Question

From a fixed pulley the masses $2\text{kg}$, $1\text{kg}$ and $3\text{kg}$ are suspended as shown in the figure. Find the extension in spring if its stiffness is $k=100\text{N/m}$. Assume the acceleration of $1\text{kg}$ and $3\text{kg}$ to be same.
(Take $g=10{\text{m/s}}^2$)

• A. $0.4\text{m}$
• B. $0.2\text{m}$
• C. $0.3\text{m}$
• D. $0.5\text{m}$

Answer 1

The correct option is B $0.2\text{m}$

The block of mass $1\text{kg}$ is connected with string and spring, hence it is acted upon by two different tension $T$ and $T^{\prime }$

Consider $2\text{kg}$ and $4\text{kg}$ as a combined system.

Thus system will consist of $2\text{kg}$ and $4\text{kg}$ mass.

Acceleration of the system, $a=\frac{M-m}{M+m}g=\frac{4-2}{4+2}g=\frac{g}{3}$

Here tension $T^{\prime }$ is due to spring force,



$\Rightarrow T^{\prime }=kx$

Substituting values in Eq. $\left(iii\right)$,

$\Rightarrow$ $T^{\prime }=3g-3a$

$\text{or,}kx=3g-3\left(\frac{g}{3}\right)=2g$

$\Rightarrow x=\frac{2g}{k}=\frac{20}{100}=0.20\text{m}$

$\begin{array}{c}\text{Why this question}?\\ \text{It intends to test your understanding of}\\ \text{constraint relation and application of newton's second law}\\ \text{Tip: Since the mass}1\text{kg}\text{is connected with}\\ 2\text{kg}\text{mass}{a}_{1\text{kg}}=a_{2\text{kg}}\text{will always hold.}\\ \\ \text{Caution: Acceleration of masses connected with spring}\\ \text{may not be equal due to inertia. However}\\ \text{in this case assumption is mentioned in question.}\end{array}$