A person of mass M is sitting on a swing of length L and swinging with and angul | TestHub

A person of mass $M$ is sitting on a swing of length $L$ and swinging with and angular amplitude $θ_{0} .$ If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance $l (l < < L),$ is close to:

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Answer:

Solution:

From angular momentum conservation
$\begin{array}{l} M V_{0} L = M V (L - l) \\ V = V_{0} (\frac{L}{L - l}) \end{array}$
Now
$W = M g l + \frac{1}{2} M [{V_{0}}^{2} {(\frac{L}{L - l})}^{2} - {V_{0}}^{2}]$
Since $l ≪ L$
$\begin{array}{l} W = M g l + \frac{1}{2} M [{V_{0}}^{2} {(1 - \frac{l}{L})}^{- 2} - {V_{0}}^{2}] \\ = M g l + \frac{1}{2} M [{V_{0}}^{2} (1 + 2 \frac{l}{L}) - {V_{0}}^{2}] \\ = M g l + M [{V_{0}}^{2} (\frac{l}{L})] \end{array}$
From SHM
$V_{0} = ω A = \sqrt{\frac{g}{L}} \times L θ_{0} = \sqrt{g L} θ_{0}$
Using this
$\begin{array}{l} W = M g l + M [g L \times {θ_{0}}^{2} \times (\frac{l}{L})] \\ W = M g l [1 + {θ_{0}}^{2}] \end{array}$

Physics - SHM Question with Solution | TestHub

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