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 𝜔. Thus, their positions at time 𝑡 are given by $x_1 (t) = (x_0 + d) + \alpha \ sin \omega t$ and $x_2t = (x_0 -d) − \alpha sin \ wt,$ respectively, where $𝑑 2𝑎.$ Particle 3 of mass 𝑚 moves towards this system with speed 𝑢0 = 𝑎𝜔/2, and undergoes instantaneous elastic collision with particle 2, at time $𝑡_0$ . Finally, particles 1 and 2 acquire a center of mass speed $𝑣_{cm}$ and oscillate with amplitude 𝑏 and the same angular frequency 𝜔.

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 𝜔. Thus, their positions at time 𝑡 are given by  $x_1 (t) = (x_0 + d) + \alpha \  sin \omega t$  and  $x_2t = (x_0 -d) − \alpha sin \ wt,$  respectively, where  $𝑑 > 2𝑎.$  Particle 3 of mass 𝑚 moves towards this system with speed 𝑢0 = 𝑎𝜔/2, and undergoes instantaneous elastic collision with particle 2, at time  $𝑡_0$ . Finally, particles 1 and 2 acquire a center of mass speed  $𝑣_{cm}$  and oscillate with amplitude 𝑏 and the same angular frequency 𝜔.

cdquestions Admin · Accepted Answer

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If the collision occurs at time \(𝑡_0 = 0\), the value of \(\frac{𝑣_cm}{(\alpha\omega)}\) will be __________

Correct Answer: 0.75

Solution and Explanation

If the collision occurs at time \(𝑡_0 =\frac{\pi}{(2\omega)}\), then the value of \(4𝑏^2/𝑎^2\) will be ________

Correct Answer: 4.25

Solution and Explanation

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