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A solid spherical ball of mass 1 kg and radius 1 m is struck horizontally by a horizontal force at a height h above the ground. As a result the ball immediately begins to roll without slipping on the smooth ground. What is the value of h?
- 7/5 m
- 5/3 m
- 7/3 m
- 4/3 m
Correct answer: 7/5 m
Solution
The horizontal force F applied at height h provides linear impulse J=F*dt giving velocity v=J/M, and angular impulse J*(h-R) about the bottom contact point giving omega = J*(h-R)/I_contact. For pure rolling v=omega*R, so h-R = (I_contact/M*R) - wait: from J*(h-R) = I_contact*omega and J=M*v=M*omega*R, we get h-R = I_contact/(M*R) = (7/5)*R²/R*... giving h = R + (2/5)*R = 7R/5 = 7/5 m.
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