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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 atarta rolling without shpping the speed of its centre is
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The correct answer is:
$\frac{5}{7} v_{0}$
Let, the final velocity be V
So, angular momentum will remain conserved along point of contact By conservation of angular momentum Angular momentum will remain conserved along point of contact
$$
\begin{aligned}
I_{\omega} &=\text { constant } \\
m v_{0} r &=m v r+\frac{2}{5} m r^{2} \times \omega \quad\left(\because \omega=\frac{v}{r}\right) \\
m v_{0} r &=m v r+\frac{2}{5} m r^{2}\left(\frac{v}{r}\right) \\
v_{0} &=v+\frac{2}{5} v \\
v_{0} &=\frac{7}{5} v \Rightarrow v=\frac{5}{7} v_{0}
\end{aligned}
$$
So, angular momentum will remain conserved along point of contact By conservation of angular momentum Angular momentum will remain conserved along point of contact
$$
\begin{aligned}
I_{\omega} &=\text { constant } \\
m v_{0} r &=m v r+\frac{2}{5} m r^{2} \times \omega \quad\left(\because \omega=\frac{v}{r}\right) \\
m v_{0} r &=m v r+\frac{2}{5} m r^{2}\left(\frac{v}{r}\right) \\
v_{0} &=v+\frac{2}{5} v \\
v_{0} &=\frac{7}{5} v \Rightarrow v=\frac{5}{7} v_{0}
\end{aligned}
$$
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