PHYSICS 231

INTRODUCTORY PHYSICS I

Lecture 10

• Elastic Collisions:

• Multi-part Collision Problems (conserve E or p)

• Angular motion

Last Lecture

!

m1v1i + m

2v2i = m

1v1 f + m

2v2 f

v1i " v2i = " v

1 f " v2 f( )

!

s = r" (" in radians)

Angular Speed

• Can also be given in• Revolutions/s• Degrees/s

• Linear (tangential) Speed at r

!

vt="s

"t=r"#

"t

!

vt= r"

!

" =#$

#t=$ f %$i

t f % ti

(! in rad/s)

(in rad/s)

Example 7.2

A race car engine can turn at a maximum rate of 12,000rpm. (revolutions per minute).

a) What is the angular velocity in radians per second.

b) If helicopter blades were attached to the crankshaftwhile it turns with this angular velocity, what is themaximum radius of a blade such that the speed of theblade tips stays below the speed of sound.DATA: The speed of sound is 343 m/s

a) 1256 rad/s

b) 27 cm

Angular Acceleration

• Denoted by !

• " in rad/s

• ! rad/s!

• Every point on rigid object has same " and !

! =" f #" i

t

Rotational/Linear Correspondence:

!" # !x

$0# v

0

$ f # v f

% # a

t# t

Rotational/Linear Correspondence, cont’d

Rotational Motion Linear Motion

!" =#0+# f( )2

t

!" =#0t +1

2$t2

! f =!0 +"t

! f2

2=!0

2

2+"#$

!" =# f t $1

2%t2

!x =v0+ v f( )2

t

v f = v0 + at

!x = v0t +1

2at2

!x = v f t "1

2at2

v f2

2=v0

2

2+ a!x

Con

stant

!Consta

nt a

Example 7.3

A pottery wheel is accelerated uniformly from restto a rotation speed of 10 rpm in 30 seconds.

a.) What was the angular acceleration? (in rad/s2)

b.) How many revolutions did the wheel undergoduring that time?

a) 0.0349 rad/s2

b) 2.50 revolutions

Linear movement of a rotating point

• Distance

• Speed

• Acceleration

Angles must be in radians!

Different pointshave differentlinear speeds!

!

at

= r"

!

vt= r"

!

"s = r"#

Special Case - Rolling

• Wheel (radius r) rolls without slipping

• Angular motion of wheel gives linear motion of car

• Distance

• Speed

• Acceleration

x = r!"

v = r!

a = r!

Example 7.4

A coin of radius 1.5 cm is initially rolling with arotational speed of 3.0 radians per second, andcomes to a rest after experiencing a slowing down of! = 0.05 rad/s2.

a.) Over what angle (in radians) did the coin rotate?

b.) What linear distance did the coin move?

a) 90 radb) 135 cm

Centripetal Acceleration

• Moving in circle at constantSPEED does not mean constantVELOCITY

• Centripetal acceleration resultsfrom CHANGING DIRECTION

of the velocity

•

• Acceleration points towardcenter of circle

!

r a =

"r v

"t

• Similar triangles:

•

• Small times:

•

• Using or

Derivation: acent = "2r = v2/r

!

a = v"#

"t= v$

!

"v

v="s

r

!

aavg ="v

"t=v

r

"s

"t

!

"s # arc length = r"$

!

v ="r

!

" = v /r

!

acent

=" 2r =

v2

r

!

"

Forces Cause Centripetal Acceleration

• Newton’s Second Law

• Radial acceleration requires radial force• Examples of forces

• Spinning ball on a string• Gravity• Electric forces, e.g. atoms

rF = m

ra

Example 7.5a

a) Vector Ab) Vector Bc) Vector C

A

B

CAn astronaut is incircular orbitaround the Earth.

Which vector mightdescribe theastronaut’s velocity?

Example 7.5b

a) Vector Ab) Vector Bc) Vector C

A

B

CAn astronaut is incircular orbitaround the Earth.

Which vector mightdescribe theastronaut’sacceleration?

Example 7.5c

a) Vector Ab) Vector Bc) Vector C

A

B

CAn astronaut is incircular orbitaround the Earth.

Which vector mightdescribe thegravitational forceacting on theastronaut?

Example 7.6a

a) Vector Ab) Vector Bc) Vector C

AB

C

Dale Earnhart drives150 mph around acircular track atconstant speed.

Neglecting airresistance, whichvector bestdescribes thefrictionalforce exerted on thetires from contactwith the pavement?

Example 7.6b

a) Vector Ab) Vector Bc) Vector C

Dale Earnhartdrives 150 mpharound a circulartrack at constantspeed.

Which vector bestdescribes thefrictional forceDale Earnhartexperiences fromthe seat?

AB

C

Ball-on-String Demo

Example 7.7

A puck (m=.25 kg), sliding on a frictionless table,is attached to a string of length 0.5 m. Theother end of the string is fixed to a point on thetable and the puck is sent revolving around thefixed point. It take 2 seconds to make a completerevolution.

a) What is the acceleration of the puck?

b) What is the tension in the string?

a) 4.93 m/s2

b) 1.23 N

DEMO: FLYING POKER CHIPS

Example 7.8

A race car speeds around a circular track.

a) If the coefficient of friction with the tires is 1.1,what is the maximum centripetal acceleration (in“g”s) that the race car can experience?

b) What is the minimum circumference of the trackthat would permit the race car to travel at 300km/hr?

a) 1.1 “g”sb) 4.04 km (in real life curves are banked)

Example 7.9

A curve with a radius ofcurvature of 0.5 km on ahighway is banked at anangle of 20°. If thehighway were frictionless,at what speed could a cardrive without sliding offthe road?

42.3 m/s = 94.5 mph

Example 7.11a

Which vector represents acceleration?

a) A b) E

c) F d) B

e) I

Example7.11b

If car moves at "design" speed, which vector representsthe force acting on car from contact with road

a) D b) E

c) G d) I

e) J

Example 7.11c

If car moves slower than "design" speed, whichvector represents frictional force acting on carfrom contact with road (neglect air resistance)

a) B b) C

c) E d) F

e) I