Angle between the centripetal acceleration and radius vector is

Options

A. `90^(@)`
B. `180^(@)`
C. `0^(@)`
D. `45^(@)`

Answer

To determine the angle between the centripetal acceleration and the radius vector in circular motion, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- **Centripetal Acceleration (a_c)**: This is the acceleration that acts on an object moving in a circular path, directed towards the center of the circle. It is given by the formula:
\[ a_c = \frac{v^2}{r} \]
where \( v \) is the tangential speed and \( r \) is the radius of the circular path.

- **Radius Vector (r)**: This is the vector that points from the center of the circle to the position of the particle moving in the circular path. It is directed away from the center.

2. **Direction of the Vectors**:
- The centripetal acceleration always points towards the center of the circular path.
- The radius vector points away from the center of the circular path.

3. **Visualize the Vectors**:
- If we draw a diagram, we can represent the radius vector \( \vec{r} \) pointing outward from the center of the circle and the centripetal acceleration \( \vec{a_c} \) pointing inward towards the center.

4. **Determine the Relationship**:
- Since the centripetal acceleration points towards the center and the radius vector points away from the center, these two vectors are in opposite directions.

5. **Calculate the Angle**:
- The angle between two vectors that are in opposite directions is \( 180^\circ \).

### Conclusion:
The angle between the centripetal acceleration and the radius vector is \( 180^\circ \).

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