The Kinetic Energy of Rolling — Question 6

Question 6

A height measurement can reject an impossible inertia model

A rigid object rolls from rest through $h=0.50\,\mathrm m$ with no losses. All its mass lies within outer radius $R$, and $I=\beta MR^2$. An instrument reports $V=2.0\,\mathrm{m/s}$. Take $g=9.8\,\mathrm{m/s^2}$.

1. Infer the apparent $\beta$ and test the assumptions.

2. Find the smallest possible final speed under those assumptions.

Question 6 - Solution

A height measurement can reject an impossible inertia model

(a) Derivation and calculation

The measured speed would imply $$\beta=\frac{2gh}{V^2}-1=\frac{2(9.8)(.50)}{(2.0)^2}-1 =\boxed{1.45}.$$ But $r\le R$ for every mass element requires $$I=\int r^2\,dm\le MR^2\Rightarrow\beta\le1.$$ The measurement and the complete stated ideal model are incompatible.

(b) Result and interpretation

The speed decreases with increasing $\beta$: $$V(\beta)=\sqrt{\frac{2gh}{1+\beta}}.$$ Its smallest admissible value occurs at $\beta=1$: $$\boxed{V_{\min}=\sqrt{gh}=\sqrt{4.90}=2.214\ \mathrm{m/s}}.$$ A loss, measurement error, or failure of the radius/no-slip assumptions must explain the lower reported speed.

Physical check

The reported 2.0 m/s is below the allowed minimum even for a hoop-like inertia distribution.