physical pendulum we obtain the equation

A more accurate determination of the value of g can be made with a physical pendulum than with a simple one. The method, however, demands some means of determining I.

121. The torsion pendulum; angular simple harmonic motion. When a rod which is supported by an elastic wire (Fig. 93, a) is moved in a

horizontal plane so as to twist the

wire, and is then released, it will oscillate back and forth through an angle, marked o in Fig. 93, b. In Fig. 93, b, the wire is perpendicular to the plane of the paper, and is fastened to the rod at the point O. The motion of the rod is simple harmonic motion, the rod moving not in a straight line but through an angle. For this reason this type is called angular simple harmonic motion. A rod vibrating in this manner is an example of a torsion pendulum. The balance wheel of a watch is a torsion pendulum; but instead of using the torsion of a straight wire, it is the torsion of the hairspring which furnishes the restoring moment of force.

a

b

FIG. 93

122. Vibrations in musical instruments. In a stringed musical instrument the strings vibrate back and forth in harmonic motion. Usually a string gives out a certain tone together with a number

of higher-pitched tones called overtones. But when a string gives out a pure tone, a tone of only one pitch, then the string vibrates as shown in Fig. 94, and each particle vibrates with simple harmonic motion.

In wind instruments there are

FIG. 94

air columns which are set in vibration. Usually the motion is quite complicated, and tones of more than one pitch are emitted. In this case and also with vibrating strings the vibrations can be regarded as due to the superposition of a large number of simple harmonic motions of different periods.

PROBLEMS

1. A body vibrating with simple harmonic motion has an amplitude of 10 cm. and makes 3 vibrations per second. Compute the velocity at the middle of its swing.

2. If the mass of the vibrating body of Problem 1 is 200 gm., find the restoring force (a) when it is displaced 2 cm. from its position of equilibrium, (b) when it is displaced 4 cm., and (c) when it is displaced 6 cm.

3. A weight of 5 lb. is hung from a spring. It is found that a force of 2 lb. is required to displace it 1 in. What period will it have when set in vibration ?

4. A mass in simple harmonic motion makes 2 vibrations per second with an amplitude of 2 cm. Compute the acceleration (a) at the end of the vibration, (b) when the displacement is 1 cm., and (c) at the middle of the swing.

5. If the mass of Problem 4 were 300 gm., how much force would be required to hold it with a displacement of 1 cm.?

6. The end of a prong of a tuning-fork which makes 200 vibrations per second has an amplitude of 1 mm. Compute the maximum velocity of the end of the prong.

7. What will be the length of a pendulum that has a period of 1 sec. if g is equal to 980 cm./sec.2?

8. A simple pendulum consisting of a small lead ball is hung by a fine thread. If the length of the pendulum is 1 m., and g is equal to 980 cm./sec.2, what should be the period of the pendulum?

9. In a laboratory experiment the period of a simple pendulum, the length of which was 90 cm., was found to be 1.90 sec. Compute the value of g.

10. A pendulum clock which runs correctly (period, 1 sec.) at a place where g is equal to 980 cm./sec.2 is taken to Denver, where g is equal to 979.6 cm./sec.2 (a) What will be the period of the pendulum? (b) How many seconds will be gained or lost in 24 hr. ?

11. Compute the kinetic energy of the mass in Problems 1 and 2 as it moves through the position of equilibrium.

12. A mass of 784 gm. is suspended by a light spring. An additional load of 20 gm. is required to displace this mass downward 1 cm. more. What will be the period when the 784-gram mass is vibrating? (g = 980 cm./sec.2)

CHAPTER XI

ROTARY MOTION *

Axis of rotation, 123. Angular speed, 124. Relation between radians per second and linear speed, 125. Relation between revolutions per minute or second and angular speed in radians per second, 126. Angular acceleration, 127. Gyroscopic motion, 128. Vector addition of angular velocities, 129. Resolution of angular velocities; components of rotation of the earth, 130. Effect of the rotation of the earth on the motion of projectiles and winds, 131. Kinetic energy of rotating bodies; moment of inertia, 132. Radius of gyration, 133. Formulas analogous to those of linear motion, 134. Conservation of angular momentum, 135. Total kinetic energy, 136.

Nearly everyone has watched a spinning top and wondered at the apparently mysterious force which kept it from falling. But this is only one instance; there are numerous cases of rotating bodies, many of practical importance, which require special methods to explain them. It is well worth while to learn the methods of analyzing problems relating to bodies in rotation.

123. Axis of rotation. In the case of a rotating body it is always possible to find a line which is not moving. This line is called the axis of rotation. In a flywheel turning on a shaft the axis is a line through the center of the shaft. Usually it is a simple matter to locate the axis of a rotating body.

124. Angular speed. The linear speed of a rotating body is an indefinite quantity; it is zero at a point on the axis, and increases the farther the point is away from the axis. It thus has all values between zero and the maximum value which points on the outside edge have. Hence it is not convenient to measure or state the speed of rotation of a body in terms of linear speed. Instead, the speed is measured in terms of the angular speed. For example, the average angular speed of the body in Fig. 95 is equal to the

* This chapter may be omitted in a short course.

angle through which the line AB, which is drawn perpendicular to the axis, moves in 1 second. When angular speed is measured in radians per second (see section 77), it will be denoted by the Greek letter w. Angular speed may be measured also by the number of revolutions per second (R.P.S.) or by the number of revolutions per minute (R.P.M.).

125. Relation between radians per second and linear speed. It is often convenient to be able to reduce linear speed to angular speed. For example, one might know the linear speed of a belt, and hence that of the rim of the pulley, and wish to compute the angular speed of the pulley.

One formula may be deduced as follows: Take some point, A, in a body which rotates about an axis through O (Fig. 96). Let B be the position of this point after an interval of time r. Then the arc AB is equal to vr, where v is the average linear speed. When the angle AOB is measured in radians (see equation (32), sect. 77),

or

But

arc AB UT

LAOB=

θ

r

θ=ωτ,

where w is the angular speed in radians per second.