Uniform Circular Motion
Non-uniform Circular Motion
r = r(cos(ωt) î + sin(ωt) ĵ)
v = ṙ = rω(-sin(ωt) î + cos(ωt) ĵ)
Calculate v · r.
Circular Motion
t - time
θ - angle traversed
ω - angular velocity (speed)
r - position vector
ṙ - derivative of r with respect to t
atan - tangential acceleration vector
ac - centripetal (radial) acceleration vector
Assumption:
ω and |r| are constant.
The particle moves counter-clockwise beginning at (|r|, 0) and t = 0.
The period T is the time it takes the particle to make one complete revolution. Calculate T.
Why is |v| also constant? Why doesn't a change |v|? What is a · v?
How long does it take for v to make a full rotation about the origin? a?
v repositioned at the origin
a repositioned at the origin
a also rotates without changing magnitude. Calculate ȧ. Where would you draw it?
Assumption:
| r | is constant,
ω changes.
Since v changes both in magnitude and direction, a now has two components: atan (tangential) and ac (centripetal).
In which direction is atan pointing when the particle is decelerating (the magnitude of v is decreasing)?
How does ac change as the magnitude of v changes?