Question

A car is moving at a rate of 65 miles per hour and the diameter of its wheels is 2.5 feet.

a) Find the number of revolutions per minute the wheels are rotating.
b) Find the angular speed of the wheels in radians per minute.

Answer 1

a) 728.293 rev/min

b) 4576 rad/min

Step-by-step explanation:

a) 65 mi/hr = 2.5π ft/rev • x rev/min •60 min/hr • 1 mi/5280 ft

Feet, revolutions, and minutes cancel out.

65 mi/hr = 150πx/5280 hr

Cross multiply:

150πx = 65 • 5280

150πx = 343200

x ≈ 728.293

b) angular speed is theta/time

theta = 728.293 • 2π

time = 1 min

Angular speed:

≈ 4576 rad/min

Answer 2

a) Let's convert the velocity of the car into m/s:

`v=65 mph= 65 mil/h \cdot (1609 m/mil)/(3600 s/h)=29.1 m/s`
Let's also convert the diameter of the wheel into meters:

`d=2.5 ft=0.76 m`
The radius of the wheel is r=d/2=0.38 m, and its circumference is

`p=2\pi r=2 \pi (0.38 m)=2.38 m`

Since the car is moving at 29.1 m/s, a point along the circumference of the wheel will cover 29.1 m every second, so the number of revolutions per second will be given by the total distance covered in one second divided by the length of the circumference:

`N= (29.1)/(2.38)=12.2 rev/s`
And since there are 60 seconds in 1 minute, the number of revolutions per minute is

`N=(12.2 rev/s)\cdot (60 s/min)=732 rev/min`

b) The angular speed of the wheel is equal to the ratio between the wheel velocity and its radius:

`w= (v)/(r)=(29.1 m/s)/(0.38 m)=76.6 rad/s`
and converting into rad/min, we get

`w=(76.6 rad/s)\cdot 60=4596 rad/min`