Question

A person of mass M is, sitting on a swing of length L and swinging with an angular amplitude ${\theta _0}$ ​. If the person stands up when the swing passes through its lowest point, what is the work done by him, assuming that his centre of mass moves by a distance $l(l < < L)$?

Answer 1

We will start our solution by taking the angular momentums conservation law. Further by putting the values and solving for $(l < < L)$ by binomial approximation, we will get the required work done by the person of mass M.

Formula used:
$M{V_0}L = M{V_{1}}(L - \ell )$
$\omega \times A = (\sqrt {\dfrac{g}{L}} )({\theta _0}L)$

Complete step by step answer:
here we know angular momentum or moment is defined as the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity, which says that the total angular momentum of a closed system remains constant.
Angular momentum conservation is given by:
\eqalign{ & M{V_0}L = M{V_{1}}(L - \ell ) \cr & \Rightarrow {V_1} = {V_0}(\dfrac{L}{{L - \ell }}) \cr & \Rightarrow {w_g} + {w_p} = \Delta KE \cr & \Rightarrow - mg\ell + {w_p} = \dfrac{1}{2}m({V_1}^2 - {V_0}^{2}) \cr & \Rightarrow {w_p} = mg\ell + \dfrac{1}{2}m{V_0}^2\left[ {{{\left( {\dfrac{{L- \ell}}{{L }}} \right)}^{ - 2}} - 1} \right] \cr & \Rightarrow {w_p} = mg\ell + \dfrac{1}{2}m{V_0}^2\left[ {{{\left( {1 - \dfrac{{l}}{L}} \right)}^{ - 2}} - 1} \right] \cr}
Now $l(l < < L)$,
From binomial approximation we get:
\eqalign{ & \Rightarrow {w_p} = mg\ell + \dfrac{{1}}{2}m{V_0}^2\left( {(1 + \dfrac{{2\ell }}{L}) - 1} \right) \cr & \Rightarrow {w_p} = mg\ell + \dfrac{{1}}{2}m{V_0}^2\left( {\dfrac{{2\ell }}{L}} \right) \cr & \Rightarrow {w_p} = mg\ell + m{V_0}^{2}\dfrac{\ell }{L} \cr}
Here, ${V_0}$ is maximum velocity.
\eqalign{ & \omega \times A = (\sqrt {\dfrac{g}{L}} )({\theta _0}L) \cr & {V_0} = {\theta _0}\sqrt {gL} \cr & \Rightarrow {w_{p}} = mg\ell + m{({\theta _0}\sqrt {gL} )^2}\dfrac{\ell }{L} \cr & \therefore {w_{p}} = mg\ell (1 + {\theta _0}^{2}) \cr}

Therefore, the work done by the man by assuming that his centre of mass moves by a distance $l(l < < L)$ is given by ${w_p}$.

Additional Information: The law of conservation of angular momentum states that when no external torque acts on an object, no change of angular momentum will occur.
The total energy of a system is the sum of kinetic energy and the potential energy. Also, we know that according to the law of conservation of energy, energy can neither be created nor be destroyed; it can only be transferred from one form to another. Electric energy is converted to heat energy by the use of a water heater.
Potential energy is defined as the energy that is stored in an object due to its position relative to some zero position. An object or a body possesses gravitational potential energy if it is positioned at a height above or below the zero height.
Further, Kinetic energy is defined as the form of energy that an object or a particle has by reason of its motion. Kinetic energy is a property of a moving object or particle.

Note: Law of conservation of angular momentum and energy is needed to be remembered i.e., when no external torque acts on an object, no change of angular momentum will occur and energy can neither be created nor be destroyed, it can only be transferred from one form to another respectively. If a particle or body is only in motion but does not have any height, the potential energy in that case will be zero.