Question

The linear velocity 'v' of rotating body is given by:

• A. $v = \frac{\omega}{r}$
• B. $v = \omega \times r$
• C. $v = \frac{r}{\omega}$
• D. $v = \sqrt{\omega \times r}$

Hint:

To convert angular variables to linear variables, simply multiply by the radius \(r\):
- Displacement: \(s = \theta \times r\)
- Velocity: \(v = \omega \times r\)
- Acceleration: \(a = \alpha \times r\)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Rotational motion can be described using both linear (translational) and angular variables.
- Linear velocity (\(v\)) is the rate of change of linear displacement over time, measured in meters per second (\(\text{m/s}\)).
- Angular velocity (\(\omega\)) is the rate of change of angular displacement over time, measured in radians per second (\(\text{rad/s}\)).

Step 2: Detailed Explanation:
Let us analyze a point on a body rotating at a constant angular velocity \(\omega\) about a fixed axis:
- The point travels in a circular path of radius \(r\) centered on the rotation axis.
- The linear distance \(s\) traveled by the point along the arc is related to the angular displacement \(\theta\) in radians by:
\[ s = \theta \times r \]
- To find the linear velocity (\(v\)), we take the derivative of this displacement with respect to time (\(t\)):
\[ v = \frac{ds}{dt} = \frac{d(\theta \times r)}{dt} \]
- Since the radius \(r\) is constant for a fixed point on the rotating body:
\[ v = r \times \frac{d\theta}{dt} \]
- By definition, the rate of change of angular displacement is the angular velocity (\(\omega = \frac{d\theta}{dt}\)):
\[ v = \omega \times r \]
In vector notation, this relationship is expressed as a cross product:
\[ \vec{v} = \vec{\omega} \times \vec{r} \]
This matches Option (B) perfectly.

Step 3: Final Answer:
The linear velocity of a rotating body is given by \(v = \omega \times r\).