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

MechanicsMechanics

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

Rotation and RevolutionRotation and Revolution Axis –Axis – straight line around which straight line around which

rotation occursrotation occurs Rotation –Rotation – object turns about an object turns about an

internalinternal axis (Earth axis (Earth rotatesrotates around its around its axis)axis)

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

RotationRotation

RevolutionRevolution

Rotational SpeedRotational Speed The The linear speedlinear speed of an object is greater of an object is greater

near the outer edge of a rotating object near the outer edge of a rotating object than on the inner edge of the rotating than on the inner edge of the rotating objectobject

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

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

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

Uniform Circular MotionUniform Circular Motion

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

Tangential Speed ~ Radial Distance x Rotational Tangential Speed ~ Radial Distance x Rotational SpeedSpeed

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

Centripetal ForceCentripetal Force Centripetal ForceCentripetal Force – – anyany force that force that

causes an object to follow a circular causes an object to follow a circular pathpath

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

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

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 – the outward

force associated with circular motionforce associated with circular motion It is not a true force, but rather the It is not a true force, but rather the

effect that inertia tries to place on effect that inertia tries to place on you as you follow a circular pathyou as you follow a circular path

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

Centrifugal ForceCentrifugal Force

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

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

geometric center of the objectgeometric center of the object For 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 the average position of

all the particles of all the particles of massmass that make up an that make up an objectobject

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

If you threw an object in the air, you’d see If you threw an object in the air, you’d see it wobble around its center of gravityit wobble 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 contribute to the overall center of mass of the solar system, so the sun wobbles off the solar system, so the sun wobbles off centercenter

This is how astronomers look for planets This is how astronomers look for planets orbiting other stars! orbiting 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 The center of gravity is the balance

point, supporting that single point point, supporting that single point supports the whole objectsupports the whole object

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

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

Locating the Center of GravityLocating the Center of Gravity

TopplingToppling If the CG of an object is above the If the CG of an object is above the

area of support, it will remain uprightarea of support, 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 topple because its CG does not extend beyond its baseextend beyond 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 an object balanced so that its center of gravity is balanced so that its center of gravity is neither raised nor lowered with neither raised nor lowered with displacementdisplacement

StabilityStability

Center of Gravity of PeopleCenter of Gravity of People

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

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

Center of Gravity of PeopleCenter of Gravity of People

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

applied with “leverage”applied with “leverage” You use leverage when you use a You use leverage when you use a

screwdriver to open a can of paintscrewdriver to open a can of paint The direction of your applied force is The direction of your applied force is

important, you would never try to open a important, you would never try to open a door with a doorknob by push or pulling door with a doorknob by push or pulling the doorknob sidewaysthe doorknob sideways

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

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

the force and lever arm are largethe force and lever arm are large

TorqueTorque

Torque and Center of GravityTorque and Center of Gravity If the direction of force is through the If the direction of force is through the

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

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

Rotational InertiaRotational Inertia An object rotating about an axis tends An object rotating about an axis tends

to keep rotating about that axis (look to keep rotating about that axis (look familiar?)familiar?)

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

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 3 The rotational inertia when your arms are extended is 3

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

Rotational Inertia and GymnasticsRotational Inertia and Gymnastics

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

of rotating objectsof rotating objects Like Like linear momentumlinear momentum, angular momentum is a , angular momentum is a

vector quantityvector quantity Rotational Velocity – Rotational Velocity – when a direction is when a direction is

assigned to rotational speedassigned to rotational 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 Law can now be restated for

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

angular momentum unless acted upon by an angular momentum unless acted upon by an unbalanced external torqueunbalanced external torque

Angular MomentumAngular Momentum

Conservation of Angular Conservation of Angular MomentumMomentum

The Law of Conservation of The Law of Conservation of Angular Momentum:Angular Momentum:

If no unbalanced external torque acts on If no unbalanced external torque acts on a rotating system, the angular a rotating system, the angular

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

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

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), Ch. 9: Do #31-38 (pg. 135),

Appendix F #1-7 (pg. 674)Appendix F #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)