Name: Pulley-Block System
Uploaded: 2019-07-19
Duration: 10 min 30 s
Description: We must notice the pulley block system to lift, move anything in real life. Do you know how we can estimate the forces and mechanism? Let us analyse this in...

Question

Find out the acceleration of the block $B$ in the following systems.

Accepted Answer

Solution for (a)

Let's take the acceleration of block $B$ as $a_{2}$ and that of the other block as $a_{1}$ as shown.

The work done by the strings on the whole system is zero.

If $x_{2}$ is the displacement of the block $B$ and $x_{1}$ is the displacement of the other block in some time then

$(T x_{2}) + (- 2 T x_{1}) = 0$

$\Rightarrow x_{2} = 2 x_{1}$

On double differentiating the above equation with respect to time, we get,

$\Rightarrow a_{2} = 2 a_{1} . . . (i)$ .

From the free-body diagram of the blocks

$\Rightarrow T = m a_{2} . . . (ii)$

$\Rightarrow 2 m g - 2 T = 2 m a_{1}$ or

$\Rightarrow m g - T = m a_{1} . . . (iii)$

From $(ii)$ and $(iii)$

$m g - m a_{2} = m a_{1}$

$\Rightarrow a_{1} + a_{2} = g$

Now using $(iii)$ ,

$\Rightarrow a_{1} + 2 a_{1} = g$

$\Rightarrow a_{1} = \frac{g}{3}$ and $a_{2} = \frac{2 g}{3}$

Therefore, the acceleration of block $B$ is $\frac{2 g}{3}$ .

Solution for (b)

Let's take the acceleration of block $B$ as $a_{2}$ and that of the other block as $a_{1}$ as shown.

The work done by the strings on the whole system is zero.

If $x_{2}$ is the displacement of the block $B$ and $x_{1}$ is the displacement of the other block in some time then

$(T x_{2}) + (- 3 T x_{1}) = 0$

$\Rightarrow x_{2} = 3 x_{1}$

On double differentiating the above equation with respect to time, we get,

$\Rightarrow a_{2} = 3 a_{1} . . . (i)$ .

From the free-body diagram of the blocks

$\Rightarrow T = m a_{2} . . . (ii)$

$\Rightarrow 3 m g - 3 T = 3 m a_{1}$ or

$\Rightarrow m g - T = m a_{1} . . . (iii)$

From $(ii)$ and $(iii)$

$m g - m a_{2} = m a_{1}$

$\Rightarrow a_{1} + a_{2} = g$

Now using $(iii)$ ,

$\Rightarrow a_{1} + 3 a_{1} = g$

$\Rightarrow a_{1} = \frac{g}{4}$ and $a_{2} = \frac{3 g}{4}$

Therefore, the acceleration of block $B$ is $\frac{3 g}{4}$ .

Find out the acceleration of the block $B$ in the following systems.

Important Questions on Laws of Motion

Learn and Prepare for any exam you want !

Find out the acceleration of the block B in the following systems.

Important Questions on Laws of Motion

Learn and Prepare for any exam you want !

Find out the acceleration of the block $B$ in the following systems.