A spherical ball of radius r initially at rest on a rough horizontal surface is hit horizontally at a point at a distance x above the central line. Due to this sharp impulse the centre of the ball acquires a velocity vc. After some time, the ball will start pure rolling with a velocity equal to

Question

Infinity Learn · Accepted Answer

Initially ball will gain both linear and angular velocity. Linear impulse: $$I0=mvC$$ Angular impulse: $$I0x=I$$ω$$0$$⇒$$mvCx=I$$ω$$0I$$ is moment of Inertia about C.Apply conservation of angular momentum about lowest point.Iω$$0+mvCr=I$$ω$$+mvr$$ where ω$$=vr$$ at the time of pure rolling ⇒ $$mvCx+mvCr=25mr2vr+mvr$$⇒ $$vC(x+r)=75rv$$⇒ $$v=57vCx+rr$$

A spherical ball of radius r initially at rest on a rough horizontal surface is hit horizontally at a point at a distance x above the central line. Due to this sharp impulse the centre of the ball acquires a velocity v_c. After some time, the ball will start pure rolling with a velocity equal to

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