Kinematics
Intermediate
Circular Motion
Source: Principles of Physics
Problem Sets:
Kinematics set 2
Problem
A boy whirls a stone in a horizontal circle of radius $r$ and at height $h$ above level ground. The string breaks, and the stone flies off horizontally. It strikes the ground after traveling a horizontal distance of $d$.
What is the magnitude of the centripetal acceleration of the stone during the circular motion?
$$a_c = \frac{d^2g}{2hr}$$
The stone's tangential speed $v$ in the circle is equal to its initial horizontal speed for the projectile motion.
First, determine the time of flight $t$ from the vertical motion under gravity $g$:
$$h = \frac{1}{2}gt^2 \implies t = \sqrt{\frac{2h}{g}}$$Next, use the horizontal distance $d$ to find the speed $v$:
$$d = vt \implies v = \frac{d}{t} = d\sqrt{\frac{g}{2h}}$$Finally, calculate the centripetal acceleration $a_c$ using the speed $v$ and radius $r$:
$$a_c = \frac{v^2}{r} = \frac{1}{r} \left(d\sqrt{\frac{g}{2h}}\right)^2$$