A thin circular plate of mass M and radius R has its density varying a

1.	A thin circular plate of mass M and radius R has its density varying as p (r) = pₒr with Pₒ as constant and r is the distance from its center. 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) 8/5(B) 1/2(C) 3/5(D) 3/2
Answer» The value of the coefficient a is equal to 3/5. Consider a ring of thickness dr at a distance R from the centre. So, the mass of this thin ring dm = p₀.r.2πr.dr Hence the mass of the disc M = ∫dm = ∫p₀.2πr²dr = 2πp₀R³/3 The moment of INERTIA is DEFINED as the product of the mass and the square of the distance from the axis. I = ∫p₀.r.2πr.dr.r² = ∫p₀.2πr⁴dr = 2πp₀R⁵/5 The RATIO of the moment of inertia to the mass of the disc is equal to 3R²/5. Hence, the moment of inertia of the disc is equal to 3MR²/5. Therefore, the value of the constant a is equal to 3/5.

A thin circular plate of mass M and radius R has its density varying as p (r) = pₒr with Pₒ as constant and r is the distance from its center. 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) 8/5(B) 1/2(C) 3/5(D) 3/2

Answer»

The value of the coefficient a is equal to 3/5.

Consider a ring of thickness dr at a distance R from the centre.
So, the mass of this thin ring dm = p₀.r.2πr.dr
Hence the mass of the disc M = ∫dm = ∫p₀.2πr²dr = 2πp₀R³/3
The moment of INERTIA is DEFINED as the product of the mass and the square of the distance from the axis.
I = ∫p₀.r.2πr.dr.r² = ∫p₀.2πr⁴dr = 2πp₀R⁵/5
The RATIO of the moment of inertia to the mass of the disc is equal to 3R²/5.
Hence, the moment of inertia of the disc is equal to 3MR²/5.
Therefore, the value of the constant a is equal to 3/5.

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