Question

A spool is on a horizontal surface (with friction) and pulled to the right with a thread attached to the center of the spool, so that the spool rolls without slipping. The spool has a mass of M, moment of inertia I, and a radius of R.
Requird:
a. What is the relationship between the magnitude of the force of tension and the force of friction?
b. What expressions relates the magnitudes of the translational acceleration of the spool's center of mass a and the spool's rotational acceleration about the center of mass a (recall the spool rolls without slipping on the surface).

Answer 1

Answer:

The correct answer is -

A) $F_{f} =(\frac{r}{R} ) F_{T}$

B) $\alpha =\frac{a}{R}$

Explanation:

As it is mention that the spool has mass M, radius R and moment of inertia I. In the first part of the question as the spool is not moving and r < R which means there is net torque = 0

so, $F_{f}{R} = F_{T}{r}$

and $F_{f} =(\frac{r}{R} ) F_{T}$

In the second part of the question from the given information, we can express the angular acceleration

 = $\alpha =\frac{a}{R}$ ( alpha = angular acceleration and a = translational acceleration)