Physics 2514Lecture 19

P. Gutierrez

Department of Physics & AstronomyUniversity of Oklahoma

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Goals

Start the discussion of circular motion.We will define the kinematic variables needed to describecircular motion.

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Circular Motion

We will start with the case of uniform circular motion. (Thisrefers to circular motion with a constant speed.)

Some definitionsPeriod (T ) time interval to complete one orbit, thereforespeed is given by

v =circumference

period =2πr

T

r is the radius of the circle.

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Circular Motion

Some definitions—Use angular variables to measureposition

θ(radians) =s

r

θfull circle = 2π

θ increases in counter-clockwisedirection decreases in clockwisedirection.θ measured from x axis

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Circular Motion

Some definitions—Use angular variables to measure speed

Angular Velocity

ωaverage =∆θ

∆t

ω = lim∆t→0

∆θ

∆t=

dθ

dt

ω > 0 motion counter-clockwiseω < 0 motion clockwise

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Angular Position vs. Time

Slope of angular position vs. time plot gives angular velocity vs.time plot.

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Clicker

A particle moves clockwise around a circle at constant speed for2.0 s. It reverses direction and moves counter clockwise at halfthe original speed until it has traveled through the same angle.Which is the particles angle-versus-time graph?

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Kinematic Equations

For uniform circular motion ω is constant.From definition of angular velocity

ω =dθ

dt⇒

∫ θf

θ0

dθ =

∫ tf

t0

ωdt = θf = θ0 + ω(tf − t0)

Take t0 = 0, tf = t, then

θ(t) = ωt + θ0

Tangential acceleration is zero (acceleration in directionobjects moves).

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Coordinate System

Tangential velocity and acceleration in terms of angular variablesRecall: s = rθ ⇒ vt = ds

dt= r

dθ

dt= rω ω is angular velocity.

Therefore: vt = rω ⇒ at = dvt

dt= r

dω

dt= rα α is angular

acceleration.Physics 2514 – p. 9/14

Kinematic Equations

The kinematic equations for uniform circular motion werederived earlier, here we consider nonuniform motionMotion along arc is 1-D with tangentialacceleration and velocity determining mo-tion

s = s0 + vott +1

2att

2

vt = v0t + att

Now divide by r

1

r(s = s0 + v0tt +

1

2att

2)

1

r(vt = v0t + att)

θ = θ0 + ω0t +1

2αt2

ω = ω0 + αt α = at/rPhysics 2514 – p. 10/14

Centripetal Acceleration

Assume constant tangential speed (|~v|)Acceleration points to center of circle (~ar ⊥ ~v)

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Centripetal Acceleration

Calculate average acceleration

CB = ∆~r2−∆~r1 = ~v2∆t−~v1∆t = ∆~v∆t

Angles

ABO: θ + α + α = 180

DAC: φ + α + α = 180

)

⇒ θ = φ

Similar triangles

CBAB

=ABAO

⇒|∆~v|∆t

v∆t=

v∆t

r

Average radial acceleration

aaverager =

|∆~v|

∆t=

v2

r

ar = lim∆t→0

|∆~v|

∆t=

v2

r

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Clicker

A ball is lodged in a hole in the floor of a merry-go-round that isturning at constant speed. Which kinematic variable or variableschange with time, assuming that the position is measuredrelative to a fixed coordinate system with its origin at the centerof the merry-go-round?

1. the position of the ball only;2. the velocity of the ball only;3. the acceleration of the ball only;4. both the position and velocity of the ball;5. the position and velocity and acceleration of the ball.

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Assignment

Continue reading Chapter 7Will discuss dynamics in next lecture

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