Physics | JEE Main previous year questions with Solutions

Q 1 :

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for the moment of inertia about their diameter axis AB as shown in the figure, is $\sqrt{8 / x}$ . The value of $x$ is [2024].

17
34
67
51

1

2

3

4

(3)

Hollow Sphere, $I_{sphere} = \frac{2}{3} M R^{2} = M k_{1}^{2}$

Solid Cylinder, $I_{cylinder} = \frac{1}{12} M (4 R^{2}) + \frac{1}{4} M R^{2} + M {(2 R)}^{2}$

$I_{cylinder} = \frac{67}{12} M R^{2} = M k_{2}^{2}$

$\frac{k_{1}}{k_{2}} = \sqrt{\frac{2}{3} \cdot \frac{12}{67}} = \sqrt{\frac{8}{67}} \Rightarrow x = 67$

Q 2 :

Three balls of masses 2 kg, 4 kg, and 6 kg respectively are arranged at the center of the edges of an equilateral triangle of side 2 m. The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of the triangle will be ____ kg $m^{2}$ . [2024]

(4)

$r \cos 30 ° = \frac{1}{2} \Rightarrow r = \frac{1}{\sqrt{3}}$

$I = m_{1} r^{2} + m_{2} r^{2} + m_{3} r^{2}$

$I = (2 + 4 + 6) r^{2} = 12 \times \frac{1}{3} = 4 \Rightarrow I = 4 {kg-m}^{2}$

Q 3 :

Four particles each of mass 1 kg are placed at four corners of a square of side 2 m. Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is _____ ${kgm}^{2}$ . [2024]

(16)

$I = 4 m a^{2} = 4 \times 1 \times {(2)}^{2} = 16$

Q 4 :

Two identical spheres each of mass 2 kg and radius 50 cm are fixed at the ends of a light rod so that the separation between the centers is 150 cm. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is $x / 20$ ${kgm}^{2}$ , where the value of $x$ is _____. [2024]

(53)

$I = (\frac{2}{5} m R^{2} + m d^{2}) \times 2$

$I = 2 (\frac{2}{5} \times 2 \times {(\frac{1}{2})}^{2} + 2 \times {(\frac{3}{4})}^{2}) = \frac{53}{20} {kg-m}^{2}$

$X = 53$

Q 5 :

A uniform circular disc of radius 'R' and mass 'M' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius R/2 is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above. [2025]

$\frac{7}{32} M R^{2}$
$\frac{9}{32} M R^{2}$
$\frac{17}{32} M R^{2}$
$\frac{13}{32} M R^{2}$

1

2

3

4

(4)

$I = I_{complete} - I_{removed}$

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

$I_{removed} = (\frac{M}{4}) {(\frac{R}{2})}^{2} \frac{1}{2} + (\frac{M}{4}) {(\frac{R}{2})}^{2} = \frac{3 M R^{2}}{32}$

$I = \frac{M R^{2}}{2} - \frac{3 M R^{2}}{32} = \frac{13}{32} M R^{2}$

Q 6 :

Moment of inertia of a rod of mass 'M' and length 'L' about an axis passing through its center and normal to its length is ' $α$ '. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is: [2025]

$α$
$α / 4$
$α / 8$
$α / 2$

1

2

3

4

(2)

Moment of inertia of the rod in $I^{st}$ case,

$α = \frac{M l^{2}}{12}$

Moment of inertia of the rod in ${II}^{nd}$ case,

$α' = 2 [\frac{\frac{M}{2} {(\frac{l}{2})}^{2}}{12}] = \frac{M l^{2}}{48} = \frac{α}{4}$

Q 7 :

The moment of inertia of a circular ring of mass M and diameter r about a tangential axis lying in the plane of the ring is: [2025]

$\frac{1}{2} M r^{2}$
$\frac{3}{8} M r^{2}$
$\frac{3}{2} M r^{2}$
$2 M r^{2}$

1

2

3

4

(2)

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

$I' = I_{1} + M R^{2} = \frac{3}{2} M R^{2}$

For r = 2R

$I' = \frac{3}{8} M r^{2}$

Q 8 :

A rod of linear mass density ' $λ$ ' and length 'L' is bent to form a ring of radius 'R'. Moment of inertia of ring about any of its diameter is: [2025]

$\frac{λ L^{3}}{16 π^{2}}$
$\frac{λ L^{3}}{12}$
$\frac{λ L^{3}}{4 π^{2}}$
$\frac{λ L^{3}}{8 π^{2}}$

1

2

3

4

(4)

$L = 2 π R \Rightarrow R = \frac{L}{2 π}$

$I = \frac{M R^{2}}{2} = \frac{λ \times L}{2} \times {(\frac{L}{2 π})}^{2} = \frac{λ L^{3}}{8 π^{2}}$

Q 9 :

The moment of inertia of a solid disc rotating along its diameter is 2.5 times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is n times higher than the moment of inertia of the given ring. Here, n = __________.

