The position vector of a particle in a circular motion about the origin sweeps out equal areal in equal time. Its

Options

A. velocity remains constant
B. speed remains constant
C. accelertion remains constant
D. tangential accelertion remains constant

Answer

To solve the problem, we need to analyze the implications of the statement that a particle in circular motion sweeps out equal areas in equal times. This is a characteristic of uniform circular motion.

### Step-by-Step Solution:

1. **Understanding the Concept of Areal Velocity**:
- The statement "sweeps out equal area in equal time" refers to the concept of areal velocity, which is defined as the area swept out by the position vector of the particle per unit time. Mathematically, it is given by:
\[ \frac{dA}{dt} = \text{constant} \]
- This implies that the particle is moving in such a way that the area swept out in a given time interval is constant.

2. **Relating Areal Velocity to Angular Velocity**:
- The area \( dA \) swept out by the position vector in a small time \( dt \) can be expressed in terms of the angle \( d\theta \) that the particle sweeps out:
\[ dA = \frac{1}{2} r^2 d\theta \]
- Therefore, the areal velocity can be written as:
\[ \frac{dA}{dt} = \frac{1}{2} r^2 \frac{d\theta}{dt} \]
- Since \( \frac{dA}{dt} \) is constant, it follows that \( \frac{d\theta}{dt} \) (angular velocity \( \omega \)) must also be constant.

3. **Conclusion about the Motion**:
- If \( \omega \) is constant, the particle is undergoing uniform circular motion. In uniform circular motion:
- The speed \( v \) of the particle is constant and is given by:
\[ v = r \omega \]
- The direction of the velocity vector changes, but its magnitude (speed) remains constant.

4. **Acceleration in Circular Motion**:
- In uniform circular motion, there is centripetal acceleration directed towards the center of the circle, given by:
\[ a_c = \frac{v^2}{r} = r \omega^2 \]
- There is no tangential acceleration since the speed is constant.

### Final Answer:
Thus, the particle is in **uniform circular motion**.