Rotation 1 Question 5

5. A thin circular plate of mass M and radius R has its density varying as ρ(r)=ρ0r with ρ0 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) 12

(b) 35

(c) 85

(d) 32

(2019 Main, 8 April I)

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Answer:

Correct Answer: 5. (*)

Solution:

  1. Consider an elementary ring of thickness dx and radius x.

Moment of inertia of this ring about a perpendicular axes through centre is

dIc=dmx2=ρ0x(2πx)dxx2=2πρ0x4dx

Moment of inertia of this elementary ring about a perpendicular axes at a point through edge, (by parallel axes theorem)

dI=dmx2+dmR2=2πρ0x4dx+2πρ0R2x2dx

Moment of inertia of complete disc is

I=0RdI=0R2πρ0x4dx+0R2πρ0R2x2dx=2πρ0R55+2πρ0R53=16πρ0R515a=1615 (No option matches) 



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