rolling constraint hard

Question

A hollow cylinder (I = m r^2) and solid cylinder (I = (1/2) m r^2) roll down same incline from rest. Ratio of their accelerations a_hollow : a_solid is:

RankGuru · Accepted Answer

Acceleration $a = \frac{g \sin 	heta}{1 + I/(m r^2)}$.
- For **Hollow Cylinder**: $I = m r^2 \implies \beta = 1$. 
  $a_h = \frac{g \sin 	heta}{1 + 1} = \frac{1}{2} g \sin 	heta$.
- For **Solid Cylinder**: $I = \frac{1}{2} m r^2 \implies \beta = 1/2$. 
  $a_s = \frac{g \sin 	heta}{1 + 1/2} = \frac{2}{3} g \sin 	heta$.
Ratio $a_h : a_s = \frac{1}{2} : \frac{2}{3} = 3 : 4$ (if simplified, or simply $1/2 : 2/3$).

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rolling constraint hard

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