Circular Motion - Florida State University

[Pages:13]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

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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()

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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

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Angular Velocity

Definition:

d =

dt

For constant and in radian we find:

v =r

Namely, for one period:

T = 2 v T = 2 r

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Surface of Rotating Water

Integration: parabola.

dy(r)

v2 2 r2 2 r

= tan() = =

=

dr

gr gr g

r2 y(r) = + y(0)

2g

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For the mathematically ambitious only:

Another Derivation of the Acceleration

Now,

Therefore, The velocity is

r(t) = r r^ with r^ = cos() y^ + sin() x^ (t) = t

r^ = cos(t) y^ + sin(t) x^

dr dr^ v = = r = -r sin(t) y^ + r cos( t) x^

dt dt

where

= -v sin(t) y^ + v cos( t) x^ = v t^ t^ = - sin(t) y^ + cos( t) x^

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is the tangential unit vector. In the same way the acceleration follows:

dv dt^ a = = v = -v cos(t) y^ - v sin(t) x^

dt dt

v2

v2

v2

= - cos(t) y^ - sin(t) x^ = - r^

r

r

r

v2 a = ar r^ with ar = - r

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Questions on Circular motion

A particle of mass m moves with constant speed v on a circle of radius R. The following holds (pick one): 1. The centripetal force is v2/R towards the center. 2. The centripetal force is m v2/R towards the center. 3. The centripetal force is m v2/R away from the center. 4. The centripetal force is v2/R away from the center. 5. The centripetal force is m v2/R downward.

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