Question

Two blocks of masses $m_1$ and $m_2$ are connected by a spring of stiffness $k$. The coefficient of friction between the blocks and the surface is $\mu$. Find the minimum constant force $F$ to be applied to $m_1$ in order to just slide the mass $m_2$

Answer 1

At any intermediate position, the net external force is equal to
$F^{\prime }=F-(\mu m_{1}g+kx)$(numerically)
The work done by the net external force $F^{\prime }$ in displacing the block $m$, through a length $x_0=W_{ext}=\int F^{\prime }.dx$
$\Rightarrow W_{ext}={\int }_0^{x_0}[F-(\mu m^{}g+kx)]dx$
where $x_0=$ compression of the spring required to just move the block $m_2$
$\Rightarrow k{x}_0=\mu m_{2}g\Rightarrow x_0\frac{\mu m_{2}g}{k}$
Since the KE of the system just before moving the block $m_1$, just before moving the block $m_2$ iz zero, the change in $KE=0$
$\Rightarrow \Delta KE=0$
$\therefore W_{ext}=\Delta KE$
$\Rightarrow {\int }_0^{x_0}[F-(\mu m^{}g+kx)]dx=0\Rightarrow F=\mu m_{1}g+\frac{k{x}_0}{2}=\mu m_{1}g+\frac{\mu m_{2}g}{2}=[m_1+\frac{m_2}{2}]\mu g$