Physics | JEE Main previous year questions with Solutions

Q 11 :

The moment of inertia of rod shown in the figure is $\frac{M L^{2}}{6 N}$ . Find N.

(2)

$I = \frac{M L^{2}}{3} \sin^{2} θ, angle is with vertical$

Q 12 :

From a uniform sphere of mass M and radius R, a cavity of diameter R is created as shown. Find the ratio of moment of inertia of the sphere left about AA′ and BB′.

$\frac{15}{28}$
$\frac{28}{15}$
$\frac{31}{30}$
$\frac{62}{57}$

1

2

3

4

(4)

$I_{A'} = \frac{2}{5} M R^{2} - \frac{2}{5} \frac{M}{8} \times \frac{R^{2}}{4}$

$I_{A'} = \frac{2}{5} M R^{2} \times \frac{31}{32}$

$I_{B'} = \frac{2}{5} M R^{2} - \frac{7}{5} \frac{M}{8} \frac{R^{2}}{4}$

$I_{B'} = \frac{2}{5} M R^{2} [1 - \frac{7}{64}]$

$I_{B'} = \frac{2}{5} M R^{2} \times \frac{57}{64}$

$\frac{I_{A'}}{I_{B'}} = \frac{31}{32} \times \frac{64}{57} = \frac{62}{57}$

Q 13 :

A thin wire of length L and uniform linear mass density $ρ$ is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX ' is

$\frac{ρ L^{3}}{8 π^{2}}$
$\frac{ρ L^{3}}{16 π^{2}}$
$\frac{5 ρ L^{3}}{16 π^{2}}$
$\frac{3 ρ L^{3}}{8 π^{2}}$

1

2

3

4

(4)

$About the diameter of the circular loop (ring)$

$I = \frac{1}{2} M R^{2}$

$Using parallel axis theorem$

$Moment of inertia of the loop about X X' axis$

$I_{x x'} = \frac{M R^{2}}{2} + M R^{2} = \frac{3}{2} M R^{2}$

$Here mass M = L ρ and radius R = \frac{L}{2 π};$

$∴$ $I_{x x'} = \frac{3}{2} (L ρ) {(\frac{L}{2 π})}^{2} = \frac{3 L^{3} ρ}{8 π^{2}}$

Q 14 :

A solid sphere of mass $M$ and radius $R$ is divided into two unequal parts. The smaller part having mass $\frac{M}{8}$ is converted into a sphere of radius $r$ and the larger part is converted into a circular disc of thickness $t$ and radius $2 R$ . If $I_{1}$ is moment of inertia of a sphere having radius $r$ about an axis through its centre and $I_{2}$ is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia $I_{2} / I_{1} =$ ______. [2026]

35
70
140
210

1

2

3

4

(2)

$I_{1} = \frac{2}{5} (\frac{M}{8}) r^{2} and I_{2} = \frac{\frac{7 M}{8} \times {(2 R)}^{2}}{4}$

$M = ρ \times \frac{4 π}{3} \times R^{3}$ $. . . (i)$

$\frac{M}{8} = ρ \times \frac{4 π}{3} \times$ $| r |^{3}$ $. . . (i i)$

$From (i) and (ii), r = \frac{R}{2}$

$\frac{I_{1}}{I_{2}} = \frac{\frac{2}{5} (\frac{M}{8}) {(\frac{R}{2})}^{2}}{\frac{7 M}{8 \times 4} \times {(2 R)}^{2}} = \frac{2}{5} \times \frac{1}{4 \times 7} = \frac{1}{70}$

$\Rightarrow \frac{I_{2}}{I_{1}} = \frac{70}{1}$

Q 15 :

Moment of inertia about an axis AB for a rod of mass 40 kg and length 3 m is same as that of a solid sphere of mass 10 kg and radius R about an axis parallel to AB axis with separation of 3 m as shown in figure below. The value of R is given as $\sqrt{\frac{α}{2}}$ . The value of $α$ is _______. [2026]

(15)

$Moment of inertia about an axis A B for a rod is same as that of a solid sphere, I_{Rod} = I_{SS}$

$\frac{M_{1} L^{2}}{3} = (\frac{2}{5} m_{2} R^{2} + m_{2} L^{2})$

$\frac{40 \times {(3)}^{2}}{3} = \frac{2}{5} \times 10 \times R^{2} + 10 \times {(3)}^{2}$

$120 = 4 R^{2} + 90$

$4 R^{2} = 30$

$R = \sqrt{\frac{15}{2}} = \sqrt{\frac{α}{2}} \Rightarrow α = 15$

Topic Question Set

Q 11 :

The moment of inertia of rod shown in the figure is $\frac{M L^{2}}{6 N}$ . Find N.

Q 12 :

From a uniform sphere of mass M and radius R, a cavity of diameter R is created as shown. Find the ratio of moment of inertia of the sphere left about AA′ and BB′.

Q 13 :

A thin wire of length L and uniform linear mass density $ρ$ is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX ' is

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Topic Question Set

Q 11 : The moment of inertia of rod shown in the figure is ML26N. Find N.

Q 12 : From a uniform sphere of mass M and radius R, a cavity of diameter R is created as shown. Find the ratio of moment of inertia of the sphere left about AA′ and BB′.

Q 13 : A thin wire of length L and uniform linear mass density ρ is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX ' is

Q 15 : Moment of inertia about an axis AB for a rod of mass 40 kg and length 3 m is same as that of a solid sphere of mass 10 kg and radius R about an axis parallel to AB axis with separation of 3 m as shown in figure below. The value of R is given as α2. The value of α is _______. [2026]

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Q 11 :

The moment of inertia of rod shown in the figure is $\frac{M L^{2}}{6 N}$ . Find N.

Q 12 :

From a uniform sphere of mass M and radius R, a cavity of diameter R is created as shown. Find the ratio of moment of inertia of the sphere left about AA′ and BB′.

Q 13 :

A thin wire of length L and uniform linear mass density $ρ$ is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX ' is