particle moves in a circular path with a continuously increasing speed. Its motion is,

The motion of a particle moving in a circular path with a continuously increasing speed is characterized as non-uniform circular motion. In this type of motion, the particle travels along a circular circumference while its speed varies.

Key Characteristics

1. Centripetal Acceleration: As the particle moves in a circular path, it experiences centripetal acceleration directed toward the center of the circle, which is essential for maintaining its circular motion. This can be expressed as:

   $a_c = \frac{v^2}{r}$

   where $a_c$ is the centripetal acceleration, $v$ is the instantaneous speed, and $r$ is the radius of the circular path.

2. Tangential Acceleration: Since the speed of the particle is increasing, it also experiences tangential acceleration, which is responsible for the change in the magnitude of the velocity. This can be expressed in terms of the change in speed over time:

   $a_t = \frac{dv}{dt}$

   where $a_t$ is the tangential acceleration and $\frac{dv}{dt}$ is the rate of change of speed.

3. Resultant Acceleration: The total acceleration of the particle can therefore be found using vector addition of centripetal and tangential accelerations:

   $a_{total} = \sqrt{a_c^2 + a_t^2}$

Thus, the motion can be described effectively through these components and equations, making it evident that the particle experiences both angular and linear dynamics due to its varying speed in a circular path.