Chapter 8Chapter 8

Rotational Rotational MotionMotion

Forces and circular motionForces and circular motion

Circular motion = Circular motion = accelerated motion accelerated motion (direction changing)(direction changing)Centripetal acceleration Centripetal acceleration presentpresentCentripetal force must be Centripetal force must be actingactingCentrifugal force - Centrifugal force - apparent outward tug as apparent outward tug as direction changesdirection changesCentripetal force ends: Centripetal force ends: motion = straight linemotion = straight line

ac = v2

r

Fc =mac = mv2

r

Direction ofMotion

Centrifugal Force

CentripetalForce

Centripetal ForceCentripetal Force……has different origin (friction, has different origin (friction, tension, gravity, etc.).tension, gravity, etc.).

Centripetal means "center seeking".Centripetal means "center seeking".

Centrifugal ForceCentrifugal Force(not a real force)(not a real force)

……results from a natural tendencyto results from a natural tendencyto keep a state of motion (inertia).keep a state of motion (inertia).

Centrifugal means "center fleeing".Centrifugal means "center fleeing".

What is that force that throws you to the right if you turn to the left in your car?

centrifugal force.

What is that force that keeps you in your seat when you turn left in your car?

centripetal force.

ExamplesExamples

water in bucketwater in bucket

moon and earthmoon and earth

car on circular pathcar on circular path

coin on a hangercoin on a hanger

jogging in a space stationjogging in a space station

Centripetal Centripetal ForceForce

BucketBucket

Earth’s Earth’s gravitygravity

Road FrictionRoad Friction

HangerHanger

Space Space Station FloorStation Floor

Centrifugal Centrifugal ForceForce

InertiaInertia

InertiaInertia

InertiaInertia

InertiaInertia

InertiaInertia

Circular MotionCircular MotionLinear speedLinear speed - the distance moved per - the distance moved per unit time. Also called simply speed.unit time. Also called simply speed.

Rotational speedRotational speed - the number of - the number of rotations or revolutions per unit time.rotations or revolutions per unit time.

Rotational speed is often measured in Rotational speed is often measured in revolutions per minuterevolutions per minute (RPM). (RPM).

Angular Position, Velocity, and Acceleration

Angular Position, Velocity, and Acceleration

Degrees and revolutions:

Angular Position, Velocity, and Acceleration

Arc length s, measured in radians:

Connections Between Linear and Rotational Quantities

Connections Between Linear and Rotational Quantities

The linear speed is directly The linear speed is directly proportional to both rotational speed proportional to both rotational speed and radial distance. and radial distance.

v = v = r r

What are two ways that you can What are two ways that you can increase your linear speed on a increase your linear speed on a rotating platform?rotating platform?– Answers: Answers:

Move away from the rotation axis.Move away from the rotation axis.

Have the platform spin faster.Have the platform spin faster.

Connections Between Linear and Rotational Quantities

Connections Between Linear and Rotational Quantities

This merry-go-round has both tangential and centripetal acceleration.

Center of MassCenter of MassThe center of mass of an object is the average The center of mass of an object is the average position of mass.position of mass.

Objects tend to rotate about their center of mass.Objects tend to rotate about their center of mass.

Examples: Examples: Meter stickMeter stick

Map of TexasMap of Texas

Rotating HammerRotating Hammer

Center of Mass and Balance

If an extended object is to be balanced, it must be supported through its center of mass.

Center of Mass and Balance

This fact can be used to find the center of mass of an object – suspend it from different axes and trace a vertical line. The center of mass is where the lines meet.

Rotational InertiaRotational Inertia An object rotating about an axis tends to An object rotating about an axis tends to remain rotating unless interfered with by some remain rotating unless interfered with by some external influence.external influence.

This influence is called This influence is called torquetorque..

Rotation adds stability to linear motion.Rotation adds stability to linear motion.– Examples: Examples:

spinning footballspinning footballbicycle tiresbicycle tiresFrisbee Frisbee

The greater the distance between the The greater the distance between the bulk of an object's mass and its axis of bulk of an object's mass and its axis of rotation, the greater the rotation, the greater the rotational rotational inertiainertia..

