Question

Angular velocity depends on:
A. Radius of rotation directly,
B. Radius of rotation inversely,
C. Linear velocity directly,
D. Linear velocity indirectly

• A. A and C only
• B. C and B only
• C. B and D only
• D. D and A only

Hint:

Remember that angular velocity is directly proportional to linear velocity and inversely proportional to the radius of rotation. This relationship helps in understanding how changes in one factor affect the others in circular motion scenarios.

Answer 1

The Correct Option is B

Step 1: Concept
Angular velocity is a measure of how quickly an object rotates or revolves relative to another point. It is typically denoted by the symbol \(\omega\) (omega). The relationship between angular velocity and linear velocity \(v\) for an object moving in a circular path with radius \(r\) is given by the formula: \[ v = r \cdot \omega \] This equation shows that linear velocity is directly proportional to both the radius of rotation and the angular velocity.

Step 2: Meaning
Radius of Rotation: The distance from the center of rotation to the point on the object. Linear Velocity: The speed at which a point on the object moves along its circular path.

Step 3: Analysis
To determine how angular velocity depends on the given factors, we analyze the relationship between linear velocity and radius: \[ v = r \cdot \omega \] From this equation, it is clear that if \(r\) (radius of rotation) increases while \(\omega\) remains constant, then \(v\) will increase. Similarly, if \(\omega\) increases while \(r\) remains constant, then \(v\) will also increase. Therefore, linear velocity \(v\) depends directly on both the radius of rotation and angular velocity. Now, let's consider how angular velocity itself is affected by these factors: Radius of Rotation: From the equation \(v = r \cdot \omega\), if we rearrange for \(\omega\): \[ \omega = \frac\{v\}\{r\} \] This shows that angular velocity \(\omega\) depends inversely on the radius of rotation. If the radius increases, the angular velocity decreases to maintain a constant linear velocity (assuming \(v\) is kept constant). Linear Velocity: From the equation \(v = r \cdot \omega\), if we rearrange for \(\omega\): \[ \omega = \frac\{v\}\{r\} \] This shows that angular velocity depends directly on linear velocity. If the linear velocity increases, the angular velocity will also increase to maintain a constant radius (assuming \(r\) is kept constant).

Step 4: Conclusion
Based on the analysis, we can conclude: Angular velocity \(\omega\) depends inversely on the radius of rotation. Angular velocity \(\omega\) depends directly on linear velocity. Final Answer: (B)