Rolling

A rolling wheel has both linear and angular velocity. If the wheel is not slipping, the two velocities are related by geometry, because the circumference of a circle is 2π times the radius r. A wheel rotates through an angle of 2π radians (a full rotation) as it moves forward by a distance of 2πr (one circumference). Read the text aloud Radius and the circumference of a circle
The linear velocity is the circumference divided by the time it takes to make one turn, i.e., v = 2πr/t. For one rotation, the quantity 2π/t is the angular velocity ω. The linear velocity is therefore equal to ω multiplied by the radius r. Linear and angular velocity are related to each other by the equation Read the text aloud
(7.2) v=ωr
v  = linear velocity (m/s)
ω  = angular velocity (rad/s)
r  = radius (m)
Linear velocity
from angular velocity
Why did old bicycles have such large wheels?The radius r that appears in equation (7.2) means that larger wheels have a higher linear velocity than smaller wheels for a given angular velocity. Early bicycles had very large wheels because the larger wheels would create a higher linear velocity. Unfortunately, they were very difficult to ride and quite unstable! Modern bicycles use gears and chains so that the pedals can turn at a different angular velocity from the wheels. Read the text aloud
Linear speed of a rolling wheel
A car has wheels with a radius of 30 cm. What is the angular velocity of the wheels in both radians per second and revolutions per minute when the car is moving at 30 m/s (67 mph)?
Asked: angular velocity ω calculated in two units: rad/s and rpm
Given: radius r = 0.3 m; velocity v = 30 m/s
Relationships: linear velocity v = ωr; one revolution is 2π = 6.28 radians
Solution: In rad/s, ω = v/r = (30 m/s)/(0.3 m) = 100 rad/s.
In rpm, ω = (100 rad/s)×(60 s/min)×(1 revolution/6.28 rad) = 955 rpm.
Answer: ω = 100 rad/s = 955 rpm
Read the text aloud
A wheel is rolling at a linear velocity of 25 m/s. If the radius of the wheel is decreased, and the linear velocity remains the same, what does this indicate about the angular velocity of the smaller wheel?
  1. It remains unchanged.
  2. It increases.
  3. It decreases.
  4. There is not enough information to answer.
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