Physics | JEE Main previous year questions with Solutions

Q 11 :

A, B and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure.

The moment of inertia of the given system about PQ is $\frac{x}{15} I$ , where $I$ is the moment of inertia of the disc about its diameter. The value of x is _________. [2025]

0%

(199)

All bodies have same mass and same radius.

Disk, $I_{A} = \frac{m R^{2}}{4} = I$

Solid sphere, $I_{B} = \frac{7}{5} m R^{2}$

Spherical shell, $I_{C} = \frac{5}{3} m R^{2}$

$I_{P Q} = \frac{m R^{2}}{4} + (\frac{2}{5} m R^{2} + m R^{2}) + (\frac{2}{3} m R^{2} + m R^{2})$

$I_{P Q} = \frac{15 m R^{2} + 24 m R^{2} + 60 m R^{2} + 40 m R^{2} + 60 m R^{2}}{60}$

$I_{P Q} = \frac{199}{60} m R^{2} = \frac{199}{15} (\frac{m R^{2}}{4}) = \frac{199}{15} I$

Q 12 :

M and R be the mass and radius of a disc. A small disc of radius R/3 is removed from the bigger dise as shown in figure. The moment of inertia of remaining part of bigger disc about an axis AB passing through the centre O and perpendicular to the plane of disc is $\frac{4}{x} M R^{2}$ . The value of x is __________. [2025]

0%

(9)

Without cavity $I_{1} = \frac{M R^{2}}{2}$

Mass of removed disc = $\frac{M}{π R^{2}} \times {(\frac{R}{3})}^{2} π = (\frac{M}{9})$

M.I. of removed disc $I_{2} = \frac{\frac{M}{9} {(\frac{R}{3})}^{2}}{2} + \frac{M}{9} \times {(\frac{2 R}{3})}^{2} = \frac{M R^{2}}{18}$

$I_{(remaining part)} = I_{1} - I_{2} = \frac{M R^{2}}{2} - \frac{M R^{2}}{18} = \frac{4 M R^{2}}{9} \Rightarrow x = 9$

Q 13 :

$I_{C M}$ is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of the disc. $I_{A B}$ is its moment of inertia about an axis AB perpendicular to the plane and parallel to axis CM at a distance $\frac{2}{3} R$ from the center, where R is the radius of the disc. The ratio of $I_{A B}$ and $I_{C M}$ is $x : 9$ . The value of $' x'$ is ________. [2023]

0%

(17)

$I_{C M} = \frac{M R^{2}}{2}$

$I_{A B} = I_{C M} + M x^{2}$

$= \frac{M R^{2}}{2} + M {(\frac{2}{3} R)}^{2}$

$= \frac{M R^{2}}{2} + M \frac{4 R^{2}}{9} = M R^{2} (\frac{1}{2} + \frac{4}{9})$

$= M R^{2} (\frac{9 + 8}{18}) = \frac{17 M R^{2}}{18}$

$I_{A B} : I_{C M} = \frac{17}{9}$

Q 14 :

If a solid sphere of mass 5 kg and a disc of mass 4 kg have the same radius, then the ratio of moment of inertia of the disc about a tangent in its plane to the moment of inertia of the sphere about its tangent will be $\frac{x}{7} .$ The value of x is __________ . [2023]

0%

(5)

$I_{1} = \frac{2}{5} m_{1} R^{2} + m_{1} R^{2} = m_{1} R^{2} (\frac{7}{5})$

$\Rightarrow I_{1} = 7 R^{2}$

$I_{2} = \frac{m_{2} R^{2}}{4} + m_{2} R^{2} = \frac{5}{4} m_{2} R^{2}$

$\Rightarrow I_{2} = 5 R^{2}$

$\frac{I_{2}}{I_{1}} = \frac{5}{7} \Rightarrow x = 5$

Q 15 :

Two discs of same mass and different radii are made of different materials such that their thicknesses are 1 cm and 0.5 cm respectively. The densities of the materials are in the ratio 3 : 5. The moment of inertia of these discs respectively about their diameters will be in the ratio of $\frac{x}{6}$ . The value of $x$ is ________. [2023]

0%

(5)

$m = ρ π R^{2} t$

$So R^{2} = \frac{m}{ρ π t}$

$I = \frac{m R^{2}}{4} = \frac{m^{2}}{4 ρ π t}$

$So \frac{I_{1}}{I_{2}} = \frac{ρ_{2} t_{2}}{ρ_{1} t_{1}} = \frac{5}{3} \times \frac{0.5}{1} = \frac{5}{6}$

$So x = 5$

Q 16 :

Moment of inertia of a disc of mass $M$ and radius 'R' about any of its diameter is $\frac{M R^{2}}{4}$ . The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, $\frac{x}{2} M R^{2}$ . The value of $x$ is __________ . [2023]

0%

(3)

