Answer

Verified

409.8k+ views

**Hint:**When any kind of mass attached to the string which has its one end fixed to a point experiences an external force the string and the mass undergo SHM (simple harmonic motion). During the successively changing harmonics, the total energy of the system always remains the same; it can just convert into the kinetic and the potential energy.

As per the given data,

Force constant of the string is $k=\dfrac{4mg}{L}$,

Acceleration of the monkey is $\dfrac{g}{2}$

**Complete answer:**

According to the monkey, it is moving from the string which is attached to a point. The objective of the question can be visualized clearing from the following diagram.

The force acting on the initiative when the monkey started to down is given as,

$F=k{{x}_{0}}\quad ...(1)$

As the monkey is moving downwards the force applied by the monkey will be,

$\begin{align}

& F=m(g-\dfrac{g}{2}) \\

& \Rightarrow F=m\dfrac{g}{2}\quad .....\left( 2 \right) \\

\end{align}$

By putting the value of the $k$ in extension (1) and combining equation (2) and (1),

$\begin{align}

& m\dfrac{g}{2}={{x}_{0}}\left( \dfrac{4mg}{L} \right) \\

& \Rightarrow {{x}_{0}}=\dfrac{L}{8} \\

\end{align}$

The potential energy store by the system in the starting is

$U=\dfrac{1}{2}k{{x}_{0}}$

By putting the values of the

$\begin{align}

& U=\dfrac{1}{2}k\dfrac{4mg}{L}{{\left( \dfrac{L}{8} \right)}^{2}} \\

& \Rightarrow U=\dfrac{mgL}{32}\quad ....(3) \\

\end{align}$

The kinetic energy of the moving monkey will be given as,

${{K}_{0}}=\dfrac{1}{2}m{{v}^{2}}$

The velocity of the free-falling body is given by,

$\begin{align}

& v=\sqrt{2gh} \\

& So,\,{{v}^{2}}=2gh \\

\end{align}$

The total distance the monkey has to cover after the first extension of the string will is,

$\begin{align}

& h=L+\dfrac{L}{8} \\

& \Rightarrow h=\dfrac{9L}{8} \\

\end{align}$

So, the kinetic energy of the monkey is given as

${{K}_{0}}=\dfrac{9}{16}mgL\quad ....(4)$

If the cord undergo further extension the balanced energy equation will be given as,

$\begin{align}

& {{K}_{total}}={{U}_{total}} \\

& \Rightarrow {{K}_{0}}+mgx=\dfrac{1}{2}{{\left( kx+{{x}_{0}} \right)}^{2}}-\dfrac{1}{2}kx_{0}^{2} \\

\end{align}$

By putting the values from the equation (1), (2), (3), and (4) and simplifying it we obtain a quadratic equation,

$32{{x}^{2}}-8Lx-9{{L}^{2}}=0$

So, at the roots of this quadratic equation, the value of the extension is obtained as,

$x=\dfrac{L}{8}+\dfrac{\sqrt{19}}{8}$

Thus the required answer to the question is, $\dfrac{L}{8}+\dfrac{\sqrt{19}}{8}$

**Note:**

Every object which is at a height from the ground has some potential energy and experiences a gravitational pull towards the ground. When the object moves from the height towards the ground it obeys the law of conservation of energy. According to this law, the total energy of a system always remains constant.

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

10 examples of law on inertia in our daily life

Write a letter to the principal requesting him to grant class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Change the following sentences into negative and interrogative class 10 english CBSE