Question 492: Centre of Mass and Rotational Motion

Question

A solid cylinder of mass ' $M$ ' and radius ' $R$ ' is rotating about its geometrical axis. A solid sphere of same mass and same radius is also rotating about its diameter with an angular speed half that of the cylinder. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder will be

Accepted Answer

Step 1: Recall the moments of inertia
Solid cylinder about its symmetry axis:
$ I_	ext{cyl} = 	frac{1}{2} M R^2 $
Solid sphere about its diameter:
$ I_	ext{sphere} = 	frac{2}{5} M R^2 $
Step 2: Rotational KE formula
$ 	ext{Rotational KE} = 	frac{1}{2} I \omega^2 $
Step 3: Compute KEs
Cylinder (angular speed = $\omega$):
$ K_	ext{cyl} = 	frac{1}{2} \cdot 	frac{1}{2} M R^2 \cdot \omega^2   = 	frac{1}{4} M R^2 \omega^2 $
Sphere (angular speed = $	frac{\omega}{2}$):
$ K_	ext{sphere} = 	frac{1}{2} \cdot 	frac{2}{5} M R^2 \cdot \Big(	frac{\omega}{2}\Big)^2 $
$ = 	frac{1}{2} \cdot 	frac{2}{5} M R^2 \cdot 	frac{\omega^2}{4} = 	frac{1}{20} M R^2 \omega^2 $
Step 4: Ratio
$ \frac{K_	ext{sphere}}{K_	ext{cyl}}  = \frac{	frac{1}{20} M R^2 \omega^2}{	frac{1}{4} M R^2 \omega^2} = \frac{1}{20} \cdot \frac{4}{1} = 	frac{1}{5} $
So the ratio = $1 : 5$.
Ã¢ÂÂ Final Answer:
Option C: $1:5$

Answer

$2: 3$

Answer

$3: 2$

Answer

$1: 5$

Answer

$5: 1$

Centre of Mass and Rotational Motion

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