Consider all the bodies have equal masses. [2025]

(4)

$I_{1} = \frac{M R_{1}^{2}}{4}, I_{2} = \frac{M R_{2}^{2}}{2}, I_{3} = \frac{2 M R_{1}^{2}}{5}$

According to problem

$\frac{I_{1}}{I_{2}} = 2.5 \Rightarrow \frac{\frac{M R_{1}^{2}}{4}}{\frac{M R_{2}^{2}}{2}} = \frac{5}{2} \Rightarrow \frac{R_{1}^{2}}{R_{2}^{2}} = 5$ ... (i)

Now we are provided with information that

$\frac{I_{3}}{I_{2}} = n \Rightarrow \frac{\frac{2 M R_{1}^{2}}{5}}{\frac{M R_{2}^{2}}{2}} = n \Rightarrow \frac{4 R_{1}^{2}}{5 R_{2}^{2}} = n$ ... (ii)

From Eq. (i) and (ii) $\Rightarrow$ n = 4

Q 10 :

Two iron solid discs of negligible thickness have radii $R_{1}$ and $R_{2}$ and moment of inertia $I_{1}$ and $I_{2}$ , respectively. For $R_{2} = 2 R_{1}$ , the ratio of $I_{1}$ and $I_{2}$ would be 1/x, where x = __________. [2025]

(16)

Given, $R_{2} = 2 R_{1}$

$M_{1} = σ \times π R_{1}^{2} = M_{o}$

$M_{2} = σ \times π R_{2}^{2} = σ \times π {[2 R_{1}]}^{2} = σ \times 4 π R_{1}^{2} = 4 M_{o}$

$\frac{I_{1}}{I_{2}} = \frac{\frac{M_{1} R_{1}^{2}}{2}}{\frac{M_{2} R_{2}^{2}}{2}} = \frac{M_{1} R_{1}^{2}}{M_{2} R_{2}^{2}} = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$

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]

(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]

(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]

(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]

(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]

(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]

(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]

(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]

(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]

(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]

(5)

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

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

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

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

$∴$ $K_{1} = K_{2}$

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

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

$∴$ $x = 5$

Q 21 :

The moment of inertia of a semicircular ring about an axis passing through the center and perpendicular to the plane of the ring is $\frac{1}{x} M R^{2},$ where R is the radius and M is the mass of the semicircular ring. The value of $x$ will be _______ . [2023]

(1)

The moment of inertia of a semicircular ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is $= M R^{2}$

$So x = 1$

Q 22 :

A solid sphere and a solid cylinder of the same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyration respectively $(k_{sph} : k_{cyl})$ is $2 : \sqrt{x},$ then the value of $x$ is _________ . [2023]

(5)

$For solid sphere, \frac{2}{5} m R^{2} = m k_{sph}^{2}$

$k_{sph} = \sqrt{\frac{2}{5}} R$

$For solid cylinder, \frac{m R^{2}}{2} = m k_{cyl}^{2}$

$\Rightarrow k_{cyl} = \frac{R}{\sqrt{2}}$

$\frac{k_{sph}}{k_{cyl}} = \frac{\sqrt{\frac{2}{5}}}{\frac{1}{\sqrt{2}}} = \frac{2}{\sqrt{5}} = \frac{2}{\sqrt{x}}$

$∴$ $x = 5$

Q 23 :

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius $R$ and length $L$ $(R < L)$ about an axis passing through the mid points of opposite sides, is ______. (Take the mass of the entire loop as M). [2026]

$\frac{3}{8} M R^{2} + \frac{1}{6} M L^{2}$
$\frac{3}{8} M R^{2} + \frac{7}{12} M L^{2}$
$\frac{3}{4} M R^{2} + \frac{7}{12} M L^{2}$
$\frac{3}{4} M R^{2} + \frac{1}{6} M L^{2}$

