Two rings of radii R and n R made from the same wire have the ratio of moments of inertia about an axis passing through their centre equal to 1 : 8. The value of n is

To find the value of n in the problem of two rings of the same wire, we must compare their moments of inertia. A ring’s moment of inertia is a function of its mass and the square of its radius; thus, two rings, one with radius R and the other with radius nR, have their moments of inertia to be compared:.

Given that the ratio of their moments of inertia is 1:8, we can write this relationship by looking at how the mass of each ring is related to its radius. Since both rings are made of the same wire, they have mass proportional to their circumferences. Thus, the mass of the first ring can be expressed in relation to its radius and similarly for the second ring.

Substituting these expressions into the moment of inertia ratio gives us a relationship that allows us to isolate n. Simplifying, we see that n³ = 8. Taking the cube root of both sides gives us the conclusion that the value of n is 2. This means that the radius of the second ring is twice that of the first ring.

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