Rotational Dynamics
6.0 Uniform pure rolling
6.0 Uniform pure rolling
Pure rolling is a situation when there is no slipping between the two bodies at the point of contact.
So, for no slipping between the two bodies at the point of contact, there should be no relative motion.
Therefore, the relative velocity and relative acceleration should be zero at the point of contact.
6.0.1 Case I
Now consider a ball of radius $R$ is moving with velocity $v$ and angular velocity $\omega $ on a horizontal ground as shown in the figure.
Point $P$ is the point of contact.
$\overrightarrow \omega = \omega \left( { - \widehat k} \right)$: Clock wise direction is taken as negative by right hand thumb rule
${\overrightarrow v _{{P_{ground}}}} = 0$ (Velocity of point $P$ on ground)
${\overrightarrow v _{{P_{sphere}}}} $ (Velocity of point $P$ on sphere)
As we know, $$\begin{equation} \begin{aligned} {\overrightarrow v _{{P_{sphere}}}} = {{\vec v}_{{P_{translation}}}} + {{\vec v}_{{P_{rotational}}}} \\ {\overrightarrow v _{{P_{sphere}}}} = v\left( {\widehat i} \right) + \overrightarrow \omega \times {\overrightarrow r _{OP}} \\ {\overrightarrow v _{{P_{sphere}}}} = v\left( {\widehat i} \right) + \omega \left( { - \widehat k} \right) \times R\left( { - \widehat j} \right) \\ {\overrightarrow v _{{P_{sphere}}}} = \left( {v - R\omega } \right)\widehat i \\\end{aligned} \end{equation} $$To avoid relative motion at the point of contact. So, relative velocity should be zero at the point of contact. $$\begin{equation} \begin{aligned} {\overrightarrow v _{{P_{sphere,\;ground}}}} = {\overrightarrow v _{{P_{sphere}}}} - {\overrightarrow v _{{P_{ground}}}} \\ 0 = \left( {v - R\omega } \right)\widehat i - 0 \\ v = R\omega \\\end{aligned} \end{equation} $$
Note:
| Situation | Condition | Diagram |
| $v = R\omega $ | Condition for pure rolling when a sphere rolls on a horizontal ground. |
|
| $v > R\omega $ | Ball will slip forward. So, the motion is known as forward slipping. It is often called as backward english. |  |
| $v < R\omega $ | Ball will slip backward. So, the motion is called as backward slipping. It is often called as forward english. |
|
6.0.2 Case II
Consider a disc of radius $R$ moving with linear velocity $v$ and angular velocity $\omega $ on a plank which is moving with velocity $v_o$ as shown in the figure.
$\overrightarrow \omega = \omega \left( { - \widehat k} \right)$: Clock wise direction is taken as negative by right hand thumb rule
Point $P$ is the point of contact, $$\begin{equation} \begin{aligned} {\overrightarrow v _{{P_{sphere}}}} = {\overrightarrow v _{{P_{translation}}}} + {\overrightarrow v _{{P_{rotational}}}} \\ {\overrightarrow v _{{P_{sphere}}}} = v\widehat i + \overrightarrow \omega \times {\overrightarrow r _{OP}} \\ {\overrightarrow v _{{P_{sphere}}}} = v\widehat i + \omega \left( { - \widehat k} \right) \times R\left( { - \widehat j} \right) \\ {\overrightarrow v _{{P_{sphere}}}} = \left( {v - R\omega } \right)\widehat i \\\end{aligned} \end{equation} $$ $${\overrightarrow v _{{P_{plank}}}} = {v_O}\widehat i$$
To avoid relative motion, relative velocity should be zero at the point of contact. $$\begin{equation} \begin{aligned} {\overrightarrow v _{{P_{sphere,\;plank}}}} = {\overrightarrow v _{{P_{sphere}}}} - {\overrightarrow v _{{P_{plank}}}} \\ 0 = \left( {v - R\omega } \right)\widehat i - {v_O}\widehat i \\ v - R\omega = {v_O} \\ v = {v_O} + R\omega \\\end{aligned} \end{equation} $$
6.0.3 Case III
When the ball is rolling on a inclined wedge with linear velocity $v$ and angular velocity $\omega $ as shown in the figure. The 
wedge is also moving with horizontal velocity $v_O$.
wedge is also moving with horizontal velocity $v_O$.Consider the cartesian co-ordinate along the incline as shown in the figure.
Point $P$ is the point of contact.
$\overrightarrow \omega = \omega \left( { - \widehat k} \right)$: Clock wise direction is taken as negative by right hand thumb rule
Since the ball is rolling along the incline, so there should be no relative motion along the incline. 
${\overrightarrow v _{{P_{wedg{e_{incline}}}}}} = {v_O}\cos \theta \widehat i$ (Velocity of point $P$ on the wedge along the incline)
$$\begin{equation} \begin{aligned} {\overrightarrow v _{{P_{ball}}}} = {\left( {{{\overrightarrow v }_{{P_{ball}}}}} \right)_{translation}} + {\left( {{{\overrightarrow v }_{{P_{ball}}}}} \right)_{rotation}} \\ {\overrightarrow v _{{P_{ball}}}} = v\widehat i + \overrightarrow \omega \times {\overrightarrow r _{OP}} \\ {\overrightarrow v _{{P_{ball}}}} = v\widehat i + \omega \left( { - \widehat k} \right) \times R\left( { - \widehat j} \right) \\ {\overrightarrow v _{{P_{ball}}}} = \left( {v - R\omega } \right)\widehat i \\\end{aligned} \end{equation} $$
${\overrightarrow v _{{P_{ball}}}}$ is the velocity of point $P$ on the ball along the incline
So, $$\begin{equation} \begin{aligned} {\left( {{{\overrightarrow v }_{{P_{ball}}}}} \right)_{incline}} = \left( {v - R\omega } \right)\widehat i \\ {\left( {{{\overrightarrow v }_{{P_{plank}}}}} \right)_{incline}} = {v_O}\cos \theta \widehat i \\\end{aligned} \end{equation} $$
To avoid relative motion, relative velocity should be zero at the point of contact, $$\begin{equation} \begin{aligned} {\left( {{{\overrightarrow v }_{{P_{ball,\;sphere}}}}} \right)_{incline}} = {\left( {{{\overrightarrow v }_{{P_{ball}}}}} \right)_{incline}} - {\left( {{{\overrightarrow v }_{{P_{sphere}}}}} \right)_{incline}} \\ 0 = \left( {v - R\omega } \right)\widehat i - {v_O}\cos \theta \widehat i \\ v = {v_O}\cos \theta + R\omega \\\end{aligned} \end{equation} $$
