The string in the figure is 50 cm long. When the ball is released from rest, it swings along the dotted arc. How fast is it going at the lowest point in its swing?

Question

Quizplus · Accepted Answer

Using conservation of energy, we can find the speed of the ball at the lowest point of its swing. At the highest point, the ball has only potential energy, which is converted entirely to kinetic energy at the lowest point. Therefore, the potential energy at the highest point ($PE_1$) equals the kinetic energy at the lowest point ($KE_2$):
$$PE_1=KE_2$$
$$mgh= \frac{1}{2}mv^2$$
where $m$ is the mass of the ball, $g$ is the acceleration due to gravity, $h$ is the height of the highest point with respect to the lowest point, and $v$ is the speed of the ball at the lowest point.
The highest point of the ball is at the same height as the string holding it up, which is equal to the length of the string. Therefore, $h=50$ cm = 0.5 m.
Plugging in $m=0.3$ kg and $g=9.8$ m/s$^2$ gives:
$$0.3	imes9.8	imes0.5= \frac{1}{2}	imes0.3	imes v^2$$
Solving for $v$, we get:
$$v=\sqrt{\frac{0.6	imes9.8}{0.3}}\approx 3.1\ 	ext{m/s}$$
Therefore, the best choice is C.

The String in the Figure Is 50 Cm Long