125. Simple Harmonic Motion Isochronous.-It will be

noticed that the expression P2=

4π2xm
F

does not contain r

and is therefore independent of the amplitude of the vibration, so that it does not make any difference in the period of vibration

x

whether the amplitude is large or small, provided the ratio is F

constant, in which case the motion is truly simple harmonic. When vibrations have this property they are said to be isochronous.

126. Energy of a Vibrating Mass.-The energy of an oscillating mass is all potential at the ends of its vibration, but in the middle where the velocity is greatest it is all kinetic and so may readily be computed. For we have seen, §121, that the maximum velocity of the 2 πη vibrating body is v。= and since kinetic energy = m V1 ? P

we find, kinetic energy at middle or total energy

2m2r2

where

P2

m is the mass of the vibrating particle, P is its period of vibration, and r is the amplitude of its motion.

127. Simple Pendulum.-A mass suspended from a fixed

As a

point so that it can swing freely in a circular arc about the fixed point as a center, is called a pendulum. simple or ideal case we may suppose the whole mass of the pendulum to be concentrated at the point B, the mass of the suspending cord or wire being so small as to be neglected. The forces acting on the mass m are its weight mg and the tension T of the suspending cord. The weight mg may be resolved into two components, one in line with the cord and opposing its tension and one at right angles to the suspending cord and in the direction in which the mass m moves. It is this latter component F (Fig. 64) which gives it motion along the circle. Since the diagram of forces is a triangle similar to BCO we have F: mg BC: BO;

N/mg

FIG. 64.

but

BO=1,

and if the angle a through which the pendulum swings to and fro is small, BC is very nearly indeed equal to the arc BA, the length of which may be represented by x.

Therefore the force F urging m along the arc toward A is proportional to the displacement a measured along the arc. But with such a law of force there is simple harmonic vibration (§123) and the relation of force to period of vibration is expressed in the formula

The period of vibration therefore depends only on the length of the pendulum and the acceleration of gravity at the place where it is swung, and is independent of its mass and of the length of the arc, provided the arc is so small that the approximation made above is justified.

The effect of the length of arc upon the period is shown by the following more exact formula,

meter long swings 5 cms. on each side of the lowest point, the arc a = 20

, so that the period is greater by one part in 6400 than if

the arc had been infinitely small.

128. Pendulum Clocks.-The pendulum affords a valuable means of regulating the motion of a clock, since when it swings through a small arc its oscillations are nearly isochronous; i.e., its period of oscillation is nearly independent of the amplitude of its swing.

When an ordinary clock driven by a spring is just wound up it gives a greater impulse to the pendulum through the escapement than when it is nearly run down, and even in clocks driven by weights the friction is not always constant and the swing of the pendulum will vary accordingly.

It must be remembered also that the pendulum of a clock is not free, but the little backward and forward impulses which it receives from the escapement hasten somewhat its motion. To secure regularity of motion, therefore, the pendulum should be heavy, so that its natural period will be only slightly affected by the pushes of the escapement.

In the finest astromonical clocks what is known as a gravity escapement is used, in which the pendulum does not receive any impulse directly from the spring or weight that drives the clock, but its motion is kept up by a small weighted lever which is set free just as the pendulum reaches the end of its swing, and in falling gives a slight push to the latter.

Between successive impulses the lever is raised and set in position by the action of the clockwork.

Full details as to some forms of gravity escapement will be found in the article Clocks in the Encyclopedia Britannica.

Problems.

1. Show that the motion of the piston of a steam engine when the crank is turning with uniform velocity is not simple harmonic. At which end of the piston's motion is the acceleration greatest and why?

2. Assuming that the motion of the piston is simple harmonic, find its velocity in the middle of its stroke when the crank is 8 inches long and makes 200 revolutions per minute. Also find acceleration at middle and end of its stroke.

3. If the piston and connecting rod weigh 100 lbs. in the last problem, find the maximum force against the crank pin due to their inertia alone, neglecting the effect of steam pressure.

4. A pendulum 1 meter long swings 10 cms. on each side of its lowest

point; find the direction and amount of the acceleration at the ends of its swing and at middle.

5. How long must a pendulum be to beat seconds at a place where g= 980.

6. A clock having a pendulum which beats seconds where g is 980, is taken to another place where g = 981; will it gain or lose, and how much in one day?

7. Each prong of a tuning fork making 100 complete vibrations per second vibrates to and fro through a distance of 1.5 mm. Find the velocity of the prong in the middle of its swing.

8. A 400-grm. weight when hung on a long and light helical spring stretches it 30 cms. What will be its period of oscillation if drawn down a little and then set free? Take g 980 and neglect mass of spring.

=

Ans. 1.099 sec.

IV. ROTATION OF RIGID BODIES

MOTION OF A RIGID BODY.

129. Translation and Rotation.-If a rigid body moves in such a way that any straight line joining two points in the body remains parallel to itself as it moves along, the motion is said to be a translation without rotation. A book slid about on a table keeping one edge always parallel to one edge of the table is a case of pure translation. If the edge of the book changes its direction there is said to be rotation.

Any motion of a rigid body may be considered as made up of the motion of its center of mass combined with rotation about an axis through that center.

Motion of the Center of Mass.-It may be proved that when any external forces act on a rigid body the center of mass of the body moves just as though the whole mass of the body were concentrated at that point and all the forces were applied directly to it, and it makes no difference at what points on the body the forces may be applied.

When a top spins on a smooth frictionless table its center of gravity remains at rest, for the external forces acting on the top are its weight due to the attraction between it and the earth and the upward pressure of the table on its point. These two forces are equal and opposite and consequently the center of

gravity has no translational acceleration even when the top is inclined as in figure 75. (p. 101.)

When a stick of wood is hurled through the air its center of mass moves in a simple parabolic curve ($110) just as a particle would move, except as affected by air resistance.

130. Angular Velocity.-When any line in a body is at rest while other points in the body move in circles about that fixed line or axis, the motion is called rotation. In case of a rigid body, like a wheel, all parts whether near the axis or far from it must rotate through equal angles in the same time. The rate at which the body is turning at any instant, measured in radians per second, is known as its angular velocity and is represented by w (the Greek letter omega).

Since the length of a radian of arc at a distance r from the axis is equal to r, we have

v = wr

where v is the linear velocity of a particle at a distance r from the axis.

Example. If a wheel of radius 15 centimeters is making 3 revolutions per second, its angular velocity is 3 ×27 radians per sec., and the linear velocity of a point on the rim is 3 ×2 π × 15 centimeters per sec.

131. Angular Acceleration.-When the angular velocity of a body is changing, the rate of change per second is known as its angular acceleration, and may be represented by A. If the angular velocity w, changes to w, in t seconds then

where A is the average rate of acceleration during the time t. The direction of the axis of rotation may change, and this also constitutes an angular acceleration, even though the speed of rotation about the axis may remain constant. This is illustrated by the motion of a spinning top when its axis is inclined, for the axis swings around in a circle keeping a constant inclination to the vertical.

132. Vector Representation of Angular Velocity.-The angular velocity of a body may be represented by a vector or arrow