Rotational Dynamics | JEE Main previous year questions with Solutions

Q 1 :
A hollow sphere is rolling on a plane surface about its axis of symmetry. The ratio of rotational kinetic energy to its total kinetic energy is $\frac{x}{5}$ . The value of $x$ is _______ . [2024]

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(2) $\frac{K E_{Rot}}{K E_{Trans}} = \frac{\frac{1}{2} I ω^{2}}{\frac{1}{2} I ω^{2} + \frac{1}{2} m v^{2}} = \frac{(\frac{1}{2}) (\frac{2}{3} m R^{2}) ω^{2}}{(\frac{1}{2}) (\frac{2}{3} m R^{2}) ω^{2} + \frac{1}{2} m {(R ω)}^{2}}$

$\Rightarrow \frac{K E_{Rot}}{K E_{Trans}} = \frac{\frac{2}{3}}{\frac{2}{3} + 1} = \frac{2}{5}$

Hence, $x = 2$

Q 2 :
A circular disc reaches from top to bottom of an inclined plane of length $l$ . When it slips down the plane, if takes $t$ s. When it rolls down the plane then it takes ${(\frac{α}{2})}^{1 / 2}$ $t$ $s$ , where $α$ is _______ . [2024]

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(3) If disc slips on inclined plane, then it's acceleration, $a_{1} = g \sin θ$

$l = \frac{1}{2} a_{1} t_{1}^{2} \Rightarrow t = \sqrt{\frac{2 l}{g \sin θ}}$ $\dots (i)$

If disc rolls on inclined plane its acceleration,

$a_{2} = \frac{g \sin θ}{1 + \frac{I}{m R^{2}}} = \frac{g \sin θ}{1 + \frac{m R^{2}}{2 m R^{2}}} = \frac{2}{3} g \sin θ$

Now $l = \frac{1}{2} a_{2} t_{2}^{2} \Rightarrow t_{2} = \sqrt{\frac{2 l}{\frac{2}{3} g \sin θ}}$

$\Rightarrow t_{2} = \sqrt{\frac{3}{2}} \sqrt{\frac{2 l}{g \sin θ}} = \sqrt{\frac{3}{2}} t$ $\dots (i i)$

On comparing, $\sqrt{\frac{3}{2}} t = \sqrt{\frac{α}{2}} t \Rightarrow α = 3$

Q 3 :
A solid sphere and a hollow cylinder roll up without slipping on the same inclined plane with the same initial speed v. The sphere and the cylinder reach up to maximum heights $h_{1}$ and $h_{2}$ , respectively, above the initial level. The ratio $h_{1} : h_{2}$ is $\frac{n}{10}$ . The value of n is _____. [2024]

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(7)

Energy conservation, Gain in P.E. = Loss in K.E.

$\frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2} = m g h$

For pure rolling; $v = ω R$

$\frac{1}{2} m v^{2} + \frac{1}{2} I \frac{v^{2}}{R^{2}} = m g h$

$\frac{1}{2} v^{2} (m + \frac{m K^{2}}{R^{2}}) = m g h$

$\Rightarrow m g h = \frac{1}{2} m v^{2} (1 + \frac{K^{2}}{R^{2}}) \Rightarrow h \propto 1 + \frac{K^{2}}{R^{2}}$

$\frac{h_{1}}{h_{2}} = \frac{1 + \frac{2}{5}}{1 + 1} = \frac{7}{5 \times 2} = \frac{7}{10}$

Hence, $n = 7$

Q 4 :
A solid sphere of mass 'm' and radius 'r' is allowed to roll without slipping from the highest point of an inclined plane of length 'L' and makes an angle 30° with the horizontal. The speed of the particle at the bottom of the plane is $v_{1}$ . If the angle of inclination is increased to 45° while keeping L constant. Then the new speed of the sphere at the bottom of the plane is $v_{2}$ . The ratio of $v_{1}^{2} : v_{2}^{2}$ is [2025]

$1 : \sqrt{2}$
1 : 3
1 : 2
$1 : \sqrt{3}$

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(1)

Loss in P.E. = gain in K.E.

$m g L \sin θ = \frac{1}{2} (\frac{7}{5}) m v^{2} \Rightarrow v^{2} \propto \sin θ$

$\Rightarrow \frac{v_{1}^{2}}{v_{2}^{2}} = \frac{\sin 30 °}{\sin 45 °} = \frac{1}{\sqrt{2}}$

Q 5 :
A uniform solid cylinder of mass 'm' and radius 'r' rolls along an inclined rough plane of inclination 45°. If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder axis will be: [2025]

$\frac{1}{\sqrt{2}} g$
$\frac{1}{3 \sqrt{2}} g$
$\frac{\sqrt{2} g}{3}$
$\sqrt{2} g$

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(3)

$a = \frac{g \sin θ}{1 + \frac{I}{m r^{2}}}$

$a = \frac{\frac{g}{\sqrt{2}}}{1 + \frac{1}{2}} = \frac{2 \frac{g}{\sqrt{2}}}{3} = \frac{\sqrt{2} g}{3}$

