There are two points P and Q on a planar rigid body. The relative velocity between the two points

When considering a planar rigid body, the motion of any two points, say P and Q, is interconnected. A rigid body is defined as a body where the distance between any two of its particles remains constant regardless of the forces acting on it or its motion. This fundamental property is crucial for understanding the relative velocity between points on such a body.

Rigid Body Kinematics Explained

For any two points P and Q on a rigid body, the position vector from P to Q, denoted as ${\vec{r}}_{PQ}$, has a constant magnitude. This means the length of the line segment PQ does not change. The velocity of point Q (${\vec{v}}_Q$) relative to an inertial frame can be expressed in terms of the velocity of point P (${\vec{v}}_P$) and the angular velocity of the body ($\vec{\omega }$) as:

$${\vec{v}}_Q={\vec{v}}_P+\vec{\omega }\times {\vec{r}}_{PQ}$$

Here, $\vec{\omega }$ is the angular velocity vector of the rigid body, and ${\vec{r}}_{PQ}$ is the position vector from P to Q. This equation is fundamental in rigid body kinematics.

Understanding Relative Velocity Components

The relative velocity of Q with respect to P is given by ${\vec{v}}_{Q/P}={\vec{v}}_Q-{\vec{v}}_P$. Substituting the rigid body velocity equation into this definition, we get:

$${\vec{v}}_{Q/P}=({\vec{v}}_P+\vec{\omega }\times {\vec{r}}_{PQ})-{\vec{v}}_P$$

$${\vec{v}}_{Q/P}=\vec{\omega }\times {\vec{r}}_{PQ}$$

The cross product ($\vec{\omega }\times {\vec{r}}_{PQ}$) results in a vector that is always perpendicular to both $\vec{\omega }$ and ${\vec{r}}_{PQ}$. Since ${\vec{r}}_{PQ}$ is the vector connecting P to Q (along the line PQ), the relative velocity ${\vec{v}}_{Q/P}$ must always be perpendicular to the line PQ. This is a direct consequence of the rigid body assumption that the distance between P and Q remains constant.

Alternatively, consider the square of the distance between points P and Q, $L^2={\vec{r}}_{PQ}\cdot {\vec{r}}_{PQ}$. Since $L$ is constant for a rigid body, its derivative with respect to time must be zero:

$$\frac{d}{dt}({\vec{r}}_{PQ}\cdot {\vec{r}}_{PQ})=0$$

Using the product rule for differentiation:

$$\frac{d{\vec{r}}_{PQ}}{dt}\cdot {\vec{r}}_{PQ}+{\vec{r}}_{PQ}\cdot \frac{d{\vec{r}}_{PQ}}{dt}=0$$

Since $\frac{d{\vec{r}}_{PQ}}{dt}$ is the relative velocity ${\vec{v}}_{Q/P}$ (velocity of Q relative to P), this simplifies to:

$${\vec{v}}_{Q/P}\cdot {\vec{r}}_{PQ}+{\vec{r}}_{PQ}\cdot {\vec{v}}_{Q/P}=0$$

$$2({\vec{v}}_{Q/P}\cdot {\vec{r}}_{PQ})=0$$

$${\vec{v}}_{Q/P}\cdot {\vec{r}}_{PQ}=0$$

The dot product being zero implies that the relative velocity vector ${\vec{v}}_{Q/P}$ is always perpendicular to PQ (the position vector ${\vec{r}}_{PQ}$).

Analyzing Options for Relative Velocity

- Option 1: should always be along PQ This statement is incorrect. If the relative velocity were along PQ, it would imply that the distance between points P and Q is changing. This contradicts the definition of a rigid body.
- Option 2: Can be oriented along any direction This statement is incorrect. As derived, the relative velocity is constrained to be perpendicular to the line connecting the two points due to the rigid body constraint.
- Option 3: should always be perpendicular to PQ This statement is correct. As explained by the cross product and dot product derivations, the relative velocity ${\vec{v}}_{Q/P}$ is always perpendicular to PQ. This relative motion is purely rotational about P (if P is considered fixed) or about some instantaneous center of rotation.
- Option 4: should be along QP when the body undergoes pure translation This statement is incorrect. In pure translation, all points on the rigid body have the exact same velocity. Therefore, the relative velocity between any two points (P and Q) in pure translation is zero (${\vec{v}}_Q={\vec{v}}_P⟹{\vec{v}}_{Q/P}=0$). If there were a non-zero relative velocity, it would still have to be perpendicular to PQ, not along QP.

Conclusion on Relative Velocity Direction

In summary, for a planar rigid body, the relative velocity between any two points P and Q must always be directed perpendicular to PQ. This is a defining characteristic of rigid body motion, ensuring that the distance between any two points on the body remains constant.