Angular KinematicsAngular Kinematics

Ch 9 NotesCh 9 Notes

What is a radian?What is a radian?

1 1 ΘΘradrad = 57.3 = 57.3 ͦ of twistͦ of twist

Angular Angular

VariablesVariablesƟ = S / rƟ = S / r

1 1 ωω = 57.3 ͦ of twist = 57.3 ͦ of twist in a second’s time in a second’s time = = ΘΘradrad / 1 s / 1 s

1 1 αα = = changechange in in angular speed angular speed equal to equal to 57.357.3 of of twist per second = twist per second = ΔωΔω / 1 s / 1 s

S

θrad

Angular Kinematics DefinitionsAngular Kinematics Definitions

Angular Velocity: “amount of twist Angular Velocity: “amount of twist per time”per time”

ώώ = = ΔΘΔΘ / / ΔΔtt

units: rad/sunits: rad/s

• Angular Acceleration: “amount of Angular Acceleration: “amount of change in twist per time”change in twist per time”

α = α = ΔώΔώ / / ΔΔtt

Units: rad/sUnits: rad/s22

Linking Linear with Angular…Linking Linear with Angular…

vvt t = = ΔΔx/x/ΔΔt = C/T = 2t = C/T = 2ππr/Tr/T

1 rev = 21 rev = 2ππ radians = 6.28 radians radians = 6.28 radians

Ɵ = S / rƟ = S / r

ωω = v = vTT / r / r

αα = a = aTT / r / r

S

Centripetal and Radial AccelCentripetal and Radial Accel

Centripetal or radial accelerationCentripetal or radial acceleration

aac = a = arr = v = v22/r in linear units /r in linear units

aac c = a= arr = r = rώώ22 in angular units in angular units

Clockwise (CW) vs. Clockwise (CW) vs. Counterclockwise (CCW)Counterclockwise (CCW)

CCW is +, CW is – throughout courseCCW is +, CW is – throughout course

2. Finding radial acceleration-2. Finding radial acceleration-angular unitsangular units

What is the radial acceleration of a What is the radial acceleration of a motorcycle proceeding in a circular motorcycle proceeding in a circular cage of diameter 14.0 m if it is cage of diameter 14.0 m if it is moving at an angular velocity of 2.0 moving at an angular velocity of 2.0 rad/s?rad/s?

28 rads/s28 rads/s22

1. Finding radial acceleration-1. Finding radial acceleration-linear unitslinear units

What is the radial acceleration of an What is the radial acceleration of an object that spins with a linear speed object that spins with a linear speed of 4.00 m/s at a 0.80 m distance of 4.00 m/s at a 0.80 m distance from the axis of rotation?from the axis of rotation?

20 m/s20 m/s22 or 20 rads/s or 20 rads/s22

A closer look at acceleration…A closer look at acceleration…

// to motion: tangential // to motion: tangential accelerationacceleration– Leads to more/less rpmLeads to more/less rpm

Perp. to motion: radial Perp. to motion: radial acceleration or acceleration or centripetal accelerationcentripetal acceleration– Gives inward Gives inward

acceleration needed to acceleration needed to make circular motionmake circular motion

Acceleration can be broken downAcceleration can be broken down

Galileo’s Formulas Work for Galileo’s Formulas Work for Rotation!!!!Rotation!!!!

Angular displacement: Angular displacement: ΘΘ (# radians spun) (# radians spun) radradAngular velocity: Angular velocity: ώώ (radians spun per second) (radians spun per second)

rad/srad/sAngular acceleration: Angular acceleration: αα ( (ΔΔ in rps per sec) in rps per sec) rad/srad/s22

VVff = v = vii + a + aΔΔtt ώώff = = ώώ + + αΔαΔtt

VVff22 = V = Vii

22 + 2ax + 2ax ώώff2 2 = = ώώii

22 + 2 + 2αΘαΘ

x = vx = viit + ½att + ½at22 ΘΘ = = ώώiit + 1/2t + 1/2ααtt22

Rotation and GraphingRotation and Graphing

Θ for x

ώ for v

α for a

SLOPES TO GO UP

AREAS UNDER CURVE TO GO DOWN

3. What is angular acceleration?3. What is angular acceleration?

An piece of tape at the edge of the An piece of tape at the edge of the disk of radius 0.19 m slows from a disk of radius 0.19 m slows from a speed of 2.00 m/s to a speed of 1.00 speed of 2.00 m/s to a speed of 1.00 m/s in a 6.00 second time frame. m/s in a 6.00 second time frame. What is the angular acceleration of What is the angular acceleration of the tape? the tape?

