A small disk of radius r is glued onto a large disk of radius R that is mounted on a fixed axle through its center. the combined moment of inertia of the disks is I. A string is wrapped around the edge of the small disk and a box of mass m is tied to the end of the string. the string does not slip on the disk. find the acceleration of the box after it is released from rest

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To find the acceleration of the box after it is released from rest, we need to analyze the forces acting on the system.

First, we need to determine the torque acting on the system. The torque is given by the product of the force applied and the lever arm distance. In this case, the force applied is the tension in the string and the lever arm distance is the radius of the small disk, r.

The torque equation can be written as: τ = r * F

The tension in the string can be calculated using Newton's second law for rotational motion. The net torque is equal to the product of the moment of inertia and the angular acceleration. Since the string does not slip on the disk, the angular acceleration is the same as the linear acceleration of the box.

Using this information, we can set up the equation: τ = I * α

Substituting the torque equation, we get: r * F = I * α

Since the angular acceleration, α, is the linear acceleration, a, divided by the radius, r, the equation becomes: r * F = I * (a / r)

Simplifying further: F = I * (a / r^2)

Now, we need to consider the forces acting on the box. There are two forces at play: the gravitational force, mg, acting downwards and the tension in the string, T, acting upwards.

The net force acting on the box can be determined using Newton's second law, F = ma, where F is the difference between the tension and the gravitational force: T - mg = ma

Using this equation and the equation we derived earlier, we can substitute T with (I * (a / r^2)) to get: I * (a / r^2) - mg = ma

Rearranging the equation to isolate the acceleration, we get: a = (mg) / ((I / r^2) + m)

Thus, the acceleration of the box after it is released from rest is given by a = (mg) / ((I / r^2) + m).