$I = I_{c m} + M d^{2}$

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

$\Rightarrow x = 3$

Q 17 :

Solid sphere A is rotating about an axis PQ. If the radius of the sphere is 5 cm, then its radius of gyration about PQ will be $\sqrt{x}$ cm. The value of $x$ is ________. [2023]

0%

(110)

$I_{c m} = \frac{2}{5} M R^{2}$

$I_{P Q} = I_{c m} + m d^{2}$

$I_{P Q} = \frac{2}{5} m R^{2} + m {(10 cm)}^{2}$

For radius of gyration, $I_{P Q} = m k^{2}$

$\Rightarrow k^{2} = \frac{2}{5} R^{2} + {(10 cm)}^{2}$

$= \frac{2}{5} {(5)}^{2} + 100 = 10 + 100 = 110$

$\Rightarrow k = \sqrt{110} cm$

$\Rightarrow x = 110$

Q 18 :

A uniform solid cylinder with radius R and length L has moment of inertia $I_{1}$ , about the axis of the cylinder. A concentric solid cylinder of radius $R' = \frac{R}{2}$ and length $L' = \frac{L}{2}$ is carved out of the original cylinder. If $I_{2}$ is the moment of inertia of the carved out portion of the cylinder, then $\frac{I_{1}}{I_{2}} =$ _______ . [2023]

0%

(32)

$I_{1} = \frac{m_{1} R^{2}}{2} I_{2} = \frac{m_{2} {(R / 2)}^{2}}{2}$

$\frac{I_{1}}{I_{2}} = \frac{4 m_{1}}{m_{2}} = \frac{4 \cdot ρ π R^{2} ℓ}{ρ \cdot \frac{π R^{2}}{4} \times \frac{ℓ}{2}}$ $\Rightarrow \frac{I_{1}}{I_{2}} = 32$

Q 19 :

Two identical solid spheres each of mass 2 kg and radii 10 cm are fixed at the ends of a light rod. The separation between the centres of the spheres is 40 cm. The moment of inertia of the system about an axis perpendicular to the rod passing through its middle point is __________ $\times 10^{- 3}$ $kg \cdot m^{2}$ . [2023]

0%

(176)

$I = 2 (I_{c m} + m d^{2}) = 2 (\frac{2}{5} m r^{2} + m d^{2})$

$= \frac{4}{5} \times 2 \times {(0.1)}^{2} + 2 (2) {(0.20)}^{2}$

$= \frac{8}{5} \times 10^{- 2} + 16 \times 10^{- 2}$

$= (1.6 + 16) \times 10^{- 2} = 17.6 \times 10^{- 2}$

$I = 17.6 \times 10^{- 3} {kg m}^{2}$

Q 20 :

A ring and a solid sphere rotating about an axis passing through their centers have the same radii of gyration. The axis of rotation is perpendicular to plane of the ring. The ratio of radius of the ring to that of the sphere is $\sqrt{\frac{2}{x}} .$ The value of $x$ is ________ . [2023]

0%

(5)

$For ring, I = m R_{1}^{2} = m K_{1}^{2}$

$∴$ $Radius of gyration K_{1} = R_{1}$

$For solid sphere, I' = \frac{2}{5} m' R_{2}^{2} = m' K_{2}^{2}$

$∴$ $Its radius of gyration K_{2} = \sqrt{\frac{2}{5}} R_{2}$

$∴$ $K_{1} = K_{2}$

$∴$ $R_{1} = \sqrt{\frac{2}{5}} R_{2}$

$∴$ $\frac{R_{1}}{R_{2}} = \sqrt{\frac{2}{5}}$

$∴$ $x = 5$

Topic Question Set

0%

0%

0%

Q 14 :

If a solid sphere of mass 5 kg and a disc of mass 4 kg have the same radius, then the ratio of moment of inertia of the disc about a tangent in its plane to the moment of inertia of the sphere about its tangent will be $\frac{x}{7} .$ The value of x is __________ . [2023]

0%

0%

Q 16 :

Moment of inertia of a disc of mass $M$ and radius 'R' about any of its diameter is $\frac{M R^{2}}{4}$ . The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, $\frac{x}{2} M R^{2}$ . The value of $x$ is __________ . [2023]

0%

Q 17 :

Solid sphere A is rotating about an axis PQ. If the radius of the sphere is 5 cm, then its radius of gyration about PQ will be $\sqrt{x}$ cm. The value of $x$ is ________. [2023]

0%

0%

0%

Q 20 :

A ring and a solid sphere rotating about an axis passing through their centers have the same radii of gyration. The axis of rotation is perpendicular to plane of the ring. The ratio of radius of the ring to that of the sphere is $\sqrt{\frac{2}{x}} .$ The value of $x$ is ________ . [2023]

0%

Want Us to Email you About Special Offers & Updates?