Question 288: Centre of Mass and Rotational Motion

Question

Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be I. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is :

Accepted Answer

Consider an element of radius x and thickness dx
Mass of element, dm = $$\sigma 2\pi x\left( {dx} \right)$$
Here, $$\sigma $$ = mass per unit area = $${m \over {\pi \left( {{R^2} - {r^2}} \right)}}$$
Moment of inertia of element, dI = (dm)x
2
$$ \Rightarrow $$ I = $$\sigma 2\pi \int\limits_r^R {{x^3}dx} $$
= $$\sigma 2\pi \left( {{{{R^4} - {r^4}} \over 4}} \right)$$
= $${m \over {\pi \left( {{R^2} - {r^2}} \right)}}{\pi  \over 2}\left( {{R^4} - {r^4}} \right)$$
= $${m \over 2}\left( {{R^2} + {r^2}} \right)$$ .....(i)
Moment of inertia of thin cylinder of same mass,
I = m$$r_0^2$$  ......(ii)
$$ \Rightarrow $$ m$$r_0^2$$ = $${m \over 2}\left( {{R^2} + {r^2}} \right)$$
$$ \Rightarrow $$ $$r_0^2$$ = 250
$$ \Rightarrow $$ r
0
$$ \simeq $$ 16 cm

Answer

16 cm

Answer

12 cm

Answer

14 cm

Answer

18 cm

Centre of Mass and Rotational Motion

Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be I. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is :

Choose the correct answer: