7 CIRCULAR MOTION 7.2 Uniform circular motion
proportional to the angle θ: but, what is the constant of proportionality? Well,
an angle of 360 ◦ corresponds to an arc length of 2π r. Hence, an angle θ must
correspond to an arc length of
2π
s =
360 ◦r θ(◦). (7.2)At this stage, it is convenient to define a new angular unit known as a radian
(symbol rad.). An angle measured in radians is related to an angle measured in
degrees via the following simple formula:
θ(rad.) =
360 ◦θ(◦). (7.3)Thus, 360 ◦ corresponds to 2 π radians, 180 ◦ corresponds to π radians, 90 ◦ corre-
sponds to π/2 radians, and 57.296◦ corresponds to 1 radian. When θ is measured
in radians, Eq. (7.2) simplifies greatly to give
s = r θ. (7.4)Henceforth, in this course, all angles are measured in radians by default.
Consider the motion of the object in the short interval between times t and
t + δt. In this interval, the object turns through a small angle δθ and traces out a
short arc of length δs, where
δs = r δθ. (7.5)Now δs/δt (i.e., distance moved per unit time) is simply the tangential velocity
v, whereas δθ/δt (i.e., angle turned through per unit time) is simply the angular
velocity ω. Thus, dividing Eq. (7.5) by δt, we obtain
v = r ω. (7.6)Note, however, that this formula is only valid if the angular velocity ω is mea-
sured in radians per second. From now on, in this course, all angular velocities
are measured in radians per second by default.
An object that rotates with uniform angular velocity ω turns through ω radi-ans in 1 second. Hence, the object turns through 2 π radians (i.e., it executes a
complete circle) in
T =2 π
(7.7)
ω2π