1

2

3

4

(1)

$I_{net} = 2 (I_{1} + I_{2})$

$= 2 (\frac{M' R^{2}}{4} + \frac{M' ℓ^{2}}{12}) + 2 (\frac{M' R^{2}}{2} + M' {(\frac{ℓ}{2})}^{2})$

$= \frac{M' R^{2}}{2} + \frac{M' R^{2}}{6} + M' R^{2} + \frac{M' ℓ^{2}}{2}$

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

$Given masses M' = \frac{M}{4}$

$So, I = \frac{3 (M / 4) R^{2}}{2} + \frac{2 (M / 4) ℓ^{2}}{3}$

$I = \frac{3}{8} M R^{2} + \frac{M ℓ^{2}}{6}$

Q 24 :

Two identical thin rods of mass M kg and length L m are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is $\frac{x}{12} M L^{2} k g m^{2} .$ The value of $x$ is __________. [2026]

(17)

$I = \frac{M L^{2}}{3} + (\frac{M L^{2}}{12} + M L^{2})$

$= \frac{4 M L^{2} + M L^{2} + 12 M L^{2}}{12}$

$I = \frac{17}{12} M L^{2}$

$∴$ $x = 17$

Q 25 :

A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is __________ $kg \cdot m^{2}$ . [2026]

0.18
0.63
0.36
0.72

1

2

3

4

(2)

$I = \frac{7}{5} [m_{1} R_{1}^{2} + m_{2} R_{2}^{2}]$

$= \frac{7}{5} [5 {(10)}^{2} + 10 {(20)}^{2}] \times 10^{- 4}$

$I = 63 \times 10^{- 2} {kg m}^{2}$

$I = 0.63 {kg m}^{2}$

Q 26 :

A circular disc has radius $R_{1}$ and thickness $T_{1}$ . Another circular disc made of the same material has radius $R_{2}$ and thickness $T_{2}$ . If the moment of inertia of both discs are same and $\frac{R_{1}}{R_{2}} = 2$ , then $\frac{T_{1}}{T_{2}} = \frac{1}{α} .$ The value of $α$ is _______. [2026]

(16)

$m_{1} = π R_{1}^{2} T_{1} ρ m_{2} = π R_{2}^{2} T_{2} ρ$

$I_{1} = \frac{m_{1} R_{1}^{2}}{2} I_{2} = \frac{m_{2} R_{2}^{2}}{2}$

$I_{1} = I_{2}$

$\frac{π R_{1}^{2} T_{1} ρ R_{1}^{2}}{2} = \frac{π R_{2}^{2} T_{2} ρ R_{2}^{2}}{2} \Rightarrow \frac{T_{1}}{T_{2}} = \frac{1}{16}$

Q 27 :

A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_{1}$ . A small co-centric cylinder of length $\frac{L}{2}$ and radius $\frac{R}{3}$ carved from this cylinder has moment of inertia about its axis equals to $I_{2}$ . The ratio $I_{1} / I_{2}$ is _________. [2026]

(162)

$Original mass (M)$

$The removed mass (m)$

$m = ρ \times π {(\frac{R}{3})}^{2} \times \frac{L}{2}$

$= \frac{ρ . π R^{2} L}{18} = \frac{M}{18}$

$I' = \frac{1}{2} \cdot \frac{M}{18} \cdot \frac{R^{2}}{9} = \frac{1}{324} M R^{2}$

$\frac{I}{I'} = \frac{\frac{1}{2} M R^{2}}{\frac{1}{324} M R^{2}} = 162$

Q 28 :

Suppose there is a uniform circular disc of mass M kg and radius r m shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis A of the disc is given by $\frac{x}{256} M r^{2}$ The value of $x$ is ______. [2026]

(109)

$M = σ π R^{2}$

$σ π R^{2} = 16 m$

$m = \frac{σ π R^{2}}{16}$

$I_{system} = \frac{M R^{2}}{2} - 2 (\frac{m R^{2}}{2 \times 16} + \frac{9 m R^{2}}{16})$

$= \frac{M R^{2}}{2} - 2 \times \frac{19 m R^{2}}{32}$

$= \frac{M R^{2}}{2} - \frac{19}{16} m R^{2}$

$= \frac{M R^{2}}{2} - \frac{19}{256} M R^{2} because m = \frac{M}{16}$

$= \frac{(128 - 19) M R^{2}}{256}$

$= \frac{109 M R^{2}}{256}$

Q 29 :

