Question

A uniform solid sphere of mass M and radius R is free to rotate about a horizontal axis through its center. A string is wrapped around the sphere and is attached to an object of mass m. Assume that the string does not slip on the sphere. (Use the following as necessary: M, m, and g.) Find the acceleration of the object.

Answer 1

Answer:

$a = \frac{mg}{m + \frac{2}{5}M}$

Explanation:

To calculate the Acceleration and the tension of the object, we start by considering the value of the Tension through its moment of Inertia and Acceleration based on the angular velocity

$\tau = I\alpha = Tension(T)*R$

And $a = \alpha R$

Replacing,

$T*R = I\alpha = (\frac{2}{5} MR^2)*\frac{a}{R})\\T*R = \frac{2}{5}MaR\\T = \frac{2}{5}Ma$

The following forces occur in the body,

$mg - T = ma$

By this way we have the acceleration

$mg - \frac{2}{5}Ma = ma$

$a(m + \frac{2}{5})M) = mg$

$a = \frac{mg}{m + \frac{2}{5}M}$