A uniform ball of radius \( R \) rolls without slipping between two rails such that the horizont...
1A uniform ball of radius \( R \) rolls without slipping between two rails such that the horizontal distance is \( d \) between the two contact point of the rails to the ball. (a) Show that at any instant, velocity of centre of mass is given as
\[
v_{\mathrm{cm}}=\omega \sqrt{R^{2}-\frac{d^{2}}{4}}
\]
Discuss the above expression in the limits \( d=0 \) and \( d=2 \mathrm{R} \). (b) For a uniform ball starting from rest and descending a vertical distance \( h \) while rolling without slipping down a ramp, \( v_{c m}=\sqrt{\frac{10 g h}{7}} \), If the ramp is replaced with two rails, show that
\[
v_{\mathrm{cm}}=\sqrt{\frac{10 g h}{5+\frac{2}{1-d^{2} / 4 R^{2}}}}
\]
Neglect friction in above cases.
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