What is the angular acceleration of What is the angular acceleration of the inside edge of the disk (r = 0.03 the inside edge of the disk (r = 0.03 m)?m)?

4. What is the final angular 4. What is the final angular velocity?velocity?

An object being swung around with a An object being swung around with a radius of 0.50 m is subject to an radius of 0.50 m is subject to an angular acceleration of 0.25 rad/sangular acceleration of 0.25 rad/s22 for a for a 3.00 second time frame. If it started 3.00 second time frame. If it started with a speed of 8.00 m/s, what is its with a speed of 8.00 m/s, what is its angular velocity after the three seconds angular velocity after the three seconds have elapsed?have elapsed?

What number of rotations will have occurred in this 3.0 sec time span?

5. What is the angular 5. What is the angular acceleration?acceleration?

A disk is spun for a total of 62 A disk is spun for a total of 62 rotations. It begins at an angular rotations. It begins at an angular velocity of 1.8 rad/s and due to velocity of 1.8 rad/s and due to friction this value drops to 1.2 rad/s friction this value drops to 1.2 rad/s by the 62by the 62ndnd rotation. What is this rotation. What is this disk’s angular acceleration?disk’s angular acceleration?

Moment of InertiaMoment of Inertia

““Inertia of Spin”Inertia of Spin”Measure of the unwillingness of a Measure of the unwillingness of a material to want to rotatematerial to want to rotateIncreases with greater overall mass, no Increases with greater overall mass, no matter what the shape of the materialmatter what the shape of the materialShape affects momentShape affects momentIf all mass on outside of orbit, If all mass on outside of orbit,

I = mrI = mr22

Others p. 298Others p. 298

What has more kinetic energy, What has more kinetic energy, a knuckleball pitched at 70 a knuckleball pitched at 70

mph, or a changeup pitched at mph, or a changeup pitched at 70 mph?70 mph?

Close but not identical!!!!!!Close but not identical!!!!!!

Rotational Kinetic EnergyRotational Kinetic Energy

““Energy of Spin” as opposed to Energy of Spin” as opposed to “Energy of Motion”“Energy of Motion”

KEKErotrot = ½ I = ½ I ωω22

Objects that are rolling tend to have Objects that are rolling tend to have both forms of KE!both forms of KE!

Example: Total KEExample: Total KE

What is the total KE of a basketball What is the total KE of a basketball (0.60 kg, diameter 30.0 cm) that is (0.60 kg, diameter 30.0 cm) that is rolling at a speed of 2.00 m/s on the rolling at a speed of 2.00 m/s on the ground?ground?

Moment of InertiaMoment of Inertia

Moment of inertia generally higher Moment of inertia generally higher the more mass is distributed further the more mass is distributed further from the axis of rotationfrom the axis of rotation

Moment of Moment of inertia inertia changes for changes for the same the same material material according according to what to what spin axis spin axis you choose you choose for it.for it.

Shapes & Moments of InertiaShapes & Moments of Inertia

The disk The disk cylinder has cylinder has a lower a lower moment due moment due fact that fact that mass closer mass closer to axis of to axis of rotation is rotation is easier to easier to rotate rotate

ΤΤ = F = F ll

T = I T = I αα

Extras: Ch 10-11Extras: Ch 10-11Angular Momentum: tendency for a Angular Momentum: tendency for a

system to rotatesystem to rotateL = I L = I ώώL = r m vL = r m vunits are kg munits are kg m22/s/s

Angular Momentum for a system is Angular Momentum for a system is conservedconserved

Figure skatersFigure skatersElliptical orbitsElliptical orbitsBike tireBike tire