If `a_(r)` and `a_(t)` represent radial and tangential accelerations, the motion of a particle will be uniformly circular if

Options

A. `a_(f)=0,a_(t)=0`
B. `a_(f)=0,a_(t)ne0`
C. `a_(f)ne0,a_(t)ne0`
D. None of these

Explanation

To determine the conditions under which the motion of a particle is uniformly circular, we need to analyze the definitions of radial and tangential accelerations.

### Step-by-Step Solution:

1. **Understanding Uniform Circular Motion**:
- In uniform circular motion, a particle moves along a circular path with a constant speed. This means that while the speed (magnitude of velocity) remains constant, the direction of the velocity vector continuously changes.

2. **Identifying Accelerations**:
- **Radial Acceleration (a_r)**: This is the acceleration directed towards the center of the circular path. It is responsible for changing the direction of the velocity vector and is given by the formula:
\[ a_r = \frac{v^2}{r} \]
where \( v \) is the speed of the particle and \( r \) is the radius of the circular path.
- **Tangential Acceleration (a_t)**: This is the acceleration that is tangent to the circular path and is responsible for changing the speed of the particle along the circular path.

3. **Condition for Uniform Circular Motion**:
- For the motion to be uniformly circular, the speed of the particle must remain constant. This implies that there is no change in the magnitude of the velocity.
- Since tangential acceleration (a_t) is responsible for changing the speed, for uniform circular motion, the tangential acceleration must be zero:
\[ a_t = 0 \]
- However, the radial acceleration (a_r) is not zero because it is necessary to keep the particle moving in a circular path.

4. **Conclusion**:
- Therefore, the condition for a particle to be in uniform circular motion is:
\[ a_t = 0 \quad \text{and} \quad a_r \neq 0 \]

### Final Answer:
The motion of a particle will be uniformly circular if \( a_t = 0 \) and \( a_r \neq 0 \).

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