PHYS16 – Lecture 22

Circular Motion and Rotation October 29, 2010

http://xkcd.com

Circular Motion and Rotation

• Circular Motion– Angular disp., velocity, and acceleration– Centripetal force– Circular motion kinematics

• Rotation– Inertia– Kinetic Energy– Angular momentum– Torque

• Simple Machines II – gears, belts, and levers

Circular Motion and Centripetal Force last time…

Polar Coordinates

http://en.citizendium.org/images/thumb/1/18/Polar_coordinates_.png/250px-Polar_coordinates_.png

)sin(

)cos(

)(tan 1

22

ry

rx

xy

yxr

Angular displacement, velocity, and acceleration

• Angular displacement – • Arc length –• Angular velocity (ω) –

• Angular acceleration (α) –

• Linear acceleration( ) –

12 rs

Tf

r

v

dt

d 22

r

a

dt

d

dt

d T2

2

a

rva

ra

rataa

C

T

CT

2

ˆˆ

Angular displacement, velocity, and acceleration

• Uniform circular motion – α = 0, only centripetal accel. (aC)

• Non-uniform circular motion – both aT and aC

http://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Nonuniform_circular_motion.svg/293px-Nonuniform_circular_motion.svg.png

aC

aT

Centripetal Force

• Force keeping an object moving in a circle

rmmaF

rmaF

C

C

2

ˆ

http://astronomy.swin.edu.au/cms/imagedb/albums/scaled_cache/centripedal-316x300.png

Practice Question #1

• What is the anglular displacement of the Earth during winter? Assume winter is ¼ of the year and that the Earth’s orbit can be approximated by a circle.A) 0.785 radB) 1.57 radC) 3.14 radD) None of the above

Practice Question #2

• A CD is spinning with a constant angular velocity. From last time we know that the linear velocity of a point on the edge of the CD is greater than a point in the middle. What about the centripetal acceleration?A) Centripetal acc. is greater at the edgeB) Centripetal acc. is greater in the middleC) Centripetal acc. is equal for both pointsD) There is not enough information

Discussion Question

• I have a ball attached to a string that runs through a tube and is attached to a mass at one end. I hold the tube and rotate the ball in uniform circular motion while holding the mass. Then I let the mass go. What happens to the angular velocity of the ball?

(Increase, Decrease, Stay the same)

Circular Motion Kinematics

Angular kinematics – same as linear

• Assume α=constant

020

2

0

200

2

2

1

t

tt

Example Question: The Centrifuge

• A centrifuge rotates with an angular speed of 3600 rpm. Then it is switched off and it rotates 60 times before coming to rest. What was the angular acceleration that made it stop?

http://upload.wikimedia.org/wikipedia/commons/0/0d/Tabletop_centrifuge.jpg

22

220

20

2

rad/s 200rad/s 188

2*60*2

)60/2*3600(

2

2

Example Question: The Discus

• A discus thrower with arm radius of 1.2 m starts from rest and then starts to rotate with an angular acceleration of 2.5 rad/s2. How long does it take for the throwers hand to reach 4.7 rad/s?

s 9.1

5.2/7.4/)( 0

0

t

t

t

Rotation

Rotational Inertia

• In linear motion we just care about mass• In rotational motion we care about how mass

is distributed so we need rotational inertia (I)

formulas pg. 320

• Which has more rotational inertia?

2

1i

n

iirmI

A) B)A does!

Rotational Kinetic Energy

• For rotational kinetic energy we use rotational inertia instead of mass and angular velocity instead of linear velocity

• What is kinetic energy of the earth? Mass = 5.98E24 kg and radius=6.37E6 m.

2

2

1 IK

J 2.6E29)day 1

2)(

5

2(

2

1

2

1 22 earthearthRMIK

Example Question: Rolling vs. Sliding

• Which has more energy: a cylinder that slides down a ramp with a speed of v0 or a cylinder that rolls down a ramp with the same speed?A) Cylinder that slidesB) Cylinder that rollsC) Both are equalD) There is not enough information

222

2

1)1(

2

1

2

1mvcImvK rolling

Angular Momentum

• Angular momentum (L) – momentum of a rotating object

• Cross product like the dot product is a way to multiply vectors, except cross product gives vector not scalar

• Direction of cross product is given by right hand rule

IrpL

prL

)sin(

Example Question: Ice Skating

• In a spin, why do ice skaters decrease their angular velocity when they hold their arms out?

http://www.corbisimages.com/images/67/7760610C-6DF3-4A39-ACD6-C3CDEFF73296/PN015983.jpg

Kristi Yamaguchi

L=IωHolding arms out increases I.If L stays the same, and I increases thenω decreases.

Torque

• Torque (τ) – a force that acts at a distance, usually causing rotation

• Units = Joules• Vector quantity, direction given by right hand

rule

IrF

Fr

)sin(

Example: Jet turbine

• The turbine of a jet engine has a moment of inertia of 25 kg∙m2. If the turbine is accelerated uniformly from rest to an angular speed of 150 rad/s in a time of 25 s, what is the torque?

J 15025/)150)(25(/)(

/)(

0

0

0

tII

t

t

Main Points

• Parameters for circular motion/ rotation basically have linear equivalents– θ is related to x, ω is related to v, α is related to a– I is related to m– L is related to p, L=Iω=rpsin(θ)– τ is related to F, τ=Iα =rFsin(θ)