A solid sphere of radius 10 cm is rotating about an axis which is at a distance 15 cm from its centre. The radius of gyration about this axis is $\sqrt{n}$ cm. The value of n is _____. [2026]

(265)

$Let radius of gyration is k$

$\Rightarrow m k^{2} = \frac{2}{3} m R^{2} + m d^{2}$

$k^{2} = \frac{2}{3} \times 10^{2} + 15^{2} = 265$

${(\sqrt{n})}^{2} = 265 \Rightarrow n = 265$

Topic Question Set

Q 1 :

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for the moment of inertia about their diameter axis AB as shown in the figure, is $\sqrt{8 / x}$ . The value of $x$ is [2024].

Q 2 :

Three balls of masses 2 kg, 4 kg, and 6 kg respectively are arranged at the center of the edges of an equilateral triangle of side 2 m. The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of the triangle will be ____ kg $m^{2}$ . [2024]

Q 3 :

Four particles each of mass 1 kg are placed at four corners of a square of side 2 m. Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is _____ ${kgm}^{2}$ . [2024]

Q 7 :

The moment of inertia of a circular ring of mass M and diameter r about a tangential axis lying in the plane of the ring is: [2025]

Q 8 :

A rod of linear mass density ' $λ$ ' and length 'L' is bent to form a ring of radius 'R'. Moment of inertia of ring about any of its diameter is: [2025]

Q 10 :

Two iron solid discs of negligible thickness have radii $R_{1}$ and $R_{2}$ and moment of inertia $I_{1}$ and $I_{2}$ , respectively. For $R_{2} = 2 R_{1}$ , the ratio of $I_{1}$ and $I_{2}$ would be 1/x, where x = __________. [2025]

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]

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]

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]

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]

Q 21 :

The moment of inertia of a semicircular ring about an axis passing through the center and perpendicular to the plane of the ring is $\frac{1}{x} M R^{2},$ where R is the radius and M is the mass of the semicircular ring. The value of $x$ will be _______ . [2023]

Q 22 :

A solid sphere and a solid cylinder of the same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyration respectively $(k_{sph} : k_{cyl})$ is $2 : \sqrt{x},$ then the value of $x$ is _________ . [2023]

Q 23 :

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius $R$ and length $L$ $(R < L)$ about an axis passing through the mid points of opposite sides, is ______. (Take the mass of the entire loop as M). [2026]

Q 24 :

Two identical thin rods of mass M kg and length L m are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is $\frac{x}{12} M L^{2} k g m^{2} .$ The value of $x$ is __________. [2026]

Q 25 :

A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is __________ $kg \cdot m^{2}$ . [2026]

Q 28 :

Suppose there is a uniform circular disc of mass M kg and radius r m shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis A of the disc is given by $\frac{x}{256} M r^{2}$ The value of $x$ is ______. [2026]

Q 29 :

A solid sphere of radius 10 cm is rotating about an axis which is at a distance 15 cm from its centre. The radius of gyration about this axis is $\sqrt{n}$ cm. The value of n is _____. [2026]

Want Us to Email you About Special Offers & Updates?

Topic Question Set

Q 1 : Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for the moment of inertia about their diameter axis AB as shown in the figure, is 8/x. The value of x is [2024].

Q 2 : Three balls of masses 2 kg, 4 kg, and 6 kg respectively are arranged at the center of the edges of an equilateral triangle of side 2 m. The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of the triangle will be ____ kg m2. [2024]

Q 3 : Four particles each of mass 1 kg are placed at four corners of a square of side 2 m. Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is _____ kgm2. [2024]

Q 7 : The moment of inertia of a circular ring of mass M and diameter r about a tangential axis lying in the plane of the ring is: [2025]

Q 8 : A rod of linear mass density 'λ' and length 'L' is bent to form a ring of radius 'R'. Moment of inertia of ring about any of its diameter is: [2025]

Q 10 : Two iron solid discs of negligible thickness have radii R1 and R2 and moment of inertia I1 and I2, respectively. For R2=2R1, the ratio of I1 and I2 would be 1/x, where x = __________. [2025]

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 x7. The value of x is __________ . [2023]

