A wheel of radius 14 inches is rotating 0.5 rad/s. What is the linear speed
, the angular speed in RPM, and the angular speed in deg/s?
The Correct Answer and Explanation is:
To solve this problem, we need to calculate three things:
- Linear Speed (v):
The linear speed of a point on the edge of the rotating wheel is given by the formula: v=r⋅ωv = r \cdot \omegav=r⋅ω where:- rrr is the radius of the wheel (14 inches)
- ω\omegaω is the angular speed in radians per second (0.5 rad/s)
- Angular Speed in RPM (Revolutions per Minute):
To convert the angular speed from radians per second (rad/s) to revolutions per minute (RPM), we can use the conversion factor: 1 rev=2π radians1 \, \text{rev} = 2\pi \, \text{radians}1rev=2πradians 1 minute=60 seconds1 \, \text{minute} = 60 \, \text{seconds}1minute=60seconds So the angular speed in RPM is: RPM=ω×602π\text{RPM} = \frac{\omega \times 60}{2\pi}RPM=2πω×60 Substituting the given value of ω=0.5 rad/s\omega = 0.5 \, \text{rad/s}ω=0.5rad/s: RPM=0.5 rad/s×602π=30π≈9.55 RPM\text{RPM} = \frac{0.5 \, \text{rad/s} \times 60}{2\pi} = \frac{30}{\pi} \approx 9.55 \, \text{RPM}RPM=2π0.5rad/s×60=π30≈9.55RPM So, the angular speed is approximately 9.55 RPM. - Angular Speed in deg/s:
To convert the angular speed from radians per second (rad/s) to degrees per second (deg/s), we can use the following conversion factor: 1 radian=180∘/π1 \, \text{radian} = 180^\circ/\pi1radian=180∘/π So, the angular speed in degrees per second is: deg/s=ω×180∘π\text{deg/s} = \omega \times \frac{180^\circ}{\pi}deg/s=ω×π180∘ Substituting the given value of ω=0.5 rad/s\omega = 0.5 \, \text{rad/s}ω=0.5rad/s: deg/s=0.5×180∘π≈28.65∘/s\text{deg/s} = 0.5 \times \frac{180^\circ}{\pi} \approx 28.65^\circ/\text{s}deg/s=0.5×π180∘≈28.65∘/s So, the angular speed is approximately 28.65 degrees per second.
Summary:
- Linear Speed: 7 inches per second
- Angular Speed in RPM: 9.55 RPM
- Angular Speed in deg/s: 28.65 degrees per second
This shows how a rotating wheel’s linear and angular speeds are related through the radius and angular velocity.
