PhysicsPhysicsChapter 8 – Rotational Motion Part 1

Circular MotionCircular Motion Tangential SpeedTangential Speed – The linear speed of something – The linear speed of something

moving along a circular path.moving along a circular path. Symbol is the usual Symbol is the usual vv and units are m/s and units are m/s

Rotational SpeedRotational Speed – Number of revolutions per unit of – Number of revolutions per unit of time.time. Symbol is omega, Symbol is omega, ωω and units are RPM (rotations per and units are RPM (rotations per

minute)minute)

Circular MotionCircular Motion Does Does tangential speedtangential speed depend on where the bug depend on where the bug

sits on the record?sits on the record?

Does Does rotational speedrotational speed depend on where the bug sits depend on where the bug sits on the record?on the record?

Circular MotionCircular Motion Is there a relationship between Is there a relationship between vv and and ωω??

v ~ r v ~ r ωω

Question: Who is moving at a greater tangential Question: Who is moving at a greater tangential speed: someone at the equator or someone in speed: someone at the equator or someone in Winchester, Virginia?Winchester, Virginia?

Circular MotionCircular Motion

Check QuestionCheck Question At an amusement park, you and a friend sit on a At an amusement park, you and a friend sit on a

large rotating disk. You sit at the edge and have a large rotating disk. You sit at the edge and have a rotational speed of 4 RPM and a linear speed of 6 rotational speed of 4 RPM and a linear speed of 6 m/s. Your friend sits halfway to the center. What is m/s. Your friend sits halfway to the center. What is her rotational speed? What is her linear speed?her rotational speed? What is her linear speed?

Wheels on TrainsWheels on Trains

Rotational InertiaRotational Inertia

The greater the distance between an object’s mass The greater the distance between an object’s mass concentration and the axis, the greater the rotational concentration and the axis, the greater the rotational inertia.inertia.

An object rotating about an axis tends An object rotating about an axis tends to remain rotating about the same axis to remain rotating about the same axis unless interfered with by some external unless interfered with by some external

influence.influence.

Rotational InertiaRotational Inertia

Rotational InertiaRotational Inertia You bend your legs when you run because bent You bend your legs when you run because bent

legs are much easier to swing quickly back and legs are much easier to swing quickly back and forth.forth.

Rotational InertiaRotational Inertia

Rotational InertiaRotational Inertia A short pendulum has a shorter period, A short pendulum has a shorter period, TT, than a , than a

longer pendulum.longer pendulum.

Rotational InertiaRotational Inertia For similar mass distributions, short legs have less For similar mass distributions, short legs have less

rotational inertia than long legs. rotational inertia than long legs.

Rotational InertiaRotational Inertia So how can we calculate rotational inertia?So how can we calculate rotational inertia?

In the In the special case when all the mass special case when all the mass mm of an object of an object is concentrated at the same distance is concentrated at the same distance rr from the from the rotational axis: rotational axis:

I I == mr mr22

When the mass is more spread out, the rotational inertia When the mass is more spread out, the rotational inertia is less and the formula is different. is less and the formula is different.

Rotational InertiaRotational Inertia

Rotational InertiaRotational Inertia

Which will roll down an incline with greater Which will roll down an incline with greater acceleration, a acceleration, a hollow cylinderhollow cylinder or a or a solid cylindersolid cylinder of of the same mass and radius? the same mass and radius?

Rotational InertiaRotational Inertia

It turns out that It turns out that anyany solidsolid cylinder will cylinder will roll down an incline with more roll down an incline with more acceleration than acceleration than anyany hollowhollow cylinder, cylinder, regardless of mass or radius. regardless of mass or radius.

A A hollowhollow cylinder has more “laziness cylinder has more “laziness per massper mass” than a solid cylinder.” than a solid cylinder.

Check QuestionCheck Question

A A heavyheavy iron cylinder and a iron cylinder and a lightlight wooden cylinder, wooden cylinder, similar in shape, roll down an incline. Which will similar in shape, roll down an incline. Which will have more acceleration?have more acceleration?

They have the same rotational inertia per mass so They have the same rotational inertia per mass so they’ll accelerate equally!they’ll accelerate equally!

Rotational InertiaRotational Inertia Rotational inertia can be used to advantage. Rotational inertia can be used to advantage. Industrial flywheelsIndustrial flywheels

TorqueTorque Torque is the rotational counterpart to ForceTorque is the rotational counterpart to Force

Apply a Apply a forceforce if you want if you want linear motionlinear motion (acceleration) (acceleration) Apply a Apply a torquetorque if you want if you want rotationrotation..

TorqueTorque Now we have to expand our definition of Mechanical Now we have to expand our definition of Mechanical

Equilibrium!Equilibrium!

∑ ∑ F = 0 …F = 0 …

… … AND…AND…

∑ ∑ ττ = 0 = 0

TorqueTorque

Let’s consider a door:Let’s consider a door:

TorqueTorque

When the force is When the force is perpendicularperpendicular, the distance from , the distance from the turning axis to the point of contact is called the the turning axis to the point of contact is called the lever armlever arm. .

TorqueTorque The same torque can be produced by a large force The same torque can be produced by a large force

with a short lever arm, or a small force with a long with a short lever arm, or a small force with a long lever arm. lever arm.

