R. Field 2/5/2013 University of Florida

PHY 2053 Page 1

Circular Motion: Angular Variables

The arc length s is related to the angle (in radians = rad) as follows:

• Arc Length:

rs r

s

if • Angular Displacement and Angular Velocity:

rdt

dsvv tt

dt

d

tt

0

lim

• Tangential Velocity and Angular Velocity:

(radians/second)

• Period and Frequency (constant ):

(360o = 2 rad)

TvrC t 2 22

tv

rT

Tf

1

(revolutions/second)

Tangential Velocityrvt

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PHY 2053 Page 2

Circular Motion: Radial Acceleration

Centripetal acceleration = “toward the center”

rvv tt ˆ)(

• Radial (centripetal) Acceleration:

rrrvrt

vt

va tt

t

tradial ˆ)(ˆ)(ˆ)(lim 2

0

r

vraa

t

radialradial

22 Magnitude = (vt)2/r

Direction = toward the center of the circle

Radial Acceleration

r̂

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PHY 2053 Page 3

Example Problem

• A puck of mass m slides in a circle of radius r = 0.5 m on a frictionless table while attached to a hanging cylinder of mass M = 2m by a cord through a hole in the table. What speed of the mass m keeps the cylinder at rest?

Answer: smm

Mgrv /13.3

Solution:

Mg

FT

FT

0 MgFT

r

vmmaF radialT

2

Mgr

vm

2

smmsm

grm

mgr

m

Mgrv

/13.3)5.0)(/8.9(2

22

2

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PHY 2053 Page 4

Example: Conical Pendulum

g

LTperiod

cos

22

2sin mrmamaT radialx

0cos ymamgT

x-component:

y-component:

• A stone of mass m is connected to a cord with length L and negligible mass. The stone is undergoing uniform circular motion in the horizontal plane. If the cord makes an angle with the vertical direction, what is the period of the circular motion?

cosL

g

g

L

g

r 22 )sin(tan

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PHY 2053 Page 5

Exam 1 Spring 2012: Problem 20

• A conical pendulum is constructed from a stone of mass M connected to a cord with length L and negligible mass. The stone is undergoing uniform circular motion in the horizontal plane as shown in the figure. If the cord makes an angle = 30o with the vertical direction and the period of the circular motion is 4 s, what is the length L of the cord (in meters)?Answer: 4.59 % Right: 34%

Rg

vR

vMMaF

MgF

xT

T

2

2

tan

sin

0cos

L

RT

Rv

v

RT

sin

4

2

2

222

L

M

L

Mg

x-axis

y-axis

R

FT

sin

4

tan

4tan

2

2

2

2

RL

gTR

gT

R

mssmgT

L 59.4)30cos(4

)4)(/8.9(

cos4 2

22

2

2

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PHY 2053 Page 6

Example Problem: Unbanked Curves

R

vtangential aradial

• A car of mass M is traveling in a circle with radius R on a flat highway with speed v. If the static coefficient of friction between the tires and the road is s, what is the maximum speed of the car such that it will not slide?

x-axis fs

N

R

y-axis

R

vMMaMaf radialxs

2

0 yMaMgN

gRMgM

RN

M

Rf

M

Rv ssss )()()( max

2max

Nfs s

gRv smax

x-component:

y-component:

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PHY 2053 Page 7

Exam 2 Spring 2011: Problem 4

• Near the surface of the Earth, a car is traveling at a constant speed v around a flat circular race track with a radius of 50 m. If the coefficients of kinetic and static friction between the car’s tires and the road are k = 0.1, s = 0.4, respectively, what is the maximum speed the car can travel without slipping?Answer: 14 m/s% Right: 74%

x-axis fs

FN

R

y-axis

R

vMMaMaf radialxs

2

0 yN MaMgF

gRMgM

RF

M

Rf

M

Rv ssNss )()()( max

2max

NsFfs

smmsmgRv s /14)50)(/8.9)(4.0( 2max

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PHY 2053 Page 8

Example Problem: Banked Curves

R

vMMaNf radials

2

sincos

0sincos ys MaMgfN

sincos s

MgN

x-component:

y-component:

)tan1(

)tan(

)sin(cos

)sincos(

)sincos(sincos)( max2max

s

s

s

s

ss

RgRg

NM

RNf

M

Rv

)tan1(

)tan(max

s

sRgv

• If the car in the previous problem is traveling on a banked road (angle ), what is the maximum speed of the car such that it will not slide?

Mg

Nf ss max)(

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PHY 2053 Page 9

Rolling Without Slipping: Rotation & Translation

R

v

s x-axis

x

• If a cylinder of radius R rolls without slipping along the x-axis then:

rsx

R

dt

dR

dt

dxv

Translational Speed

Rotational Speed