Rotational Dynamics
4.0 Combined translational and rotational motion of a rigid body
4.0 Combined translational and rotational motion of a rigid body
In this section we will discuss the translation and rotational motion of a body.
Combined translational and rotational motion is also known as a plane motion of rigid body.
When we say pure rotational motion, it means that a rigid body rotates about a fixed axis of rotation.
When we say combined translational and rotational motion, it means that a rigid body rotates about an axis of rotation and this axis of rotation is undergoing a translational motion.
Consider a circular wheel which has linear velocity $v_o$ in the horizontal direction and angular velocity $\omega $ about its center $O$ as shown in the figure.
Each point on the wheel has translational as well as rotational motion.
Translational velocity of every point in a rigid body is same and is equal to the velocity of the center of mass.
Deriving a generalized velocity equation for any point $P$
Now consider a point $P$ on the wheel at an angle $\theta $ with the horizontal as shown in the figure.
So, motion of point $P$ is due to both translational and rotational motion of a rigid body. $$\overrightarrow \omega = \omega \left( { - \widehat k} \right)$$
Note: By right hand thumb rule clock wise direction is taken as negative
Position vector of point $P$ with respect to center $O$ is, ${\overrightarrow r _{OP}} = R\left( {\cos \theta \widehat i + \sin \theta \widehat j} \right)$
Translational velocity of point $P$ is equal to the velocity of center of mass. So, ${\overrightarrow v _{{P_{_T}}}} = {\overrightarrow v _o}$
Rotational velocity of point $P$, ${\overrightarrow v _{{P_R}}} = \left( {\overrightarrow \omega \times {{\overrightarrow r }_{OP}}} \right)$
Mathematically, $$\begin{equation} \begin{aligned} {\text{Motion of Point }}P{\text{ = Translation motion of point }}P{\text{ + Rotational motion of point }}P \\ {\text{Velocity of Point }}P{\text{ = translation velocity of point }}P{\text{ + rotational velocity of point }}P \\ \\ {\overrightarrow v _P} = {\overrightarrow v _o} + \left( {\overrightarrow \omega \times {{\overrightarrow r }_{OP}}} \right) \\ {\overrightarrow v _P} = {v_o}\widehat i + \omega \left( { - \widehat k} \right) \times R\left( {\cos \theta \widehat i + \sin \theta \widehat j} \right) \\ {\overrightarrow v _P} = {v_o}\widehat i + \omega \left( { - \widehat k} \right) \times R\left( {\cos \theta \widehat i + \sin \theta \widehat j} \right) \\ {\overrightarrow v _P} = \left( {{v_o} + \omega R\sin \theta } \right)\widehat i - \omega R\cos \theta \widehat j \\\end{aligned} \end{equation} $$
| S. No. | Point | Diagram | Velocity | Velocity diagram |
| 1. | $A$ |
| $$\begin{equation} \begin{aligned} {\overrightarrow v _A} = \left( {{v_o} + \omega R\sin \pi } \right)\widehat i - \omega R\cos \pi \widehat j \\ {\overrightarrow v _A} = {v_o}\widehat i + \omega R\widehat j \\\end{aligned} \end{equation} $$ |
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| 2. | $B$ |
| $$\begin{equation} \begin{aligned} {\overrightarrow v _A} = \left( {{v_o} + \omega R\sin \frac{\pi }{2}} \right)\widehat i - \omega R\cos \frac{\pi }{2}\widehat j \\ {\overrightarrow v _A} = \left( {{v_o} + \omega R} \right)\widehat i \\\end{aligned} \end{equation} $$ |
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| 3. | $C$ |
| $$\begin{equation} \begin{aligned} {\overrightarrow v _A} = \left( {{v_o} + \omega R\sin 0} \right)\widehat i - \omega R\cos 0\widehat j \\ {\overrightarrow v _A} = {v_o}\widehat i - \omega R\widehat j \\\end{aligned} \end{equation} $$ |
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| 4. | $D$ |
| $$\begin{equation} \begin{aligned} {\overrightarrow v _A} = \left[ {{v_o} + \omega R\sin \left( { - \frac{\pi }{2}} \right)} \right]\widehat i - \omega R\cos \left( { - \frac{\pi }{2}} \right)\widehat j \\ {\overrightarrow v _A} = \left( {{v_o} - \omega R} \right)\widehat i \\\end{aligned} \end{equation} $$ |
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