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A solid sphere is placed on a horizontal surface with **initial linear velocity $v_0$ (forward) and zero angular velocity** (it spins-skids forward). Coefficient of friction $\mu$. The **final speed** when pure rolling sets in is:
A$v_0$
B$5v_0/7$
C$2v_0/7$
D$0$
Answer & Solution
Correct answer: B. $5v_0/7$
Friction decelerates linear motion ($a = -\mu g$) and angularly accelerates spin ($\alpha = \mu g R/I = 5\mu g/(2R)$). Pure rolling when $v = R\omega$: $v_0 - \mu g t = R \cdot (5\mu g/(2R))\cdot t = (5/2)\mu g t$ ⇒ $v_0 = (7/2)\mu g t$ ⇒ $t = 2v_0/(7\mu g)$. $v_f = v_0 - \mu g t = v_0 - 2v_0/7 = 5v_0/7$.
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