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Moment of Inertia of a Rod

Two loops P a...

Question

Two loops P and Q are made from a same uniform wire. The radii of P and Q are $r_{1}$ and $r_{2}$ respectively and their moments of inertia are $l_{1}$ and $l_{2}$ respectively. If $\frac{l_{2}}{l_{1}}$ = 4, then $\frac{r_{2}}{r_{1}}$ is equal to:

A

4^{2 / 3}

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B

4^{1 / 3}

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C

4^{- 2 / 3}

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D

4^{- 1 / 3}

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Solution

The correct option is B $4^{1 / 3}$
$\frac{I_{2}}{I_{1}} = 4$
$\Rightarrow \frac{M_{2} R_{2}^{2}}{M_{1} R_{1}^{2}} = 4$ .....(i)
$λ ∴$ linear mean density
$M_{1} = (2 π R_{1}) λ, M_{2} = 2 π R_{2} λ_{2}$
by eq (i)
$\frac{2 π R_{2} λ R_{2}^{2}}{2 π R_{1} λ R_{1}^{2}} = 4$
$\frac{R_{1}^{3}}{R_{1}^{3}} = 4 \Rightarrow \frac{R_{2}}{R_{1}} = (4)^{1 / 3}$

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Similar questions

Q. Two loops P and Q are made from a uniform wire. The radii of P and Q are

r_{1}

r_{2}

respectively, and their moments of inertia about their own axises are

I_{1}

I_{2}

respectively. If

I_{2} = 4 I_{1}

r_{1} / r_{2}

Q. Two tubes of radii

r_{1}

r_{2}

l_{1}

l_{2}

, respectively, are connected in series and a liquid flows through each of them in stream line conditions.

P_{1}

P_{2}

are pressure differences across the two tubes. If

P_{2}

4 P_{1}

l_{2}

\frac{l_{1}}{4}

, then the radius

r_{2}

will be equal to :

Q. A planet of mass M, has two natural satellites with masses

m_{1}

m_{2}

. The radii of their circular orbits are

R_{1}

R_{2}

respectively. Ignore the gravitational force between the satellites. Define

v_{1}, L_{1}, K_{1}

T_{1}

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v_{2}, L_{2}, K_{2}

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to be the corresponding quantities of satellite

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m_{1} / m_{2} = 2

R_{1} / R_{2} = 1 / 4

, match the ratios in List - I to the numbers in List-II.

List-I	List-II
P. $\frac{v_{1}}{v_{2}}$	1. $\frac{1}{8}$
Q. $\frac{L_{1}}{L_{2}}$	2. $1$
R. $\frac{K_{1}}{K_{2}}$	3. $2$
S. $\frac{T_{1}}{T_{2}}$	4. $8$

Two cylindrical rods of lengths $l_{1}$ and $l_{2}$ , radii $r_{1}$ and $r_{2}$ have thermal conductivities $k_{1}$ and $k_{2}$ respectively. The ends of the rods are maintained at the same temperature difference. If $l_{1} = 2 l_{2}$ and $r_{1} = \frac{r_{2}}{2}$ , the rates of heat flow in them will be the same if $\frac{k_{1}}{k_{2}}$ is

Q. A planet of mass

M

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m_{1}

m_{2}

. The radii of their circular orbits are

R_{1}

R_{2}

respectively. Ignore the gravitational force between the satellites. Define

v_{1}

L_{1}

K_{1}

T_{1}

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v_{2}, L_{2}, K_{2}

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R_{1} / R_{2} = 1 / 4

, match the ratios in List - I to the numbers in List-II.

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