Which of the following statements is FALSE for a paricle moving in a circle with a constant angular sppeed?

Options

A. The velocity vector is tangent to the circle
B. The acceleration vector is tangent to the circle
C. The acceleration vector points to the centre of the circle
D. The velocity and acceleration vectors are perpendicular to each other

Answer

To solve the question regarding which statement is false for a particle moving in a circle with a constant angular speed, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Circular Motion**:
- A particle moving in a circle with constant angular speed implies that the angular velocity (ω) is constant. The particle moves along a circular path with a fixed radius (r).

2. **Linear Velocity**:
- The linear velocity (v) of the particle can be expressed as:
\[ v = \omega \cdot r \]
- Since both ω and r are constant, the linear velocity (v) is also constant.

3. **Acceleration in Circular Motion**:
- In circular motion, there are two types of acceleration to consider:
- **Centripetal Acceleration (a_c)**: This is directed towards the center of the circle and is given by:
\[ a_c = \frac{v^2}{r} = \omega^2 \cdot r \]
- **Tangential Acceleration (a_t)**: This is associated with changes in the speed of the particle along the circular path. Since the speed is constant, the tangential acceleration is zero:
\[ a_t = 0 \]

4. **Direction of Vectors**:
- The velocity vector (v) is always tangent to the circular path.
- The centripetal acceleration vector (a_c) points towards the center of the circle.

5. **Analyzing the Statements**:
- Now, we need to evaluate the provided statements:
- **Statement A**: Velocity vector is tangent to the circle. (True)
- **Statement B**: Acceleration vector is tangent to the circle. (False, because the acceleration vector points towards the center)
- **Statement C**: Acceleration vector points to the center of the circle. (True)
- **Statement D**: Velocity and acceleration vectors are perpendicular to each other. (True, as the velocity is tangent and acceleration is radial)

6. **Conclusion**:
- The false statement among the options is **Statement B**, which claims that the acceleration vector is tangent to the circle.

### Final Answer:
The false statement for a particle moving in a circle with a constant angular speed is **Statement B**: "Acceleration vector is tangent to the circle."

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