Question

A system consists of two cubes of masses m₁ and m₂ respectively connected by a spring of force constant k. The force (F) that should be applied to the upper cube for which the lower one just lifts after the force is removed, is

• A. $m_{1}g$
• B. $m_{2}g$
• C. $(m_1+m_2)g$
• D. $\frac{m_1+m_2}{m_1+m_2}g$

Answer 1

The correct option is C $(m_1+m_2)g$

Initially in equilibrium
$F+m_{1}g=k{x}_i$ $(x_i=initialcompression)$
$\therefore$ $x_i=\frac{F+m_{1}g}{k}$ ...(1)
Let $x_f$ be the elongation in spring when lower block just lifts. Then,
$k{x}_f=m_{2}g$ or $x_f=\frac{m_{2}g}{k}$ ....(2)
Now from conservation of mechanical energy, increase in gravitational potential energy of $m_1$ = decrease in elastic potential energy of spring.
$\therefore$ $m_{1}g(x_i+x_f)=\frac{1}{2}k(x_i^2-x_f^2)$
substituting the values of $x_i$ and $x_f$ from Eqs. (1) and (2), we get
$F=(m_1+m_2)g$