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 ) [2023]

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Solution

Q.
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]

Ans.
(75)

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

$Initial K E = Translational KE + Rotational KE$

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

$In case of rolling v = R ω$

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

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

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

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

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