acceleration given its angular velocity or acceleration, or vice versa. Let’s take a look at how this
is done.
Distance
We saw earlier that the angular position, , of a rotating particle is related to the length of the arc,
l, between the particle’s present position and the positive x-axis by the equation = l/r, or l = r.
Similarly, for any angular displacement, , we can say that the length, l, of the arc made by a
particle undergoing that displacement is
Note that the length of the arc gives us a particle’s distance traveled rather than its displacement,
since displacement is a vector quantity measuring only the straight-line distance between two
points, and not the length of the route traveled between those two points.
Velocity and Acceleration
Given the relationship we have determined between arc distance traveled, l, and angular
displacement, , we can now find expressions to relate linear and angular velocity and
acceleration.
We can express the instantaneous linear velocity of a rotating particle as v = l/t, where l is the
distance traveled along the arc. From this formula, we can derive a formula relating linear and
angular velocity:
In turn, we can express linear acceleration as a = v/t, giving us this formula relating linear and
angular acceleration:
EXAMPLE
The radius of the Earth is approximately m. What is the instantaneous velocity of a point on
the surface of the Earth at the equator?We know that the period of the Earth’s rotation is 24 hours, or seconds. From the
equation relating period, T, to angular velocity, , we can find the angular velocity of the Earth:
Now that we know the Earth’s angular velocity, we simply plug that value into the equation for
linear velocity: