A massless rope is wrapped around a uniform cylinder that has radius R and mass M,as shown in the figure.Initially,the unwrapped portion of the rope is vertical and the cylinder is horizontal.The linear acceleration of the cylinder is
A)(2/3)g
B)(1/2)g
C)(1/3)g
D)(1/6)g
E)(5/6)g

Question

A massless rope is wrapped around a uniform cylinder that has radius R and mass M,as shown in the figure.Initially,the unwrapped portion of the rope is vertical and the cylinder is horizontal.The linear acceleration of the cylinder is 
A)(2/3)g 
B)(1/2)g 
C)(1/3)g 
D)(1/6)g 
E)(5/6)g

Quizplus · Accepted Answer

According to the principle of moments, the torque produced by the tension in the rope around the cylinder must be equal to the moment of inertia of the cylinder times its angular acceleration.

Let T be the tension in the rope, a be the linear acceleration of the cylinder, and α be the angular acceleration of the cylinder. Then, we have:

T R = I α

where I is the moment of inertia of the cylinder. Since the cylinder is uniform, its moment of inertia is I = (1/2) M R^2.

The tension in the rope can also be related to the linear acceleration of the cylinder by considering the forces acting on the cylinder. The weight of the cylinder and the tension in the rope must balance each other, so we have:

T - M g = M a

Combining these two equations, we can eliminate T and find an expression for the linear acceleration a:

a = (2/3) g

Therefore, the correct answer is A) (2/3)g.

A Massless Rope Is Wrapped Around a Uniform Cylinder That