Question

In the pulley system shown, if radii of the bigger and smaller pulley are $2m$ and $1m$, respectively and the acceleration of the block A is $5m/s^2$ in the downward direction, the acceleration of block B will be:

$0m/s^2$
$5m/s^2$
$10m/s^2$
${\displaystyle \frac{5}{2}}m/s^2$

Answer 1

Hint: Since the bodies will be acting with gravity on both sides of the pulleys, the pulleys are going to experience angular acceleration. Since the bodies are connected to each other, the angular acceleration of the bodies is going to be equal.

Formulas used: Since the pulley appears to be going through a circular motion from the origin, we will use the formula $\alpha ={\displaystyle \frac{a}{r}}$ where $\alpha$ will be the angular acceleration of the pulley, $a$ is the linear acceleration of the body connected to the pulley, $r$ is the radius of the pulley.

Complete Answer:
We know that when a body is suspended by a pulley it experiences a tension that opposes gravity and the gravity itself acting on it. But considering two such pulleys connected to each other, both the pulleys will be undergoing angular acceleration.
The angular acceleration that both the bodies undergo will be given by the formula,
For body A, ${\alpha }_A={\displaystyle \frac{a_A}{r_A}}$ .
For body B, $${\alpha }_B={\displaystyle \frac{a_B}{r_B}}$$

But as we can infer from the diagram that the pulleys are connected to each other and thus the rotation of the pulleys and by them the bodies A and B are limited. And thus, the angular acceleration of the bodies is going to equal. That is,
$\Rightarrow {\alpha }_A={\alpha }_B$ which means that ${\displaystyle \frac{a_A}{r_A}}={\displaystyle \frac{a_B}{r_B}}$ .
We know from the problem, $a_A=5m/s^2$, $r_A=2m$ and $r_B=1m$. Substituting these values, we get,
${\displaystyle \frac{5}{2}}={\displaystyle \frac{a_B}{1}}$
$\Rightarrow a_B=1\times {\displaystyle \frac{5}{2}}$
Thus, the acceleration of the block B will be $a_B={\displaystyle \frac{5}{2}}m/s^2$ .
Hence the correct answer will be option D.

Note:
The formula $\alpha ={\displaystyle \frac{a}{r}}$ can be used only when the motion of the body is considered circular about the centre. Otherwise, the angular acceleration is given by the ratio between the tangential acceleration experienced by the body to their radius $(\alpha ={\displaystyle \frac{a_{\mathrm{\perp }}}{r}})$ .