You are designing a pendulum for a science museum. The pendulum is made by attaching a brass sphere with mass \(m\)to the lower end of a long, light metal wire of (unknown) length\(L\). A device near the top of the wire measures the tension in the wire and transmits that information to your laptop computer. When the wire is vertical and the sphere is at rest, the sphere’s center is\(0.800{\rm{ }}m\)above the floor and the tension in the wire is\(265{\rm{ }}N\). Keeping the wire taut, you then pull the sphere to one side (using a ladder if necessary) and gently release it. You record the height\(h\)of the center of the sphere above the floor at the point where the sphere is released and the tension\(T\)in the wire as the sphere swings through its lowest point. You collect your results:

\(h\left( M \right)\)	\(0.800\)	\(2.00\)	\(4.00\)	\(6.00\)	\(8.00\)	\(10.0\)	\(12.0\)
\(T\left( N \right)\)	\(265\)	\(274\)	\(298\)	\(313\)	\(330\)	\(348\)	\(371\)

Assume that the sphere can be treated as a point mass, ignore the mass of the wire, and assume that mechanical energy is conserved through each measurement.

1. Plot\(T\)versus\(h\), and use this graph to calculate\({\rm{L}}\).
2. If the breaking strength of the wire is\(822{\rm{ }}N\),From what maximum height\(h\)can the sphere be released if the tension in the wire is not to exceed half the breaking strength?
3. The pendulum is swinging when you leave at the end of the day. You lock the museum doors, and no one enters the building until you return the next morning. You find that the sphere is hanging at rest. Using energy considerations, how can you explain the behavior?

According to the principle of energy conservation, the total energy of the system is stored, that is, energy cannot be created or destroyed; it can only be internally converted from one state to another if the forces acting on the system have a natural sequence.

Any net force that causes the same circular motion is called the central force. The direction of the central force is centered on the curvature, similar to the centripetal acceleration method. According to Newton's second law of motion, absolute power is accelerated many times.

Graph of \({\rm{T}}\) versus \({\rm{h}}\) from the given data is given by,

Now, we need to find an equation for \(L\).

For that you have to use the concept of mechanical energy conservation and the centripetal force law.

Now, the Pendulum’s sphere is moving in circle with radius of \(L\), where \(L\) is the length of pendulum’s string. So, we use centripetal force law.

\(\begin{aligned}{}{F_r} = m{a_r}\\T - mg &= \frac{{m{v^2}}}{R}\end{aligned}\)

\(T - mg = \frac{{m{v^2}}}{L}\) ….. (1)

Now, the initial speed of brass wire is zero, and its height from the ground is \(h\). When the sphere reaches the lowest point, its height from the ground is \(0.8{\rm{ m}}\) and the speed is unknown.

\(\begin{aligned}{}{k_i} + {U_i} &= {K_f} + {U_f}\\0 + mgh &= \frac{1}{2}m{v^2} + mg\left( {0.8} \right)\\gh &= \frac{1}{2}{v^2} + g\left( {0.8} \right)\end{aligned}\)

\({v^2} = 2g\left( {h - 0.8} \right)\) ….. (2)

Now, from equation (1) and (2),

\(\begin{aligned}{}T - mg &= \frac{m}{L}2g\left( {h - 0.8} \right)\\T &= \frac{{2mg}}{L}h - \frac{{0.8m}}{L}2g + mg\end{aligned}\)

\(T = \frac{{2mg}}{L}h + mg\left( {1 - \frac{{1.6}}{L}} \right)\) ….. (3)

Now, by the graph you can see that graph of \(T\) versus \(h\) is linear.

So you get \(\frac{{2mg}}{L}\) is slope and \(mg\left( {1 - \frac{{1.6}}{L}} \right)\;\)is y-intercept.

Now, from the graph you get the slope is \(9.293{\rm{ }}{N \mathord{\left/ {\vphantom {N m}} \right.\\} m}\) and y-intercept is \(257.4{\rm{ }}N\).

\(\begin{aligned}{}\frac{{2mg}}{L} &= 9.293\\L &= \frac{{2mg}}{{9.293}}\end{aligned}\)

Now, we know that the tension in the spring at the lower point is when the pendulum in the rest is same as the weight of the sphere of the pendulum.

\(\begin{aligned}{}mg &= 265N\\L &= \frac{{2 \times 265}}{{9.293}}\\L &= 57{\rm{ }}m\end{aligned}\)

So, length of pendulum string is \(L = 57{\rm{ }}m\).

Here we have to find maximum height \(h\) if the tension in the wire is not to exceed half the breaking strength.

So, you have to take

\(\begin{aligned}{}T &= \frac{{822}}{2}\\ &= 411N\end{aligned}\)

Now, from the equation (3) can be written as

\(\begin{aligned}{}T &= \left( {9.293} \right)h + 257.4\\h &= \frac{{T - 257.4}}{{9.293}}\\ &= \frac{{411 - 257.4}}{{9.293}}\\ &= 16.5{\rm{ }}m\end{aligned}\)

So, maximum height if the tension in the wire is not to exceed half the breaking strength is \(h = 16.5{\rm{ }}m\).

The change in mechanical energy is equal to the work done on the pendulum by non-conservative forces.

Here, the air resistance doing negative work on the pendulum. So, with time, mechanical energy of pendulum is decreasing and it will eventually lose all its mechanical energy and come to rest.