A ball is thrown in such a way that it slides with a speed $v_{0}$ initially without rolling on rough horizontal plane. Prove that it will roll without sliding when its speed falls to $(5 / 7) v_{0}$ .

Open in App

Solution

We can see that the frictional force $f$ , weight $m g$ and normal contact force $N$ pass through the point of contact $P$ . Therefore, their toque about $P$ will be equal to zero.
$\Rightarrow$ Angular momentum of the sphere remains constant before and after start of pure rolling about $P$ .
Conserving the angular momentum of the sphere at position $1$ & $2$ , about the instantaneous point of contact, We obtain
$m v_{0} r = m v r + (\frac{2}{5} m r^{2}) ω$
Putting $r ω = v$ for rolling, we get $v = 5 v_{0} / 7$ .

Suggest Corrections

Similar questions

Q. A heavy ball is thrown on a rough horizontal surface in such a way that it slides with a speed

V_{0}

initially without rolling. It will roll without sliding when its speed fall to :

A bowling ball is thrown down the alley in such a way that it slides with a speed $v_{0}$ initially without rolling. What will it it's speed be when (and if) it starts pure rolling? The transition from pure sliding to pure rolling is gradual, so that both sliding and rolling takes place during this interval.

Q. A solid uniform sphere resting on a rough horizontal plane is given a horizontal impulse directed through its centre so that it starts sliding with an initial velocity

v_{0}

. When it finally starts rolling without slipping the speed of its centre is:

Q. A heavy solid sphere is thrown on a horizontal rough surface with initial velocity

u

without rolling. What will be its speed, when it starts pure rolling motion?

Join BYJU'S Learning Program

A ball is thrown in such a way that it slides with a speed v0 initially without rolling on rough horizontal plane. Prove that it will roll without sliding when its speed falls to (5/7)v0.

A ball is thrown in such a way that it slides with a speed $v_{0}$ initially without rolling on rough horizontal plane. Prove that it will roll without sliding when its speed falls to $(5 / 7) v_{0}$ .