Match the following

$\begin{array}{|l|l|}
\hline \text { a) } \mathrm{h}=\mathbf{R} / 2 & \text { i) sphere rolls without slipping with a constant velocity and no loss of energy } \\
\hline \text { b) } \mathbf{h}=\mathbf{R} & \text { ii) sphere spins clockwise, loses energy by friction } \\
\hline \text { c) } \mathrm{h}=3 \mathrm{R} / 2 & \text { iii) sphere spins anti-clockwise, loses energy by friction } \\
\hline \text { d) } \mathrm{h}=7 R / 5 & \text { iv) sphere has only a translational motion, loses energy by friction } \\
\hline
\end{array}$

Solution:

a) matches with iii)

b) matches with iv)

c) matches with ii)

d) matches with i)

Explanation

Let v be the sphere’s velocity after applying F.

The law of conservation of angular momentum is then applied.

$
\begin{array}{l}
m v(h-R)=I \omega \\
m v(h-R)=\frac{2}{5} m R^{2} \frac{v}{R} \\
h-R=\frac{2}{5} R \\
h=\frac{2}{5} R+R=\frac{7}{5} R
\end{array}
$

Hence, the sphere rolls without slipping with a constant velocity and no loss of energy. Thus (d) -(i)
Torque due to force $F=\tau=(h-R) \times F$
If $\tau=0, h-R=0$ and thus $h=R$
In this case, the sphere will only have a translation motion and slip against the force of friction. Thus (b)-(iv
For clockwise rotation of sphere $\tau>0$
$(h-R) \times F>0$
Or $h>R$, Thus (c) – (ii)
For anti-clockwise rotation $\tau<0$
$(h-R) \times F<0$
$h<R$, Thus (a) – (iii)