PHYS 201

PHYS 201

Chapter 8 +9: Rotational Kinematics

Angular DisplacementAngular VelocityAngular AccelerationTorqueAngular MomentumAngular Momentum Conservation

CAPA 9 due next Monday (11/15/10) at 11:59 PM

Final Exam on 11/19/10 (Friday) at 4:40 to 6:40 pm(Room to be announced)Send email if you have an exam conflict([email protected])

Only official OU excuse will be given a make-up exam.

Angular Displacement (Radians)

(rad)

1 rev = 2π rad = 360˚

The change in angle due to rotation.

A wheel undergoes an angular displacement of π/3. What is this in degrees?

(1) 15 (2) 30 (3) 45 (4) 60 (5) 75(6) 90 (7) 105 (8) 120 (9) 135 (0) 150

CLICKER! 1 rev = 2π rad = 360˚

Convert 45 degree to radian and revolution.

1) 0.4 rev, 0.4 rad

2) 0.25 rev, 0.25 rad

3) 0.125 rev, 0.125 rad

4) 0.125 rev, 0.25 rad

CLICKER!

45 deg = 45 deg x (1 rev/ 360 deg) = 0.125 rev.45 deg = 45 deg x (2 π rad/360 deg) = 0.25 π rad

1 rev = 2π rad = 360˚

Angular Velocity

The rate of change of angular displacement.  (Unit: rev/min, rad/s etc.)

Angular Acceleration

The rate of change of angular velocity.  (Unit: rev/min2, rad/s2 etc.)

Two objects are sitting on a rotating turntable. One is much further out from the axis of rotation. Which one has the larger angular velocity?

1) the one nearer the disk center2) the one nearer the disk edge3) they both have the same angular velocity

All points on rigid object have same angular displacement (Δθ),same angular velocity (ω), and same angular acceleration (α)

CLICKER!

An object rotates along a circular path for 60 degree in 10s. What is the angular speed?

(1) /3 rad/s (2) /10 rad/s (3) /30 rad/s

(4) 3 rad/s (5) 10 rad/s

CLICKER!

An object starts at rest and undergoes an average angular acceleration of 0.5 rad/s2 for 10 seconds. What is the angular speed after 10 seconds?

(1) 0.05 rad/s (2) 0.5 rad/s (3) 5 rad/s

(4) 10 rad/s (5) 20 rad/s (6) 50 rad/s

α = Δω / ΔtΔω = α Δt = (0.5 rad/s2) 10 s = 5 rad/sSince ω0 = 0, ωf = 5 rad/s

CLICKER!

PHYS 201

Chapter 8 +9: Rotational Kinematics/Dynamics

CAPA 9 due next Monday (11/15/10) at 11:59 PM

No help session on Thursday (holiday)

Final Exam on 11/19/10 (Friday) at 4:40 to 6:40 pm in Walter Hall(Room to be announced)

Send email if you have an exam conflict([email protected])

Only official OU excuse will be given a make-up exam.

Linear Vs. Angular

s = r

v = r

a = r

x ↔ θv ↔ ωa ↔ αF ↔ τm ↔ I

ac = v2 / r = (rω) 2 /r = r ω2

ac = v2 / r or ac = r ω2

Centripetal Acceleration

Two objects are sitting on a rotating turntable. One is much further out from the axis of rotation. Which one has the larger linear velocity?

1) the one nearer the disk center2) the one nearer the disk edge3) they both have the same linear velocity

CLICKER!

Although are the same, different ‘r’, so different ‘v’. The outer object will have a higher linear velocity.

Rolling Motion

• If object rolls without slipping, linear distance traveled is equal to arc length of rotation, so:

s = r θ

Rolling Motion

• If object rolls without slipping, linear distance traveled is equal to arc length of rotation, so:

s = r θ

s

Example 1.

• The wheels of a bike has a radius of 0.5 m, and the wheel is rotating with a constant angular speed of 3rev/s.

• a). find the linear speed of the bike. • b). Find the distance travelled in 10s.

Example 2.

• The wheels of a bike has a radius of 0.5 m. The bike starts from rest and reached an angular speed of 3rev/s in 3s.

• a). Find the angular acceleration. • b). Find the linear acceleration correspond to the first 3s.

A pulley of radius 0.10m has a string wrapped around the rim. If the pulley undergoes a total angular displacement of 25rad, what is the length of the string that comes off the reel?

