First slide

Conservation of mechanical energy

Question

A system consisting of two blocks of masses m₁ and m₂ connected by a spring of spring constant K. A force F is applied on m₁ along its weight and removed such that m₂ is just lifted. Then F is

Moderate

A

$m_{2} g$
B

${(m}_{1} {+m}_{2})g$
C

$\frac{{(m}_{1} m_{2})g}{m_{1} {+m}_{2}}$
D

$m_{1} g$

Solution

For $m_{1}$ : $F + m_{1} g = k x_{0}$
$\frac{F + m_{1} g}{k} = x_{0} \to 1$
For $m_{1}$ : $m_{2}$ to lift
$k x = m_{2} g \to 2$
Loss of potential energy of spring= gain of potential energy of blocks
$\begin{array}{l} \frac{1}{2} k ({x^{2}}_{0} - x^{2}) = m_{1} g (x_{0} + x) \\ \frac{1}{2} k (x_{0} - x) = m_{1} g \\ k x_{0} - k x = 2 m_{1} g \\ F + m_{1} g - m_{2} g = 2 m_{1} g \\ F = (m_{1} + m_{2}) g \\ F = (m_{1} + m_{2}) g \end{array}$

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