Q 16 : Moment of inertia of a disc of mass M and radius 'R' about any of its diameter is MR24. The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, x2MR2. The value of x is __________ . [2023]

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 x cm. The value of x is ________. [2023]

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 2x. The value of x is ________ . [2023]

Q 21 : The moment of inertia of a semicircular ring about an axis passing through the center and perpendicular to the plane of the ring is 1xMR2, where R is the radius and M is the mass of the semicircular ring. The value of x will be _______ . [2023]

Q 22 : A solid sphere and a solid cylinder of the same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyration respectively (ksph:kcyl) is 2:x, then the value of x is _________ . [2023]

Q 23 : The moment of inertia of a square loop made of four uniform solid cylinders, each having radius R and length L (R<L) about an axis passing through the mid points of opposite sides, is ______. (Take the mass of the entire loop as M). [2026]

Q 24 : Two identical thin rods of mass M kg and length L m are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is x12ML2 kg m2. The value of x is __________. [2026]

Q 25 : A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is __________ kg·m2. [2026]

Q 26 : A circular disc has radius R1 and thickness T1. Another circular disc made of the same material has radius R2 and thickness T2. If the moment of inertia of both discs are same and R1R2=2, then T1T2=1α. The value of α is _______. [2026]

Q 27 : A uniform solid cylinder of length L and radius R has moment of inertia about its axis equal to I1. A small co-centric cylinder of length L2 and radius R3 carved from this cylinder has moment of inertia about its axis equals to I2. The ratio I1/I2 is _________. [2026]

Q 28 : Suppose there is a uniform circular disc of mass M kg and radius r m shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis A of the disc is given by x256Mr2 The value of x is ______. [2026]

Q 29 : A solid sphere of radius 10 cm is rotating about an axis which is at a distance 15 cm from its centre. The radius of gyration about this axis is n cm. The value of n is _____. [2026]

Want Us to Email you About Special Offers & Updates?

Q 1 :

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for the moment of inertia about their diameter axis AB as shown in the figure, is $\sqrt{8 / x}$ . The value of $x$ is [2024].

Q 2 :

Three balls of masses 2 kg, 4 kg, and 6 kg respectively are arranged at the center of the edges of an equilateral triangle of side 2 m. The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of the triangle will be ____ kg $m^{2}$ . [2024]

Q 3 :

Four particles each of mass 1 kg are placed at four corners of a square of side 2 m. Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is _____ ${kgm}^{2}$ . [2024]

Q 7 :

The moment of inertia of a circular ring of mass M and diameter r about a tangential axis lying in the plane of the ring is: [2025]

Q 8 :

A rod of linear mass density ' $λ$ ' and length 'L' is bent to form a ring of radius 'R'. Moment of inertia of ring about any of its diameter is: [2025]

Q 10 :

Two iron solid discs of negligible thickness have radii $R_{1}$ and $R_{2}$ and moment of inertia $I_{1}$ and $I_{2}$ , respectively. For $R_{2} = 2 R_{1}$ , the ratio of $I_{1}$ and $I_{2}$ would be 1/x, where x = __________. [2025]

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]

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]

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]

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]

Q 21 :

The moment of inertia of a semicircular ring about an axis passing through the center and perpendicular to the plane of the ring is $\frac{1}{x} M R^{2},$ where R is the radius and M is the mass of the semicircular ring. The value of $x$ will be _______ . [2023]

Q 22 :

A solid sphere and a solid cylinder of the same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyration respectively $(k_{sph} : k_{cyl})$ is $2 : \sqrt{x},$ then the value of $x$ is _________ . [2023]

Q 23 :

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius $R$ and length $L$ $(R < L)$ about an axis passing through the mid points of opposite sides, is ______. (Take the mass of the entire loop as M). [2026]

Q 24 :

Two identical thin rods of mass M kg and length L m are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is $\frac{x}{12} M L^{2} k g m^{2} .$ The value of $x$ is __________. [2026]

Q 25 :

A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is __________ $kg \cdot m^{2}$ . [2026]

Q 28 :

Suppose there is a uniform circular disc of mass M kg and radius r m shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis A of the disc is given by $\frac{x}{256} M r^{2}$ The value of $x$ is ______. [2026]

Q 29 :

A solid sphere of radius 10 cm is rotating about an axis which is at a distance 15 cm from its centre. The radius of gyration about this axis is $\sqrt{n}$ cm. The value of n is _____. [2026]