Physics Problems & Solutions: Classical Mechanics

Two small spheres of putty, A and B of mass M and 3M, respectively, hang from the ceiling on strings of equal length l. Sphere A is drawn aside so that it is raised to a height h₀ as shown above and then released. Sphere A collides with sphere B; they stick together and swing to a maximum height h equal to

A. (1/16) h₀
B. (1/8) h₀
C. (1/4) h₀
D. (1/3) h₀
E. (1/2) h₀

(GR8677 #5)

Solution:

Conservation of energy of A before and when it hits B:
$\begin{aligned} Mgh_0&=\frac{1}{2} Mv_A^2\\v_A&=\sqrt{2gh_0}\end{aligned}$

Conservation of momentum when and after collision:
$\begin{aligned} m_A v_A+m_B v_B&=(m_A+m_B)v_{AB}\\M\sqrt{2gh_0}+0&=4Mv_{AB}\\v_{AB}&=\frac{\sqrt{2gh_0}}{4}\end{aligned}$

Conservation of energy of A and B at h = 0 and h_max:
$\begin{aligned} \frac{1}{2} m_{AB} v_{AB}^2&=m_{AB} gh_{\max}\\\frac{1}{2} 4M\left (\frac{\sqrt{2gh_0}}{4} \right )^2&=4Mgh_{\max}\\h_{\max}&=\frac{1}{16} h_0\end{aligned}$

Answer: A

Physics Problems & Solutions

Classical Mechanics - Pendulum

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