Circular Motion, Center of Circular Motion, Center of Gravity, & Rotational MechanicsGravity, & Rotational Mechanics

Chapters 9, 10, & 11Chapters 9, 10, & 11

Rotation and RevolutionRotation and Revolution

Axis –Axis – straight line around which rotation straight line around which rotation occursoccurs

Rotation –Rotation – object turns about an object turns about an internalinternal axis (Earth axis (Earth rotatesrotates around its axis) around its axis)

Revolution –Revolution – when an object turns about when an object turns about an an externalexternal axis (Earth axis (Earth revolvesrevolves around around the sun)the sun)

RotationRotation

RevolutionRevolution

Rotational SpeedRotational Speed

The The linear speedlinear speed of an object is greater near the of an object is greater near the outer edge of a rotating object than on the inner outer edge of a rotating object than on the inner edge of the rotating objectedge of the rotating object

Tangential SpeedTangential Speed – the speed of something – the speed of something moving along a circular path (the direction of moving along a circular path (the direction of motion is always tangent to the circle)motion is always tangent to the circle)

Rotational SpeedRotational Speed ( angular speed) – the ( angular speed) – the number of rotations per unit of time (expressed number of rotations per unit of time (expressed in revolutions per minute or RPM)in revolutions per minute or RPM)

All parts of a rotating object rotate about their All parts of a rotating object rotate about their axis axis in thein the same amount of timesame amount of time!!

Uniform Circular MotionUniform Circular Motion

Tangential speed and rotational speed are Tangential speed and rotational speed are related related

Tangential Speed ~ Radial Distance x Rotational SpeedTangential Speed ~ Radial Distance x Rotational Speed

As you move away from the center of a rotating As you move away from the center of a rotating object, the tangential speed will increase while object, the tangential speed will increase while your rotational speed stays the sameyour rotational speed stays the same

Centripetal ForceCentripetal Force

Centripetal ForceCentripetal Force – – anyany force that causes force that causes an object to follow a circular pathan object to follow a circular path

When a car goes around a corner, the When a car goes around a corner, the friction between the tires and the road friction between the tires and the road provides the centripetal force needed to provides the centripetal force needed to keep the car going around the curvekeep the car going around the curve

If not for the friction of the tires, the car If not for the friction of the tires, the car would continue moving in the straight-line would continue moving in the straight-line pathpath

Centripetal ForceCentripetal Force

To Have or Not to Have Centripetal To Have or Not to Have Centripetal ForceForce

Without Centripetal Force With Centripetal Force

Centrifugal ForceCentrifugal Force

Centrifugal ForceCentrifugal Force – the outward force – the outward force associated with circular motionassociated with circular motion

It is not a true force, but rather the effect It is not a true force, but rather the effect that inertia tries to place on you as you that inertia tries to place on you as you follow a circular pathfollow a circular path

From Newton’s First Law, the natural path From Newton’s First Law, the natural path of an object is a straight-line, the of an object is a straight-line, the centripetal forcecentripetal force is what keeps you going is what keeps you going in a circlein a circle

Centrifugal ForceCentrifugal Force

Center of GravityCenter of GravityCenter of Gravity – Center of Gravity – point located at the point located at the

object’s average position of weightobject’s average position of weightFor a symmetrical object, it is the For a symmetrical object, it is the

geometric center of the objectgeometric center of the objectFor an irregularly shaped object, there is For an irregularly shaped object, there is

more weight on end than the other, so the more weight on end than the other, so the center of gravity is toward the heavier endcenter of gravity is toward the heavier end

Objects not made of the same material Objects not made of the same material throughout (different densities), may have throughout (different densities), may have the center of gravity very far from the the center of gravity very far from the geometric centergeometric center

Center of GravityCenter of Gravity

Center of MassCenter of Mass

Center of Mass – Center of Mass – the average position of all the the average position of all the particles of particles of massmass that make up an object that make up an object

For almost all objects on or near Earth, center of For almost all objects on or near Earth, center of gravity and center of mass are interchangeablegravity and center of mass are interchangeable

If you threw an object in the air, you’d see it If you threw an object in the air, you’d see it wobble around its center of gravitywobble around its center of gravity

The sun wobbles also!The sun wobbles also! As the planets move around the sun, they As the planets move around the sun, they

contribute to the overall center of mass of the contribute to the overall center of mass of the solar system, so the sun wobbles off centersolar system, so the sun wobbles off center

This is how astronomers look for planets orbiting This is how astronomers look for planets orbiting other stars! other stars!

