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A small metal ball of mass m hangs from a pivot by a rigid, light
metal rod of length R as shown in the figure on the right. The ball is
swinging back and forth with an amplitude that remains small throughout
its motion, *q*_{max} £ 5^{o} .
Ignore all damping.

(a) The equation of motion of this ideal pendulum can be derived in a variety of ways and is

For small angles, show how this can be replaced by an approximate equation of motion that can be solved more easily than the one given.

(b) Write a general solution for the approximate equation of motion you
obtained above that works for any starting angle and angular velocity (as
long as the angles stay in the range where the approximation is OK). Demonstrate
that what you have written *is* a solution and show that at a time
*t* = 0 your solution can have any given starting position and velocity.

(c) If the length of the rod is 0.3 m, the mass of the ball is 0.2 kg,
and the clock is started at a time when the ball is passing through the
center (*q* = 0) and is moving with an angular
speed of 0.1 rad/s, find the maximum angle your solution says the ball
will reach. Can you use the approximate equation of motion for this
motion?

(e) If the pendulum is released from a starting angle of q_{0},
what will be the maximum speed it travels at any point on its swing?

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Page last modified October 30, 2002: O12