Experiment 10: Circular Motion
Previously we discussed two dimensional projectile motion, but another example of motion in two
dimensions is that of circular motion. The simplest case is when the radius and velocity of the moving body
are held constant. However, even though the speed is constant, the direction of the velocity is continually
changing and the body is, therefore, always undergoing acceleration. Imagine a body, originally at some
point on a circle designated by radius roand angle θ, with a velocity, −→
v, tangential to the circle (see Figure
1). Its position may be written as:
−→
r=rocosθ
b
i+rosinθ
b
j(1)
θ
r
v
y
x
Figure 1: A particle moving at constant speed v in a circular path of radius r.
Since the particle is under uniform circular motion, the angle θchanges with time as:
θ=ωt (2)
where ωis the (constant) angular velocity. Thus, combining Equations 1 and 2, we obtain the time
dependence of the body’s position
−→
r=rocos(ωt)
b
i+rosin(ωt)
b
j(3)
The velocity and the acceleration of the body are then obtained by applying Equation 3 with v=d~r/dt
and a=d~v/dt:
−→
v=−ωrosin(ωt)
b
i+ωrocos(ωt)
b
j(4)
−→
a=−ω2rocos(ωt)
b
i−ω2rosin(ωt)
b
j(5)
By looking at Equations 4 and 5, it is clear that the position, velocity and acceleration vectors all have
constant magnitudes as a function of time (ro, ωro, ω2ro, respectively). The direction of the acceleration is
always perpendicular to the direction of the velocity, which is always perpendicular to the direction of the
position vector. The acceleration always points towards the center of the circle.
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