Physics | JEE Main previous year questions with Solutions

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]

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(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]

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(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]

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

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(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]

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(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

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(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]

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(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]

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(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]

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(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]

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(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 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]

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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]

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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]

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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]

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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]

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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]

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Topic Question Set

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]

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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]

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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]

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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]

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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]

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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]

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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]

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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]