A rock of mass m is attached to a string of negligible mass and length L, as sho

Question

A rock of mass m is attached to a string of negligible mass and length L, as shown above. The rock is released from rest at the position shown (the rope string is horizontal). A short time later, the pendulum has reached the position shown for the first time, which is an angle theta down from where it began. Draw a free body diagram for the rock at each of the two points shown. For the first position, assume that the rock has just been released. At the later instant, find the following quantities in terms of L, theta, m, and relevant constants. (Your expressions might end up not including all of these quantities.) Find the speed of the rock, (c) Find the tension in the string. (d) The magnitude of the tangential acceleration of the rock. Answer the following questions: At which point (i.e. for what theta) is the tension/speed/tangential acceleration the largest? What is its value? Evaluate the tension for theta rightarrow pi/2. Does the resulting expression make sense? Explain Evaluate the tangential acceleration for theta rightarrow pi/2. Does the resulting expression make sense? Explain.

Explanation / Answer

b)from energy conservation:

KE = PE

1/2 m v^2 = m g l (1 - cos(theta))

v = sqrt {2 g l [ 1 - cos(theta)] }

c)Balancing forces in Y direction:

T - mg cos(theta) = m v^2/l

T - mg cos(theta) = m 2 g l [ 1 - cos(theta)] /l

T = mg cos(theta) + 2 m g - 2 m g cos(theta)

T = mg (2 - cos(theta))

d)mg sin(theta) = ma(T)

a(T) = g sin(theta)

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