Circular Motion - Florida State University
Circular Motion
v2 ar = r
The Period T
The time T required for one complete revolution is called the period. For
constant speed
2 r
v=
holds.
T
1
Example: Circular Pendulum: Figures 5-22 and 5-23 of Tipler-Mosca. Relation between angle and velocity.
T + Fcf + m g = 0
T = Tr r^ + Ty y^ v2
Tr = -T sin() = -m r Ty = T cos() = m g sin() v2 = cos() g r v2 tan() = gr v = g r tan()
2
Example: Forces on a car in a banked curve: Figures 5-26 of Tipler-Mosca. The optimal angle is the one for which the centrifugal force is balanced by the inward component of the normal force (i.e. without friction). Then:
Fn cos() - m g = 0 v2
Fn sin() - m r = 0 v2
tan() = gr
3
Angular Velocity
Definition:
d =
dt
For constant and in radian we find:
v =r
Namely, for one period:
T = 2 v T = 2 r
4
Surface of Rotating Water
Integration: parabola.
dy(r)
v2 2 r2 2 r
= tan() = =
=
dr
gr gr g
r2 y(r) = + y(0)
2g
5
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