Check QuestionCheck Question If you cannot exert enough torque to turn a stubborn If you cannot exert enough torque to turn a stubborn

bolt, would more torque be produced if you fastened bolt, would more torque be produced if you fastened a length of rope to the wrench handle as shown?a length of rope to the wrench handle as shown?

Check QuestionCheck Question Extending the length of the wrench handle by Extending the length of the wrench handle by

placing a pipe over it will get you more torque for the placing a pipe over it will get you more torque for the same force because you increased the lever arm!same force because you increased the lever arm!

Balanced TorquesBalanced Torques A pair of torques can balance each other. Balance is A pair of torques can balance each other. Balance is

achieved if the torque that tends to produce achieved if the torque that tends to produce clockwise rotation by the boy equals the torque that clockwise rotation by the boy equals the torque that tends to produce counterclockwise rotation by the tends to produce counterclockwise rotation by the girl.girl.

Check QuestionCheck Question This meter stick is suspended in mechanical This meter stick is suspended in mechanical

equilibrium. What is the weight of the block hanging equilibrium. What is the weight of the block hanging at the 10 cm mark?at the 10 cm mark?

Balanced TorquesBalanced Torques

Scale balances that Scale balances that work with sliding work with sliding weights are based weights are based on on balanced torquesbalanced torques, , not balanced not balanced masses. masses.

Center of MassCenter of Mass A baseball thrown into the air follows a smooth A baseball thrown into the air follows a smooth

parabolic path.parabolic path.

Center of MassCenter of Mass A bat thrown in the air, however, A bat thrown in the air, however, wobbles about a wobbles about a

special pointspecial point. . This point stays on a parabolic pathThis point stays on a parabolic path, , even though the rest of the bat does not.even though the rest of the bat does not.

Center of MassCenter of Mass

Center of Mass is the average Center of Mass is the average position of all position of all massmass that makes that makes

up the objectup the object

Center of Gravity is the average Center of Gravity is the average position of all the particles of position of all the particles of

weightweight that make up an object that make up an object

Center of MassCenter of Mass Location of the Center of MassLocation of the Center of Mass

Center of MassCenter of Mass

Objects not made of the same material Objects not made of the same material throughout may have the center of throughout may have the center of mass quite far from the geometric mass quite far from the geometric center. center.

Consider a hollow ball half filled with Consider a hollow ball half filled with lead. The center of mass would be lead. The center of mass would be located somewhere within the lead part. located somewhere within the lead part.

The ball will always roll to a stop with its The ball will always roll to a stop with its center of mass as low as possible. center of mass as low as possible.

Center of MassCenter of Mass Motion About the Center of MassMotion About the Center of Mass

As an object slides across a surface, its center of As an object slides across a surface, its center of mass follows a straight-line path. mass follows a straight-line path.

Spin Spin A force must be applied to the edge of an object for it to

spin. a.a. If the football is kicked If the football is kicked in line with its center of massin line with its center of mass, ,

it will move without rotating.it will move without rotating.b.b. If it is kicked If it is kicked above or below its center of massabove or below its center of mass, it will , it will

rotate. rotate.

Center of GravityCenter of Gravity So how can we precisely locate the center of gravity So how can we precisely locate the center of gravity

(CG)?(CG)? The CG of a uniform object is at the midpoint, its The CG of a uniform object is at the midpoint, its

geometric center. geometric center. • The CG is the balance point. The CG is the balance point. • Supporting that single point supports the whole object. Supporting that single point supports the whole object.

Center of GravityCenter of Gravity The CG of an object may be located where no The CG of an object may be located where no

actual material exists.actual material exists.

Check QuestionCheck Question A uniform meter stick supported at the 25-cm mark A uniform meter stick supported at the 25-cm mark

balances when a 1 kg rock is suspended at the 0-balances when a 1 kg rock is suspended at the 0-cm end. What is the mass of the meterstick?cm end. What is the mass of the meterstick?

Torque & Center of GravityTorque & Center of Gravity If the center of gravity of an object is above the area of

support, the object will remain upright.

If the CG extends outside the area of support, an unbalanced torque exists, and the object will topple.

Torque & Center of GravityTorque & Center of Gravity

Torque & Center of GravityTorque & Center of Gravity The Leaning Tower The Leaning Tower

of Pisa does not of Pisa does not topple because its topple because its CG does not extend CG does not extend beyond its base. beyond its base.

Torque & CGTorque & CG

BalancingBalancing Question: Which way do you suppose it will be Question: Which way do you suppose it will be

easier to balance a baseball bat on your finger?easier to balance a baseball bat on your finger?

QuestionQuestion

Where is the CG of a person?

Standing normally, it is usually 2 to 3 cm below your navel, and midway between your front and back.

But it can change!

Your CG may be located outside your body altogether.

Center of GravityCenter of Gravity

Center of GravityCenter of Gravity

A high jumper executes a “Fosbury flop” to clear the bar while his CG nearly passes beneath the bar.

Center of GravityCenter of Gravity

When you stand, your CG is somewhere above your support base, the area bounded by your feet.

StabilityStability When an object is toppled, the center of gravity of

that object is raised, lowered, or unchanged.

a. Equilibrium is unstable when the CG is lowered with displacement.

b. Equilibrium is stable when work must be done to raise the CG.

c. Equilibrium is neutral when displacement neither raises nor lowers the CG.

StabilityStability

StabilityStability The CG of a building is

lowered if much of the structure is below ground level.

This is important for tall, narrow structures.

Rolling SpoolsRolling Spools