Question 37: Oscillations

Question

Two particles, 1 and 2, each of mass $m$, are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at $x_0$, are oscillating with amplitude $a$ and angular frequency $\omega$. Thus, their positions at time $t$ are given by $x_1(t)=\left(x_0+d\right)+a \sin \omega t$ and $x_2(t)=\left(x_0-d\right)-a \sin \omega t$, respectively, where $d>2 a$. Particle 3 of mass $m$ moves towards this system with speed $u_0=a \omega / 2$, and undergoes instantaneous elastic collision with particle 2 , at time $t_0$. Finally, particles 1 and 2 acquire a center of mass speed $v_{\mathrm{cm}}$ and oscillate with amplitude $b$ and the same angular frequency $\omega$.

Accepted Answer

$$\mathrm{t}_0=\frac{\pi}{2 \omega}=\frac{\mathrm{T}}{4}$$
Particles are at extreme position
After collision
in C-frame
using WET,
$$\begin{aligned}
& \mathrm{W}_{	ext {spring }}=\Delta \mathrm{K} \
& \frac{1}{2} \mathrm{k}(2 \mathrm{~b})^2-\frac{1}{2} \mathrm{k}(2 \mathrm{a})^2=2 	imes \frac{1}{2} \mathrm{~m} 	imes\left(\frac{\mathrm{a} \omega}{4}ight)^2 \quad(\mathrm{k}=	ext { spring constant }) \
& 4 \mathrm{~kb}^2-4 \mathrm{ka}^2=2 	imes \mathrm{m} 	imes \frac{\mathrm{a}^2}{16} 	imes \frac{2 \mathrm{k}}{\mathrm{m}} \
& 4 \mathrm{~b}^2=\frac{17}{4} \mathrm{a}^2 \
& \frac{4 \mathrm{~b}^2}{\mathrm{a}^2}=4.25
\end{aligned}$$

Oscillations

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