Think of a uniform sphere. Sometimes if a force is applied to the sphere it does not only move but also spins. So there is a torque.
But is it possible to calculate which part of the applied force is responsible for movement which part works as torque to give it a spin.
I will be explaining with respect to the below free body diagram.
You can see that I have split the applied force into its components as per the axis shown.
For translational motion
For this you have to consider the sphere as a object with mass concentrated at the center of mass O(center)
You can then apply $a=\frac{F}{m} $ where a is the translational acceleration
For rotational motion
You must be knowing that $\tau=I\alpha$
Here $\tau$ is provided by the $F\sin\theta $ at a distace $r$ from the center of mass, and $\alpha$ is the rotational(angular) acceleration.
Thus $\tau=(F\sin\theta)(r)$
Initially there might be slipping but eventually when pure rolling starts $a$ will be equal to $r\alpha$ i.e. $a=r\alpha$
The solution highlights which part of the applied force is responsible for movement and which part works as torque to give it the spin. I hope that this clarifies your doubt . Any suggestion or query, use the comment section :)
