Question

video tutorial: https://www.aplusphysics.com/courses/honors/videos/circularspeed/circularspeed.html

c = 2πr

circular speed

so how do you find the speed of an object as it travels in a circular path? the formula for speed that you learned in kinematics still applies.

\\(overline{v}=\frac{d}{t}\\)

you have to be careful in using this equation, however, to understand that an object traveling in a circular path is traveling along the circumference of a circle. therefore, if an object were to make one complete revolution around the circle, the distance it travels is equal to the circles circumference.

7.07 q: miranda drives her car clockwise around a circular track of radius 30m. she completes 10 laps around the track in 2 minutes. find mirandas total distance traveled, average speed, and centripetal acceleration.

Explanation:

Step1: Calculate total distance

The circumference of a circle is \(C = 2\pi r\). Given \(r = 30m\), \(C=2\pi\times30 = 60\pi m\).
She does \(n = 10\) laps. So total distance \(d=n\times C=10\times60\pi=600\pi m\approx 1884m\).

Step2: Calculate average speed

Time \(t = 2\) minutes \(=2\times60 = 120s\).
Average speed \(\bar{v}=\frac{d}{t}\). Substituting \(d = 600\pi m\) and \(t = 120s\), \(\bar{v}=\frac{600\pi}{120}=5\pi m/s\approx 15.7m/s\).

Step3: Calculate centripetal acceleration

Centripetal acceleration \(a_{c}=\frac{\bar{v}^{2}}{r}\). Substituting \(\bar{v}=5\pi m/s\) and \(r = 30m\), \(a_{c}=\frac{(5\pi)^{2}}{30}=\frac{25\pi^{2}}{30}=\frac{5\pi^{2}}{6}m/s^{2}\approx 8.2m/s^{2}\).

Answer:

• Total distance: \(1884m\)
• Average speed: \(15.7m/s\)
• Centripetal acceleration: \(8.2m/s^{2}\)