Two massive spheres are mounted on a light rod that can be rotated by a string w

Question

Two massive spheres are mounted on a light rod that can be rotated by a string wrapped around a central cylinder, forming a winch as shown in the figure. (Figure 1) A force of magnitude F is applied to the string to turn the system. With respect to the variables given in the figure, the equation for the magnitude of the angular acceleration is

=rF3mR2.

Assume that the spheres are small enough that they may be considered point masses and that the masses of the rod and cylinder can be neglected.

Part A

If the sphere on the left is moved closer to the central cylinder and placed at a distance R/2 from the axis of rotation, what is the magnitude of the angular acceleration of the modified system? Assume that the rest of the system doesn't change.

Explanation / Answer

moment of inrtia of system = (2m R^2) + (mR^2) = 3mR^2

and usig torque = I x alpha

r x F = (3m R^2) alpha

alpha = rF / 3mR^2

A) moment of inertia of modified system ,

I = (2m ( R/2)^2) + (m R^2) = 3m R^2 /2

r x F = ( 3mR^2 /2) alpha

alpha = 2rF / 3mR^2

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