(1) 0.025 m (2) 0.25 m (3) 2.5 m(4) 25.0 m (5) 250 m (6) 2500 m

The arc length through which a point on the rim travels is the exact same as length of string which comes off the reel.

s= r Δθ = (0.10m) (25 rad) = 2.5 m

CLICKER!

A pulley of radius 0.10m has a string wrapped around the rim. If the pulley is rotating on a fixed axis at an angular speed of 0.5rad/s, what is the length of the string that comes off the reel in 10 seconds?(1) 0.005 m (2) 0.05 m (3) 0.5 m(4) 5.0 m (5) 50 m (6) 500 m

Δθ = ω (Δt) = 5 rads= r Δθ = (0.10m) (5 rad) = 0.5 m

CLICKER!

Linear Vs. Angular

F = ma

= I

I = moment of inertia

p = mv

L = I

Force Momentum

Moment of Inertia (I)

A measure of an object's resistance to changes to its rotation.

Unit: kg m2

Moment of Inertia – Multiple or Compound Objects

Angular Momentum Conservation

Initial Angular Momentum = Final Angular Momentum

Li = Lf Ii iIf f

A man is standing on a center of the disc that is rotating with 5 rev/s. He holds 1kg mass at each hand and initially the hands are stretched out as shown. At this position, the two masses are 1.5 m apart. Then he brings the two masses to a 0.5m distance in order to increase the rotation speed. Find the new angular speed.

Example 3

Multiple Objects – Add moments of Inertia

• For example, consider the following: moment of inertia of disk plus moment of inertia of two point particles.– This is all spinning about the center of the disk

• ITotal = Icylinder + IA + IB

• ITotal = ½ MCYLR2 + MARA2 + MBRB

2

RA RB

A

B

Three erasers are on a turntable. Eraser A is near the edge, eraser C is the closest to the center, and erase B is in the middle. The surface has friction. Starting from rest, the turntable slowly accelerates. Which eraser flies off first?

(1) A(2) B(3) C(4) All at the same time

All have same ω, A has greatest radius. ac = r ω2

A has greatest ac, so it requires the greatest force to stay in circle.

CLICKER!

A 0.50kg mass is hung from a massive, frictionless pulley of mass 1.5kg and radius 0.10m. Starting from rest, how long will it take for the mass to fall 1.0 m?

Example 4

0.1m

1m

1.5 kg

0.5 kg

Two forces are exerted on a wheel which has a fixed axle at the center. Force A is applied at the rim. Force B is applied halfway between the axle and the rim. |FA| = ½|FB| Which best describes the direction of the angular acceleration?1. Counterclockwise2. Clockwise3. Zero

FA

FB

FA is trying to twist CCW, FB is trying to twist CW.

The torques are the same. Torque from A has half the force, but twice the lever arm.

Torque – Circles/"Cams"

• Used boards a lot with Static Equilibrium and torques• What about circle or other extended shape?

F1

r

F2

τ1=F1·r

τ2=F2·ℓℓ

FF

Cams on exercise equipment

A

BTakes less force at point B to exert similar torque(longer lever arm)

Each Red dot represents a 1kg mass on a turntable. Which of the three turntables requires the least torque to get it from rest to an angular speed of 3 rad/s over 10 s?

(4) All the same

(1) (2) (3)

All three same angular acceleration αSmallest moment of inertia requires least torqueMasses closest to the center – smallest Moment of Inertia

τ = I α

If needed to calculate I:ITOTAL = 1/2 MCYLR2 + MARA

2 + MBRB2 + MCRC

2

Torque - CD

Στ = IαIf know all the torques and α, can find moment of inertia, I.If know I (from geometry) and α, can find net torque.In this case, know ang accel from change in ang speed.Also know CD is cylinder

For Bucket, know torque and ang accel. - Can find angular acceleration of pulley from linear acceleration of bucket.

Dizzy Stroll

• Don't forget that there are two objects in the system – the carousel and the person

• Moment of Inertia includes sum of both

Pulsars• Supernova remants

• Star collapses into VERY dense object neutron star

• Typical radius about 10km, but typical mass 1.5 times mass of Sun

•Teaspoon of neutron star material would weight 1 billion tons.

• Spinning pretty quickly, especially for such a small object

• Huge magnetic fields

http://science.nasa.gov/NEWHOME/help/tutorials/pulsar.htm