Problem — Ranking ω for equal angular momentum

Problem

Four objects all have angular momentum $L = 10\;\mathrm{kg\,m^2/s}$: (A) solid sphere $M = 5\;\mathrm{kg}$, $R = 0{,}1\;\mathrm{m}$; (B) ring $M = 2\;\mathrm{kg}$, $R = 0{,}2\;\mathrm{m}$; (C) solid cylinder $M = 1\;\mathrm{kg}$, $R = 0{,}5\;\mathrm{m}$; (D) thin disc $M = 0{,}5\;\mathrm{kg}$, $R = 1\;\mathrm{m}$. Rank the values of $\omega = L/I$ from smallest to largest.

Solution

Since $\omega = L/I$ with $L$ equal for all, the order of $\omega$ is the inverse of the order of $I$: large moment of inertia → small $\omega$.

• A: $I = \tfrac{2}{5}\cdot 5\cdot 0{,}1^2 = 0{,}02$ → $\omega = 500\;\mathrm{rad/s}$;
• B: $I = 2\cdot 0{,}2^2 = 0{,}08$ → $\omega = 125\;\mathrm{rad/s}$;
• C: $I = \tfrac{1}{2}\cdot 1\cdot 0{,}5^2 = 0{,}125$ → $\omega = 80\;\mathrm{rad/s}$;
• D: $I = \tfrac{1}{2}\cdot 0{,}5\cdot 1^2 = 0{,}25$ → $\omega = 40\;\mathrm{rad/s}$.

$\ev{\text{D} < \text{C} < \text{B} < \text{A}}$

Links

Topics: Rotational dynamics Concepts: Angular momentum · Moment of inertia Skills: Ranking reasoning