Question

The blocks of mass $m_1=1\text{kg}$ and $m_2=2\text{kg}$ are connected by a spring and rest on a rough horizontal surface. The spring is unstressed. The spring constant of spring is $K=2\text{N/m}$. The coefficient of friction between blocks and horizontal surface is 0.5. Now the left block is imparted a velocity $u$ towards right as shown, then what is the largest value of $u$ (in $\text{m/s}$) such that the block of mass $m_2$ never moves (Take $g=10{\text{m/s}}^2$).

Answer 1

Let $x$ be the compression in the spring.
For the $2\text{kg}$ block to move $kx\ge 2\mu g$ ---(1)
$\Rightarrow x\ge 5\text{m}$

For the minimum velocity given to block, it comes to stop when compression is $5m$.

Applying WE theorem :
Change in K.E = Work done by friction + P.E stored in spring
$\frac{1}{2}\times 1\times v^2=\mu \times 1\times g\times x+\frac{1}{2}k{x}^2$
$\Rightarrow \frac{1}{2}{v}^2=\frac{1}{2}g\times 5+\frac{1}{2}\times 2\times 5^2$
$\Rightarrow v=10\text{m/s}$