Uniform Circular Motion — Question 5

Question 5

Changing radius has different effects under different constraints

Two points move on concentric circles with radii R and 2R. Consider case I, where their angular speeds are equal, and case II, where their linear speeds are equal. Motion on each circle is uniform.

1. For case I, find the outer-to-inner ratios of speed, acceleration magnitude and period.

2. Find the same three ratios for case II.

3. Explain why saying that a larger circle always means smaller acceleration is incomplete.

Question 5 - Solution

Changing radius has different effects under different constraints

(a) Hold angular speed fixed.

$$\omega_o=\omega_i:\quad\boxed{v_o/v_i=2,\quad a_o/a_i=2,\quad T_o/T_i=1}$$ Using $v=\omega R$, $a=\omega^2R$ and $T=2\pi/\omega$, the ratios are 2, 2 and 1. The outer point travels twice as far per revolution in the same time, so it moves twice as fast and has twice the acceleration.

(b) Hold linear speed fixed instead.

$$v_o=v_i:\quad v_o/v_i=1,\quad a_o/a_i=R_i/R_o=1/2,\quad T_o/T_i=2$$ Using $a=v^2/R$ and $T=2\pi R/v$, the ratios are 1, $1/2$ and 2. The larger circle bends the same-speed motion more gently and requires twice the time per revolution.

(c) State what remains constant.

$$a=\frac{v^2}{R}=\omega^2R,\quad\left.\frac{\partial a}{\partial R}\right|_v=-\frac{v^2}{R^2}<0,\quad\left.\frac{\partial a}{\partial R}\right|_\omega=\omega^2>0$$ Acceleration depends on both radius and speed. Increasing radius at fixed v decreases acceleration; increasing it at fixed $\omega$ increases acceleration because v increases too. The formulas agree once their held-fixed quantities are identified.