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, $n = 7$

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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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 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]