Rotational Dynamics | JEE Main previous year questions with Solutions

Q 11 :

A hollow spherical ball of uniform density rolls up a curved surface with an initial velocity 3 m/s (as shown in figure). Maximum height with respect to the initial position covered by it will be ________ cm (take g = 10 $m / s^{2}$ ) [2023]

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(75)

$At highest point K E_{f} = 0$

$Initial K E = Translational KE + Rotational KE$

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

$In case of rolling v = R ω$

$= \frac{1}{2} m v^{2} + \frac{1}{2} \times \frac{2}{3} m R^{2} \times \frac{v^{2}}{R^{2}} = \frac{5}{6} m v^{2}$

$Apply energy conservation, K E_{i} + P E_{i} = K E_{f} + P E_{f}$

$\frac{5}{6} m v^{2} = m g h$

$h = \frac{5}{6 \times 10} \times 9 m = \frac{15}{20} m = 75 cm$

Q 12 :

For a rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is $\frac{x}{5}$ . The value of $x$ is ________. [2023]

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(2)

$\frac{K_{rot}}{K_{total}} = \frac{\frac{1}{2} (\frac{2}{3} m R^{2}) {(\frac{v}{R})}^{2}}{\frac{1}{2} m v^{2} + \frac{1}{2} (\frac{2}{3} m R^{2}) {(\frac{v}{R})}^{2}}$

$\Rightarrow \frac{x}{5} = \frac{2}{5} \Rightarrow x = 2$

Q 13 :

A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about the axis of rotation of the sphere to the total energy of the moving sphere is $π : 22$ , then the value of its angular speed will be ________ rad/s. [2023]

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(4)

$L = (I_{com}) (ω) and K = \frac{1}{2} (I_{com}) (ω^{2}) + \frac{1}{2} M V_{com}^{2}$

$L = \frac{2}{5} M R^{2} \frac{V_{com}}{R}$

$K = \frac{1}{2} (\frac{2}{5} M R^{2}) \frac{v_{com}^{2}}{R^{2}} + \frac{1}{2} M V_{com}^{2}$

$L = \frac{2 M R V_{com}}{5} K = \frac{7}{10} M V_{com}^{2}$

$Ratio \frac{L}{K} = \frac{4}{7} \frac{R}{v_{com}} = \frac{π}{22} \Rightarrow ω = \frac{4}{7} \times \frac{22}{22} \times 7 = 4$

Topic Question Set

Q 11 :

A hollow spherical ball of uniform density rolls up a curved surface with an initial velocity 3 m/s (as shown in figure). Maximum height with respect to the initial position covered by it will be ________ cm (take g = 10 $m / s^{2}$ ) [2023]

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Q 12 :

For a rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is $\frac{x}{5}$ . The value of $x$ is ________. [2023]

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Q 13 :

A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about the axis of rotation of the sphere to the total energy of the moving sphere is $π : 22$ , then the value of its angular speed will be ________ rad/s. [2023]

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