Two circular loops P and Q of radii ' r 'and ' nr ' are made respectively from a uniform wire. Moment of inertia of loop Q about its axis is four times that of loop P about its axi

Question

Two circular loops P and Q of radii ' r 'and ' nr ' are made respectively from a uniform wire. Moment of inertia of loop Q about its axis is four times that of loop P about its axis. The value of ' n ' is

TestHub · Accepted Answer

Let the linear mass density of the uniform wire be $λ$ .

Mass of loop P, $M_{P} = λ (2 π r)$

Moment of inertia of loop P about its axis, $I_{P} = M_{P} r^{2} = λ (2 π r) r^{2} = 2 πλ r^{3}$

Mass of loop Q, $M_{Q} = λ (2 πn r)$

Moment of inertia of loop Q about its axis, $I_{Q} = M_{Q} (n r)^{2} = λ (2 πn r) (n r)^{2} = 2 πλ n^{3} r^{3}$

Given that $I_{Q} = 4 I_{P}$

$2 πλ n^{3} r^{3} = 4 (2 πλ r^{3})$

$n^{3} = 4$

$n = 4^{1/3} = (2^{2})^{1/3} = 2^{2/3}$

Answer: $(2)^{2/3}$

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