1.3 notes Math 1060
1.3 Angular and Linear Velocity
If the speedometer on your car says 50 mph, then your velocity is 50 mph. Velocity is the rate at which the
location of an object is changing with respect to time. We will discuss two types of velocity. The angular
velocity of a point is the rate at which the angle is changing. The linear velocity of a point in motion is the
rate at which the distance is changing.
Angular Velocity
400 revolutions per minute could be considered angular velocity, however we will express angular velocity
in radians per unit of time. In order to convert from revolutions to radians, we use the fact that 2π rad=1 rev.
((400 rev)/(1 min))=((400 rev)/(1 min))⋅((2π rad)/(1 rev))=((800π rad)/(1 min))≈2513 rad/min
We use the Greek letter ω (omega) to represent angular velocity.
Definition: If a point is in motion on a circle through an angle of α radians in time t, then its angular velocity
ω is given by
ω = (α/t)
Example 1: Changing the units
Convert the angular velocity of 3000 rad/hr to rad/sec.
Example 2: Finding angular velocity
A typical lawnmower blade rotates at a rate of 2350 revolutions per minute (rpm). What is the angular
velocity in radians per second of a point on the tip of the blade?
Linear Velocity --If a point is in motion on a circle then the distance traveled by the point in a unit of time
is the length of an arc on the circle. We use the letter v to represent linear velocity for a point in circular
motion.
Definition: If a point is in motion on a circle of radius r through an angle of α radians in time t, then its linear
velocity is given by v = (s/t) = ((αr)/t)
Example 3: Linear Velocity of a propeller
A propeller with a radius of 8 meters is rotating at 6 revolutions per second. What is the linear velocity in
meters per second for a point on the tip of the propeller?
Example 4: What is the linear velocity in miles per hour of the tip of a 14 inch lawnmower blade that is
rotating at 1000 revolutions per minute?
Linear Velocity in Terms of Angular Velocity -- Linear velocity is arc length over time (v=s/t) and
angular velocity is angle over time (ω=(α/t)). Since s=αr, we have v = (s/t) = ((αr)/t) = r⋅(α/t) = rω
We have found the relationship between the two velocities. Theorem: If v is the linear velocity of a point on
a circle of radius r, and ω is its angular velocity, then v = rω
Any point on the surface of the earth (exept at the poles) makes one revolution (2π radians) about the axis of
the earth is 24 hours. So the angular velocity of a point on the earth is ((2π)/(24)) or (π/(12)) radians per hour.
The linear velocity of a point on the surface of the earth depends on its distance for the axis of the earth.
Example 5: Linear velocity on the surface of the earth
What is the linear velocity in feet per second for a point on the equator?