Q 6 :
A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be $t_{1}$ and $t_{2}$ , respectively, then [2025]

$t_{1} < t_{2}$
$t_{1} = t_{2}$
$t_{1} = 2 t_{2}$
$t_{1} > t_{2}$

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(1)

The time taken to reach the bottom, $t = \sqrt{\frac{2 l}{a_{cm}}}$

Acceleration, $a_{cm} = \frac{g \sin θ}{1 + \frac{I_{cm}}{M R^{2}}}$

$a_{1} = a_{{cm}_{1}} = \frac{5 g \sin θ}{7}$ ... Solid

$a_{2} = a_{{cm}_{2}} = \frac{3 g \sin θ}{5}$ ... Hollow

As $a_{1} > a_{2}$ hence, $t_{1} < t_{2}$

Q 7 :
A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg, kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is [2025]

3.5 $m / s^{2}$
0.35 $m / s^{2}$
2.5 $m / s^{2}$
0.25 $m / s^{2}$

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(1)

Torque about bottom point, $F \times 2 r = I α$

$49 \times 2 r = \frac{7}{5} m r^{2} α$

$14 = 4 r α$ ... (i)

As sphere rolls without slipping

$a = r α$ ... (ii)

$\Rightarrow a = \frac{14}{4} = \frac{7}{2} = 3.5 m / s^{2}$

Q 8 :
A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is $\sqrt{\frac{x}{5}}$ where x = __________. [2025]

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(4)

Applying Mechanical Energy conservation:

$k_{i} + U_{i} = k_{f} + U_{f}$

$\Rightarrow 0 + m g h = \frac{1}{2} m v^{2} (1 + \frac{k^{2}}{R^{2}}) + 0$

$\Rightarrow V = \sqrt{\frac{2 g h}{1 + \frac{k^{2}}{R^{2}}}}$

So, Ratio of velocities

$\frac{V_{Ring}}{V_{Solid sphere}} = \sqrt{\frac{1 + \frac{2}{5}}{1 + 1}} = \sqrt{\frac{7}{10}}$

x = 3.5 $\Rightarrow$ Rounding off x = 4.

Topic Question Set

Q 1 :
A hollow sphere is rolling on a plane surface about its axis of symmetry. The ratio of rotational kinetic energy to its total kinetic energy is $\frac{x}{5}$ . The value of $x$ is _______ . [2024]

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Q 2 :
A circular disc reaches from top to bottom of an inclined plane of length $l$ . When it slips down the plane, if takes $t$ s. When it rolls down the plane then it takes ${(\frac{α}{2})}^{1 / 2}$ $t$ $s$ , where $α$ is _______ . [2024]

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Q 5 :
A uniform solid cylinder of mass 'm' and radius 'r' rolls along an inclined rough plane of inclination 45°. If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder axis will be: [2025]

Q 6 :
A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be $t_{1}$ and $t_{2}$ , respectively, then [2025]

Q 7 :
A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg, kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is [2025]

Q 8 :
A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is $\sqrt{\frac{x}{5}}$ where x = __________. [2025]

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Topic Question Set

Q 1 : A hollow sphere is rolling on a plane surface about its axis of symmetry. The ratio of rotational kinetic energy to its total kinetic energy is x5. The value of x is _______ . [2024]

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Q 2 : A circular disc reaches from top to bottom of an inclined plane of length l. When it slips down the plane, if takes t s. When it rolls down the plane then it takes (α2)1/2 t s, where α is _______ . [2024]

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Q 3 : A solid sphere and a hollow cylinder roll up without slipping on the same inclined plane with the same initial speed v. The sphere and the cylinder reach up to maximum heights h1 and h2, respectively, above the initial level. The ratio h1:h2 is n10. The value of n is _____. [2024]

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Q 5 : A uniform solid cylinder of mass 'm' and radius 'r' rolls along an inclined rough plane of inclination 45°. If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder axis will be: [2025]

Q 6 : A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be t1 and t2, respectively, then [2025]

Q 7 : A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg, kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is [2025]

Q 8 : A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is x5 where x = __________. [2025]

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Q 1 :
A hollow sphere is rolling on a plane surface about its axis of symmetry. The ratio of rotational kinetic energy to its total kinetic energy is $\frac{x}{5}$ . The value of $x$ is _______ . [2024]

Q 2 :
A circular disc reaches from top to bottom of an inclined plane of length $l$ . When it slips down the plane, if takes $t$ s. When it rolls down the plane then it takes ${(\frac{α}{2})}^{1 / 2}$ $t$ $s$ , where $α$ is _______ . [2024]

Q 5 :
A uniform solid cylinder of mass 'm' and radius 'r' rolls along an inclined rough plane of inclination 45°. If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder axis will be: [2025]

Q 6 :
A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be $t_{1}$ and $t_{2}$ , respectively, then [2025]

Q 7 :
A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg, kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is [2025]

Q 8 :
A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is $\sqrt{\frac{x}{5}}$ where x = __________. [2025]