A thin circular plate of mass M and radius R has its density varying as textitρ(. r .)( text=ρ) text0 textitr with ρ 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=aMR2 . The value of the coefficient a is:

Question

A thin circular plate of mass M and radius R has its density varying as textitρ(. r .)( text=ρ) text0 textitr with ρ 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=aMR2 . The value of the coefficient a is: (A)  (3/5) (B)  (1/2) (C)  (8/5) (D)  (3/2)

Tardigrade Admin · Accepted Answer

dm=(2 π r d r)× ρ =2π (ρ )0r2dr M=2π ρ 0∫ limits0R r2 d r=(2/3)π ρ 0R3 I CM =ρ( dm ) r 2=2 π ρ0 ∫ r 4 dr =(2/5) π ρ0 R 5 Ic m=(3/5)MR2 I=Ic m+MR2=(8/5)MR2

$\frac{3}{5}$

$\frac{1}{2}$

$\frac{8}{5}$

$\frac{3}{2}$

$d m = (2 π r d r) \times ρ = 2 π (ρ)_{0} r^{2} d r$
$M = 2 π ρ_{0} 0 \int R r^{2} d r = \frac{2}{3} π ρ_{0} R^{3}$
$I_{CM} = ρ (d m) r^{2} = 2 π ρ_{0} \int r^{4} d r = \frac{2}{5} π ρ_{0} R^{5}$
$I_{c m} = \frac{3}{5} M R^{2}$
$I = I_{c m} + M R^{2} = \frac{8}{5} M R^{2}$

53​

21​

58​

23​

dm=(2πrdr)×ρ=2π(ρ)0​r2dr M=2πρ0​0∫R​r2dr=32​πρ0​R3 ICM​=ρ(dm)r2=2πρ0​∫r4dr=52​πρ0​R5 Icm​=53​MR2 I=Icm​+MR2=58​MR2

$\frac{3}{5}$

$\frac{1}{2}$

$\frac{8}{5}$

$\frac{3}{2}$

$d m = (2 π r d r) \times ρ = 2 π (ρ)_{0} r^{2} d r$
$M = 2 π ρ_{0} 0 \int R r^{2} d r = \frac{2}{3} π ρ_{0} R^{3}$
$I_{CM} = ρ (d m) r^{2} = 2 π ρ_{0} \int r^{4} d r = \frac{2}{5} π ρ_{0} R^{5}$
$I_{c m} = \frac{3}{5} M R^{2}$
$I = I_{c m} + M R^{2} = \frac{8}{5} M R^{2}$