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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