Examples: Examples: – Tightrope walkerTightrope walker– Inertia BarsInertia Bars– Ring and Disk on an InclineRing and Disk on an Incline– MetronomeMetronome

Torque

From experience, we know that the same force will be much more effective at rotating an object such as a nut or a door if our hand is not too close to the axis.

This is why we have long-handled wrenches, and why doorknobs are not next to hinges.

TorqueTorque

Torque is the product of the force and Torque is the product of the force and lever-arm distance, which tends to lever-arm distance, which tends to produce rotation.produce rotation.

Torque = force Torque = force lever arm lever arm– Examples: Examples:

wrencheswrenches

see-saws see-saws

We define a quantity called torque:

The torque increases as the force increases, and also as the distance increases.

Only the tangential component of force causes a torque:

StabilityStabilityFor stability center of gravity must be over area of For stability center of gravity must be over area of support.support.

Examples: Examples: Tower of PisaTower of Pisa

Touching toes with back to wallTouching toes with back to wall

Meter stick over the edgeMeter stick over the edge

Rolling Double-ConeRolling Double-Cone

Conservation of Angular Conservation of Angular MomentumMomentum

angular momentum = rotational inertia angular momentum = rotational inertia rotational velocity rotational velocity

L = I L = I

Newton's first law for rotating systems: Newton's first law for rotating systems: – ““A body will maintain its state of angular A body will maintain its state of angular

momentum unless acted upon by an unbalanced momentum unless acted upon by an unbalanced external torque.”external torque.”

Conservation of Angular Momentum

If the net external torque on a system is zero, the angular momentum is conserved.

The most interesting consequences occur in systems that are able to change shape:

Examples: Examples: – 1. ice skater spin1. ice skater spin– 2. cat dropped on back2. cat dropped on back– 3. Diving into water3. Diving into water– 4. Collapsing Stars (neutron stars)4. Collapsing Stars (neutron stars)

Example QuestionExample QuestionTwo ladybugs are sitting on a phonograph Two ladybugs are sitting on a phonograph record that rotates at 33 1/3 RPM. record that rotates at 33 1/3 RPM.

1. Which ladybug has a great linear speed?1. Which ladybug has a great linear speed?

A. The one closer to the center.A. The one closer to the center.

B. The one on the outside edge.B. The one on the outside edge.

C. The both have the same linear C. The both have the same linear speedspeed

Example QuestionExample QuestionTwo ladybugs are sitting on a phonograph Two ladybugs are sitting on a phonograph record that rotates at 33 1/3 RPM. record that rotates at 33 1/3 RPM.

1. Which ladybug has a great linear speed?1. Which ladybug has a great linear speed?

A. The one closer to the center.A. The one closer to the center.

B. The one on the outside edge.B. The one on the outside edge.

C. The both have the same linear C. The both have the same linear speedspeed

Example QuestionExample Question

Two ladybugs are sitting on a phonograph record Two ladybugs are sitting on a phonograph record that rotates at 33 1/3 RPM. that rotates at 33 1/3 RPM.

2. Which ladybug has a great rotational speed?2. Which ladybug has a great rotational speed?A. The one closer to the center.A. The one closer to the center.B. The one on the outside edge.B. The one on the outside edge.C. The both have the same rotational speedC. The both have the same rotational speed

You sit on a rotating platform halfway between You sit on a rotating platform halfway between the rotating axis and the outer edge.the rotating axis and the outer edge.

You have a rotational speed of 20 RPM and a You have a rotational speed of 20 RPM and a tangential speed of 2 m/s.tangential speed of 2 m/s.

What will be the linear speed of your friend What will be the linear speed of your friend who sit at the outer edge?who sit at the outer edge?

Example QuestionExample Question

You sit on a rotating platform halfway between the You sit on a rotating platform halfway between the rotating axis and the outer edge.rotating axis and the outer edge.

You have a rotational speed of 20 RPM and a You have a rotational speed of 20 RPM and a tangential speed of 2 m/s.tangential speed of 2 m/s.

What will be the linear speed of your friend who sit at What will be the linear speed of your friend who sit at the outer edge?the outer edge?

A. 4m/sA. 4m/sB. 2m/sB. 2m/sC. 20 RPMC. 20 RPMD. 40 RPMD. 40 RPME. None of theseE. None of these

Example QuestionExample Question