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Explains the relationship between linear velocity and period in circular motion?
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relationship between linear velocity and period in circular motion

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

Key Concepts:

  1. Angular Velocity ( ω\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:

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

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

v=rωv = r \omega

where rr is the radius of the circular path, and ω\omega is the angular velocity.

Relationship between Linear Velocity and Period:

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

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

v=2πrTv = \frac{2\pi r}{T}

This equation shows that the linear velocity is directly proportional to the radius rr and inversely proportional to the period TT.

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.

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