A metal ball of mass m = 30 kg hangs on a rope of length l =…

A metal ball of mass $m = 30\ \mathrm{kg}$ hangs on a rope of length $l = 10\ \mathrm{m}$. It is given a speed $v_0$ perpendicular to the rope, and the rope swings out to $\alpha = 60^\circ$. Find $v_0$ and the energy given to the ball.

Explanation of Method

Push needed to swing a ball out to 60°.

Step 1: Speed

\[ \tfrac12 mv_0^2 = mgl(1 - \cos\alpha), \]

\[ v_0 = \sqrt{2gl(1 - \cos\alpha)} = \sqrt{98.1}. \]

\[v_0 \approx 9.9\ \mathrm{m/s}\]

Step 2: Energy

\[E = mgl(1 - \cos\alpha) \approx 1.47\ \mathrm{kJ}\]

$v_0 = 9.9\ \mathrm{m/s}$ $60^\circ$ $h = l(1 - \cos60^\circ) = 5\ \mathrm{m}$ $l = 10\ \mathrm{m}$

The push gives the ball kinetic energy that is all turned into potential energy at the 60° turning point, 5 m higher.

Result

\[\boxed{\,v_0 \approx 9.9\ \mathrm{m/s}\,}\]

\[\boxed{\,E = mgl(1 - \cos\alpha) \approx 1.47\ \mathrm{kJ}\,}\]

Remark

Check. The original results are correct.