A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing through O and P is I₀ and Iₚ, respectively. Both these axes are perpendicular to the plane of the lamina. Find the ratio Iₚ/I₀ to the nearest integer.
Physics · Mixed Chapter
JEE Mains

Question

A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing through O and P is I₀ and Iₚ, respectively. Both these axes are perpendicular to the plane of the lamina. Find the ratio Iₚ/I₀ to the nearest integer.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Let the removed disc have mass M and radius R, so the original larger disc of radius 2R has mass 4M. The smaller disc centre is distance R vertically above O. Step 2: About O, original disc inertia is (1/2)(4M)(2R)²=8MR². Removed-disc inertia about O is (1/2)MR²+MR²=(3/2)MR². Thus I₀=(13/2)MR². Step 3: Axis P is 2R horizontally from O. The remaining lamina mass is 3M and its horizontal centre of mass is x=0; parallel-axis translation gives Iₚ=I₀+(3M)(2R)²=(13/2+12)MR²=(37/2)MR². Step 4:…