Question

A reel of mass m and radius of gyration k is rolling down smoothly from rest with one end of the thread wound on it held in the ceiling as depicted in the figure. consider the thickness of the thread and its mass negligible in comparison with the radius r of the hub and the reel maas m. Symbol g represents the acceleration due to gravity.

The linear acceleration of the reel is

• A. $\frac{g{r}^2}{r^2+k^2}$
• B. $\frac{g{k}^2}{r^2+k^2}$
• C. $\frac{grk}{r^2+k^2}$
• D. $\frac{mg{r}^2}{r^2+k^2}$

Answer 1

The correct option is A $\frac{g{r}^2}{r^2+k^2}$


Let a = linear acceleration
for a vertical translation motion.

$mg-T=ma...\left(i\right)$

For rotational motion
Tr = I $\alpha$

$[a=r\alpha \text{(pure rolling)}]$

$Tr=m{k}^2\frac{a}{r}$

$\therefore T=\frac{m{k}^2}{r^2}.a...\left(ii\right)$

from eqs. (i) and (ii), we get

$mg-\frac{m{k}^2}{r^2}.a=ma$

or $a=\frac{g{r}^2}{k^2+r^2}$