Center of Mass in a Star SystemCenter of Mass in a Star System

Locating the Center of GravityLocating the Center of Gravity

The center of gravity is the balance point, The center of gravity is the balance point, supporting that single point supports the supporting that single point supports the whole objectwhole object

If you suspend any object at a single point, If you suspend any object at a single point, the center of gravity for that object will the center of gravity for that object will hang directly below (or at) the point of hang directly below (or at) the point of suspensionsuspension

The center of gravity may be located The center of gravity may be located where no actual material exists (i.e. a ring)where no actual material exists (i.e. a ring)

Locating the Center of GravityLocating the Center of Gravity

TopplingToppling

If the CG of an object is above the area of If the CG of an object is above the area of support, it will remain uprightsupport, it will remain upright

If the CG extends outside the area of If the CG extends outside the area of support, the object will topplesupport, the object will topple

The Leaning Tower of Pisa does not The Leaning Tower of Pisa does not topple because its CG does not extend topple because its CG does not extend beyond its basebeyond its base

TopplingToppling

StabilityStability

Unstable Equilibrium – Unstable Equilibrium – an object an object balanced so that any displacement lowers balanced so that any displacement lowers its center of gravityits center of gravity

Stable Equilibrium – Stable Equilibrium – an object balanced an object balanced so that any displacement raises its center so that any displacement raises its center of gravity (requires work)of gravity (requires work)

Neutral Equilibrium – Neutral Equilibrium – an object balanced an object balanced to that its center of gravity is neither raised to that its center of gravity is neither raised nor lowered with displacementnor lowered with displacement

StabilityStability

Center of Gravity of PeopleCenter of Gravity of People

When you stand upright with your arms hanging When you stand upright with your arms hanging at your sides, your CG is within your body, at your sides, your CG is within your body, typically 2 to 3 cm below your belly buttontypically 2 to 3 cm below your belly button

The CG is slightly lower in women than in men, The CG is slightly lower in women than in men, because women tend to be proportionally larger because women tend to be proportionally larger in the pelvis and smaller in the shouldersin the pelvis and smaller in the shoulders

When you stand, your CG is somewhere above When you stand, your CG is somewhere above the support base of your feet, we spread them the support base of your feet, we spread them further apart in unstable situations (the bus)further apart in unstable situations (the bus)

When you bend over to touch your toes, you are When you bend over to touch your toes, you are unconsciously extending the lower part of your unconsciously extending the lower part of your body, putting your CG outside of your body (so body, putting your CG outside of your body (so you won’t topple over!)you won’t topple over!)

Center of Gravity of PeopleCenter of Gravity of People

TorqueTorque A torque is produced when a force is applied A torque is produced when a force is applied

with “leverage”with “leverage” You use leverage when you use a screwdriver to You use leverage when you use a screwdriver to

open a can of paintopen a can of paint The direction of your applied force is important, The direction of your applied force is important,

you would never try to open a door with a you would never try to open a door with a doorknob by push or pulling the doorknob doorknob by push or pulling the doorknob sidewayssideways

You apply your force PERPENDICULAR to the You apply your force PERPENDICULAR to the plane of the doorplane of the door

Torque = force┴ x lever armTorque = force┴ x lever arm Greater torques are produced when both the Greater torques are produced when both the

force and lever arm are largeforce and lever arm are large

TorqueTorque

Torque and Center of GravityTorque and Center of Gravity

If the direction of force is through the CG If the direction of force is through the CG of the projectile, all the force can do is of the projectile, all the force can do is move the object as a whole; there will be move the object as a whole; there will be no torque to turn the projectileno torque to turn the projectile

If the force is directed “off center”, then in If the force is directed “off center”, then in addition to motion of the CG, the projectile addition to motion of the CG, the projectile will rotate about its CGwill rotate about its CG

Rotational InertiaRotational Inertia

An object rotating about an axis tends to An object rotating about an axis tends to keep rotating about that axis (look keep rotating about that axis (look familiar?)familiar?)

