Answer to Question #111408 in Classical Mechanics for Jenny

Question

A small block of mass M = 0.201 kg hangs on the end of a light, inextensible string, length R = 25 cm. A small dart (of mass m = 0.10 kg and small enough to be considered a particle) travelling in the horizontal direction collides with and remains fixed in the block. What is the minimum speed v of the dart such that the combined object completes a circular path around the support point of the block? Minimum speed = _m.s-1.

Answer 1

According to the conservation of momentum, initially (at the bottom) onle the dard had momentum, after the interactioon the dart continued moving and gave its momentum to the block:

$$mv_\text{min}=(M+m)u_b,\\ v_\text{min}=\frac{m+M}{m}u_b.$$

At the top, however, this speed will make the block and the dart make a complete circle:

$$(m+M)\frac{v^2_t}{R}=(m+M)g,\\ v_t=\sqrt{2gR}.$$

According to law of conservation of energy for the block with the dart fixed in it, we have

$$\frac{1}{2}(m+M)v_b^2=\frac{1}{2}(m+M)v_t^2+(m+M)g\cdot2R,\\ v_b=\sqrt{5gR}.$$

Substitute this to the second equation:

$$v_\text{min}=\frac{m+M}{m}\sqrt{5gR}=10.5\text{ m/s}.$$