Question

Block $m$ is released from rest when spring is in its natural length. Assume pulley is ideal and block does not strike on ground during its motion in vertical plane. Then:

• A. maximum elongation in spring is $4mg/k$
• B. maximum elongation in spring is $2mg/k$
• C. maximum speed of block is $2g\sqrt{\frac{m}{k}}$
• D. maximum speed of block is $g\sqrt{\frac{m}{k}}$

Answer 1

Answer: (A), (C)

maximum elongation in spring is $4mg/k$
maximum speed of block is $2g\sqrt{\frac{m}{k}}$

By energy conservation $\frac{1}{2}k{x}^2=mg\left(2x\right)$

$\Rightarrow x=\frac{4mg}{k}$(maximum elongation)

at equilibrium: $kx=2mg\Rightarrow x=\frac{2mg}{k}$

So $(K.M.{)}_{max}=mg(2x)-\frac{1}{2}k{x}^2=2mg\left(\frac{2mg}{k}\right)-\frac{1}{2}k{\left(\frac{2mg}{k}\right)}^2$

$\frac{1}{2}m{v}_{max}^2=\frac{2m^2{g}^2}{k}\Rightarrow v_{max}=2g\sqrt{\frac{m}{k}}$