Rotational Inertia – Rotational Inertia – the resistance of an the resistance of an object to changes in its rotational motionobject to changes in its rotational motion

A torque is required to change the A torque is required to change the rotational state of motion of an objectrotational state of motion of an object

Rotational inertia depends on the Rotational inertia depends on the distribution of the mass of an objectdistribution of the mass of an object

Rotational InertiaRotational Inertia

Rotational Inertia and GymnasticsRotational Inertia and Gymnastics Rotational inertia is least about the vertical head-to-toe Rotational inertia is least about the vertical head-to-toe

axis (longitudinal) on any person, because most of the axis (longitudinal) on any person, because most of the mass is concentrated theremass is concentrated there

A rotation of your body along this axis is easiestA rotation of your body along this axis is easiest The rotational inertia when your arms are extended is The rotational inertia when your arms are extended is

3 times greater than when your arms are pulled in3 times greater than when your arms are pulled in You rotate about your transverse axis when you do a You rotate about your transverse axis when you do a

flip or somersaultflip or somersault The rotational inertia of a gymnast is up to 20 times The rotational inertia of a gymnast is up to 20 times

greater when she is swinging in a fully extended greater when she is swinging in a fully extended position from a horizontal bar than after dismount position from a horizontal bar than after dismount when she somersaults in a tucked position (when she when she somersaults in a tucked position (when she let goes and tucks, she is automatically increasing her let goes and tucks, she is automatically increasing her rate of rotation by 20 times!)rate of rotation by 20 times!)

Rotational Inertia and GymnasticsRotational Inertia and Gymnastics

Angular MomentumAngular Momentum Angular Momentum – Angular Momentum – the “inertia of rotation” of the “inertia of rotation” of

rotating objectsrotating objects Like Like linear momentumlinear momentum, angular momentum is a vector , angular momentum is a vector

quantityquantity Rotational Velocity – Rotational Velocity – when a direction is assigned to when a direction is assigned to

rotational speedrotational speedAngular momentum = rotational inertia (I) x rotational velocity (Angular momentum = rotational inertia (I) x rotational velocity (ωω))

Angular momentum = mass (m) x velocity (v) x radius (r)Angular momentum = mass (m) x velocity (v) x radius (r) Newton’s 1Newton’s 1stst Law can now be restated for angular Law can now be restated for angular

momentum:momentum:An object or system of objects will maintain its angular An object or system of objects will maintain its angular

momentum unless acted upon by an unbalanced momentum unless acted upon by an unbalanced external torqueexternal torque

Angular MomentumAngular Momentum

Conservation of Angular Conservation of Angular MomentumMomentum

The Law of Conservation of The Law of Conservation of Momentum:Momentum:If no unbalanced external torque acts on a If no unbalanced external torque acts on a rotating system, the angular momentum of rotating system, the angular momentum of

that system is constant.that system is constant.With no external torque, the product of With no external torque, the product of

rotational inertia and rotational velocity at rotational inertia and rotational velocity at one time will be the same at any other one time will be the same at any other timetime

Conservation of Angular Conservation of Angular MomentumMomentum

AssignmentAssignment

Read Ch. 9-11 (pg. 122-164)Read Ch. 9-11 (pg. 122-164)Ch. 9: Do #31-38 (pg. 135), Appendix F Ch. 9: Do #31-38 (pg. 135), Appendix F

#1-7 (pg. 674)#1-7 (pg. 674)Ch. 10: Do #21-34 (pg. 148-149)Ch. 10: Do #21-34 (pg. 148-149)Ch. 11: Do #33-40 (pg. 167)Ch. 11: Do #33-40 (pg. 167)