Question

Establish the relation between linear speed and angular speed of a particle executing

circular motion.​

Answer 1

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Answer 2

The relation between linear speed and angular speed of a particle executing circular motion is established through the formula, $v=r\omega$, that is the linear speed ($v$) of a particle is equal to the product of the radius of the path traversed by the particle ($r$) and the angular speed ($\omega$) of the particle.

What are linear speed and angular speed?

• Linear speed refers to the rate of change of linear displacement of a particle while angular speed refers to the rate of change of angular displacement of the particle.
• The linear speed of a particle executing circular motion occurs along the circumference of the circular path and is constant throughout.
• The angular speed of a particle executing circular motion occurs along the axis of the circular path and it varies at different points of the path.

Relation between linear speed and angular speed

• The formula for linear speed is

 $v=\frac{\triangle s}{\triangle t}$, where $\triangle s$ is the change in linear displacement and $\triangle t$ is the change in time.

• The formula for angular speed is

$\omega=\frac{\triangle \theta}{\triangle t}$, where $\triangle \theta$ is the change in angular displacement and $\triangle t$ is the change in time.

• The relation between the linear displacement and angular displacement is given by, $s=r\theta$, where $r$ is the radius of the path traversed.

Hence, using the above three relations, we get the formula for linear speed as,

$v=\frac{\triangle r\theta}{\triangle t}$

$\implies v=r\frac{\triangle \theta }{\triangle t}$    ( $r$ can be taken out since it has a constant value)

$\implies v=r \omega$

Therefore, the linear speed of a particle in circular motion is the product of the radius of the path covered and the angular speed of the particle. It is mathematically defined using the formula, $v=r\omega$