Kinematics
Intermediate
Circular Motion
Source: High School Physics Olympiad
Problem Sets:
kinetics - olympiad
Problem
An object starts from rest and moves along a circle of radius $R = 3.0$ m with constant tangential acceleration $a_\tau = 3.0$ m/s$^2$.
- After how much time does the total acceleration $\vec{a}$ make a 30° angle with the radius?
- What distance $s$ has the object traveled in this time?
$t = 3^{1/4} \approx 1.3$ s; $s = \frac{3\sqrt{3}}{2} \approx 2.6$ m
Q1: The speed grows as $v = a_\tau t$, so the normal (radial) acceleration is $a_n = \dfrac{v^2}{R} = \dfrac{a_\tau^2 t^2}{R}$.
The total acceleration makes a 30° angle with the radius (the normal direction) when:
$$\tan 30° = \frac{a_\tau}{a_n} = \frac{a_\tau R}{a_\tau^2 t^2} \quad\Rightarrow\quad t^2 = \frac{\sqrt{3}R}{a_\tau} = \frac{\sqrt{3}(3.0)}{3.0} = \sqrt{3}$$ $$t = 3^{1/4} \approx 1.3 \text{ s}$$Q2: The distance traveled:
$$s = \frac{1}{2}a_\tau t^2 = \frac{1}{2}(3.0)\sqrt{3} \approx 2.6 \text{ m}$$