In rotational motion, the linear velocity ( $v$ ) of a point on a rotating object is related to the period ( $T$ ) of rotation through the relationship between linear velocity, angular velocity, and the radius of the circular path.

Key Concepts:

1. Angular Velocity ( $\omega$ ): This is the rate at which the object rotates, typically measured in radians per second (rad/s). It is related to the period by:

$$\omega = \frac{2\pi}{T}$$
where $T$ is the period, or the time it takes for one complete rotation.

2. Linear Velocity ( $v$ ): This is the velocity of a point on the object as it moves along its circular path. It is given by:

$$v = r \omega$$

where $r$ is the radius of the circular path, and $\omega$ is the angular velocity.

Relationship between Linear Velocity and Period:

To express linear velocity in terms of the period $T$, substitute the expression for $\omega$ from the first equation into the second equation:

$$v = r \left(\frac{2\pi}{T}\right)$$
Simplifying gives:

$$v = \frac{2\pi r}{T}$$

This equation shows that the linear velocity is directly proportional to the radius $r$ and inversely proportional to the period $T$.

Summary:

• Direct Relationship: The linear velocity is directly proportional to the radius of the circular path. Larger radii result in higher linear velocities for a given period.
• Inverse Relationship: The linear velocity is inversely proportional to the period. A shorter period (faster rotation) leads to a higher linear velocity.

In essence, the faster the object rotates (shorter period), or the larger the radius, the greater the linear velocity of a point on the object.