Question 276: Centre of Mass and Rotational Motion

Question

A thin circular plate of mass M and radius R
has its density varying as $$\rho $$(r) = $$\rho $$
0
r with $$\rho $$
0
as
constant and r is the distance from its centre.
The moment of Inertia of the circular plate about
an axis perpendicular to the plate and passing
through its edge is I = aMR
2
. The value of the
coefficient a is :

Accepted Answer

$$M = \int\limits_0^R {{ho _0}r 	imes 2\pi rdr = {{2\pi {ho _0}{R^3}} \over 3}} $$
$${I_C} = \int\limits_0^R {{ho _0}r 	imes 2\pi rdr 	imes {r^2} = {{2\pi {ho _0}{R^5}} \over 3}} $$
$$ 	herefore $$ $$I = {I_C} + M{R^2} = 2\pi {ho _0}{R^5}\left( {{1 \over 3} + {1 \over 5}} ight)$$
$$ \Rightarrow $$$${{16\pi {ho _0}{R^5}} \over {15}} = {8 \over 5}\left[ {{2 \over 3}\pi {ho _0}{R^3}} ight]{R^2} = {8 \over 5}M{R^2}$$

Answer

$${1 \over 2}$$

Answer

$${3 \over 2}$$

Answer

$${8 \over 5}$$

Answer

$${3 \over 5}$$

Centre of Mass and Rotational Motion

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