Question

The direction of the linear velocity of any point on the kinematic link relative to any other point on the same kinematic link is

• A. Parallel to the line joining the points
• B. Perpendicular to the line joining the points
• C. At 45° to the line joining the points
• D. Dependent on the angular speed of rotation of the link

Hint:

For a rotating rigid body, the relative velocity between two points is perpendicular to the line joining them, given by \( \mathbf{v}_{B/A} = \mathbf{\omega} \times \mathbf{r}_{B/A} \).

Answer 1

The Correct Option is B

Step 1: Understand the motion of a kinematic link.
A kinematic link in a mechanism is typically a rigid body undergoing planar motion, which can be a combination of translation and rotation. The question asks for the relative linear velocity of one point on the link with respect to another point on the same link. Step 2: Analyze relative velocity in a rigid body.
For two points \( A \) and \( B \) on the same rigid link:
If the link is purely translating, all points move with the same velocity, so the relative velocity \( \mathbf{v}_{B/A} = \mathbf{v}_B - \mathbf{v}_A = 0 \), which has no direction.
If the link is rotating about a point (or has a rotational component), the relative velocity is due to rotation.
The velocity of point \( B \) relative to point \( A \) due to rotation is given by: \[ \mathbf{v}_{B/A} = \mathbf{\omega} \times \mathbf{r}_{B/A}, \] where:
\( \mathbf{\omega} \) is the angular velocity of the link (perpendicular to the plane of motion, i.e., along the \( z \)-axis in 2D),
\( \mathbf{r}_{B/A} \) is the position vector from \( A \) to \( B \). Step 3: Determine the direction of the relative velocity.
\( \mathbf{r}_{B/A} \) lies along the line joining \( A \) to \( B \).
In 2D planar motion, \( \mathbf{\omega} = \omega \hat{k} \) (out of the plane).
The cross product \( \mathbf{\omega} \times \mathbf{r}_{B/A} \) results in a vector perpendicular to both \( \mathbf{\omega} \) and \( \mathbf{r}_{B/A} \).
Since \( \mathbf{\omega} \) is perpendicular to the plane, \( \mathbf{v}_{B/A} \) is perpendicular to \( \mathbf{r}_{B/A} \), meaning the relative velocity is perpendicular to the line joining the points \( A \) and \( B \).
Step 4: Evaluate the options.
(1) Parallel to the line joining the points: Incorrect, as the relative velocity due to rotation is perpendicular, not parallel.
(2) Perpendicular to the line joining the points: Correct, as derived.
(3) At 45° to the line joining the points: Incorrect, the angle is 90°, not 45°.
(4) Dependent on the angular speed of rotation of the link: Incorrect, the direction is always perpendicular regardless of the magnitude of \( \omega \); angular speed affects the magnitude, not the direction.
Step 5: Select the correct answer.
The direction of the relative linear velocity is perpendicular to the line joining the points, matching option (2).