You set your pencil on the very edge of a swing and start rotating the swing to

Question

You set your pencil on the very edge of a swing and start rotating the swing to see how far you can wind it up before the pencil slips off while unwinding. The swing is simply a rope hanging from a tree and passing through the center of a flat plywood board. The rope has a torsional constant k = 1 x 10-2 Nm, and the board is 15 cm x 30 cm with a mass of 1 kg. Assume the pencil is at a point 15 cm from the center, and has a coefficient of static friction = .3. Through what maximum angle (in radians) can you rotate the swing so that when you release it the pencil does not slip? Assume the pencil will slip before it begins to roll.

Explanation / Answer

Ff = u m g static friction where u = .3
F = m a max force that can be applied to pencil before it slips
a = u g maximum acceleration of pencil
L = K theta where L is torque on board
alpha = L / I angular acceleration of board
a = R * alpha = R L / I linear acceleration of pencil
u g = R L / I using the two equations for A
u g = R K theta / I
theta = u g I / (R * K)
I = M (l*2 + w^2) / 12 = 1 * (.15^2 + .3^2) / 12 = .752 kg m^2
Now just substitute and solve for theta
I get 2.21 radians for theta

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