P01-270 – Relating Angular and Linear Quantities

Relating Angular and Linear Quantities

Arc length and angular displacement:

Consider a circle of radius $R$ and an arc of length $s$ as shown below

The arc of length $s$ subtends an angle $\theta $ and is related to this angle through the radius $R$ of the circle by the formula

\boxed{s=R\theta}

Tangential velocity and angular velocity:

A particle moving along a circular path of radius $R$ with angular velocity $\omega $ has a tangential speed $v$ that is proportional to $\omega $ and is given by

\boxed{v=R\omega}

In the figure below, point A and point B both rotate at angular speed $\omega $ and complete a full revolution in the same amount of time. However, since point B is further out from the center than point A (twice as far), it must cover a greater distance than point A in the same amount of time to cover a full revolution. Consequently, its linear speed $v_B$ must therefore be greater than the linear speed $v_A$ (twice as large).

Indeed, we have

v_A=\frac{R}{2}\ \omega \ \ \ \ \ \ \ \ \ \ and\ \ \ \ \ \ \ \ \ \ v_B=R\omega =2v_A

Tangential acceleration and angular acceleration:

A particle moving along a circular path and experiencing an angular acceleration $\alpha $ has a tangential acceleration $a_t$ that is proportional to $\alpha $ and is given by

\boxed{a_t=R\alpha}

In the figure below, point A and point B both rotate at the same angular speed $\omega $ and experience the same angular acceleration $\alpha $. However, since point B is further out from the center than point A (twice as far), it must experience a greater linear acceleration than points B. Its linear acceleration $a_B$ must therefore be greater than the linear acceleration $a_A$ (twice as large).

Indeed, we have

a_A=\frac{R}{2}\ \alpha \ \ \ \ \ and \ \ \ \ \ a_B=R\alpha =2a_A