Two pendulums with identical bobs and lengths are suspended from a common support such that in rest position the two bobs are in contact (figure). One of the bobs is released after being displaced by $10^o$ so that it collides elastically head-on with the other bob.
$(a)$ Describe the motion of two bobs.
$(b)$ Draw a graph showing variation in energy of either pendulum with time, for $0\, \leqslant \,t\, \leqslant \,2T$, where $T$ is the period of each pendulum.
Two pendulums with identical bobs and lengths are suspended from a common support such that in rest position the two bobs are in contact (figure). One of the bobs is released after being displaced by $10^o$ so that it collides elastically head-on with the other bob.
$(a)$ Describe the motion of two bobs.
$(b)$ Draw a graph showing variation in energy of either pendulum with time, for $0\, \leqslant \,t\, \leqslant \,2T$, where $T$ is the period of each pendulum.
$(a)$ Consider the adjacent diagram in which the bob B is displaced through an angle $\theta$ and released.
At $t=0$, suppose bob B is displaced by $\theta=10^{\circ}$ to the right. It is given potential energy $\mathrm{E}_{1}=\mathrm{E}$. Energy of $\mathrm{A}, \mathrm{E}_{2}=0$.
When B is released, it strikes A at $t=\mathrm{T} / 4$. In the head-on elastic collision between B and A comes to rest and A gets velocity of B. Therefore E $_{1}=0$ and E $_{2}=\mathrm{E}$. At $t=2 \mathrm{~T} / 4, \mathrm{~B}$ reaches its extreme right position when $\mathrm{KE}$ of $\mathrm{A}$ is converted into $\mathrm{PE}=\mathrm{E}_{2}=\mathrm{E}$. Energy of $\mathrm{B}, \mathrm{E}_{1}=0 .$
At $t=3 \mathrm{~T} / 4$, A reaches its mean position, when its $\mathrm{PE}$ is converted into $\mathrm{KE}=\mathrm{E}_{2}=\mathrm{E}$. It collides elastically with $\mathrm{B}$ and transfers whole of its energy to $\mathrm{B}$. Thus, $\mathrm{E}_{2}=0$ and $\mathrm{E}_{1}=\mathrm{E}$. The entire process is repeated.
$(b)$ The value of energies of $\mathrm{B}$ and $\mathrm{A}$ at different time intervals are tabulated here. The plot of energy with time $0 \leq t \leq 2 \mathrm{~T}$ is shown separately for B and A in the figure below.
| Time $(t)$ | Energy of $\mathrm{A}\left(\mathrm{E}_{1}\right)$ | Energy of $\mathrm{A}\left(\mathrm{E}_{2}\right)$ |
| $0$ | $E$ | $0$ |
| $\mathrm{~T} / 4$ | $0$ | $E$ |
| $2 \mathrm{~T} / 4$ | $0$ | $E$ |
| $3 \mathrm{~T} / 4$ | $E$ | $0$ |
| $4 \mathrm{~T} / 4$ | $E$ | $0$ |
| $5 \mathrm{~T} / 4$ | $0$ | $E$ |
| $6 \mathrm{~T} / 4$ | $0$ | $E$ |
| $7 \mathrm{~T} / 4$ | $E$ | $0$ |
| $8 \mathrm{~T} / 4$ | $E$ | $0$ |
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