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The fifth column gives the values of k2.

Determine the dimensions of the block, and calculate the moments by the formula

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where a, b, c are the half lengths of the sides of the block. Record as follows and enter in preceding table.

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The oscillations of a bifilar suspension like those of a pendulum are only approximately isochronous, and the arc of rotation should therefore be small if great accuracy is required. If the arcs are large the correction given in the next exercise may be applied.

SECTION XII.

THE COMPOUND PENDULUM.

A Compound Pendulum is one in which the mass is distributed over a finite volume and not concentrated at a single point as in the simple pendulum. Strictly speaking every pendulum is a compound pendulum, and we can only approach the ideal simple pendulum by reducing the

volume of the heavy bob of an actual pendulum as much as possible.

In Fig. 23 let G be the centre of mass of a heavy body of mass m capable of turning about an axis through O at right angles to the plane of the paper.

G

Fig. 23.

In the position of equilibrium, the centre of mass G will be vertically under 0. If the body is displaced through an angle 0, the resultant gravitational force passing through G, will have a moment mgh sin 0 about O, g being the acceleration of gravity and h the distance OG between the centre of mass and the axis of rotation. If is small sin is nearly equal to 0 in circular measure, and the couple tending to bring the body back into its position of rest will be nearly mghe, i.e. nearly proportional to the angle of displacement. Under these circumstances the body will oscillate about the point O, and the time of oscillation will for small displacements be independent of the amount of the displacement, i.e. the oscillations will be isochronous.

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where I is the moment of inertia of the pendulum about the axis of oscillation, and mgh the couple when the displacement is a right angle.

If K is the radius of gyration of the body about 0, I = mK2 and the equation becomes

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Definition. The length of the equivalent simple pendulum is the length of the simple pendulum having the same time of oscillation as the given body.

As a simple pendulum of length l has a time of oscillation

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equal to 2, it follows that if I be the length of the equivalent simple pendulum

1 = K2/h.

If mk2 be the moment of inertia of the compound pendulum about an axis parallel to the axis of oscillation, and passing through the centre of mass G, we have

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If a point P be taken in the line OG, Fig. 23, such that OP is equal to the length of the equivalent simple pendulum, P is called the centre of oscillation, while O is called the centre of suspension.

Hence the radius of gyration with respect to the centre of mass, is the geometrical mean between the distances of the centre of mass from the centres of suspension and oscillation respectively.

If the body is suspended from P, 0 will become the centre of oscillation, for k2 being a constant, h and l—h may be interchanged without interfering with the truth of the equation.

Let a pendulum be constructed, Fig. 24, such that it is capable of oscillating about either of two knife-edges A and B, the line joining the edges passing through G the centre of gravity of the pendulum. If the knife-edges can be adjusted so that the time of oscillation is the same whether the body oscillate about A or about B, then unless A and B are equidistant from G, the length AB is that of the equivalent simple pendulum, and as that distance can be accurately measured the value of g can be found accurately from the equation

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This is the principle of Kater's Pendulum.

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B

Fig. 24.

In general there are four possible centres of oscillation on every straight line, such that the times of oscillation of the body about them are equal.

Let A be any point of the body, G the centre of mass, and let the axis of rotation be at right angles to the plane of the paper (Fig. 25). With G as

centre, draw a circle through A, join AG and produce to cut the circle again in A'. Then since GA'GA the time of oscillation about A' will be the same as that about A. If AG=h and k is the radius of gyration about the axis passing through G, the length of the equivalent pendulum will be (k2 + h2)/h. Make AB equal to this length. Then if the centre of the suspension is at B, the time of oscillation is

B'

G

B

'A'

Fig. 25.

the same as before, and therefore the same also for any point B' on the circle passing through B and having G as centre. Two circles may therefore be drawn such that the times of oscillation about all points of them are the same. A straight line such as CC, not necessarily passing through G but cutting

the inner circle, will intersect these circles in four points, about which the times of oscillation will be equal.

The isochronism of the pendulum has been stated to depend on the equality of sin 0 and 0 for small values of 0. If 0 becomes large the quantities are no longer equal, and the time of oscillation will vary with the angle of displacement. In that case a closer approximation is given by

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Hence if To is the time of an infinitely small oscillation and

T the observed time we may w

write To

=

T(1-1). The

16

following table gives the value of 02/16 for different arcs of

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2. A brass rod about a point half-way between its ends and its centre.

3. A sphere about a tangent.

4. A cube about one of its edges.

5. A cylinder about one of its generating lines.

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