An auto race takes place on a circular track. A car completes one lap in a time of 26.4 s, with an average tangential speed of 42.9 m/s. Find (a) the average angular speed and (b) the radius of the track.
To find the average angular speed, we use the formula:
\[\text{Average Angular Speed} = \dfrac{\text{Angle}}{\text{Time}}\]
Since the car completes one lap, it travels a full circle, which is equivalent to an angle of \(2\pi\) radians. The time taken is 26.4 s. Therefore,
\[\text{Average Angular Speed} = \dfrac{2\pi \,\text{rad}}{26.4 \,\text{s}}\]
To find the radius of the track, we use the formula:
\[\text{Average Tangential Speed} = \text{Average Angular Speed} \times \text{Radius}\]
The average tangential speed is given as 42.9 m/s. The average angular speed is \(\dfrac{2\pi \,\text{rad}}{26.4 \,\text{s}}\). We let the radius be \(r\). Therefore,
\[42.9 \,\text{m/s} = \left( \dfrac{2\pi \,\text{rad}}{26.4 \,\text{s}} \right) \times r\]
Simplifying, we have:
\[r = \dfrac{42.9 \,\text{m/s} \times 26.4 \,\text{s}}{2\pi \,\text{rad}}\]