1Physics 1100 – Spring 2012 Chapter 8 – Rotational Motion Circular MotionCircular Motion Rotational InertiaRotational Inertia TorqueTorque Center of Mass

1Physics 1100 – Spring 2012

Chapter 8 – Rotational MotionChapter 8 – Rotational Motion

• Circular MotionCircular Motion• Rotational InertiaRotational Inertia• TorqueTorque• Center of Mass and Center of GravityCenter of Mass and Center of Gravity

– Locating Center of Gravity

– Stability

• Centripetal ForceCentripetal Force• Centrifugal ForceCentrifugal Force• Centrifugal Force in a Rotating Reference FrameCentrifugal Force in a Rotating Reference Frame• Simulated GravitySimulated Gravity• Angular MomentumAngular Momentum• Conservation of Angular MomentumConservation of Angular Momentum


Circular MotionCircular Motion

• Linear speedLinear speed - the distance moved per unit time. Also called - the distance moved per unit time. Also called simply speed.simply speed.

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

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


RotationRotation


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

v = v = r r

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



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

• v = v = r r• What are two ways that you can increase your linear speed on What are two ways that you can increase your linear speed on

a rotating platform?a rotating platform?

– Answers:

• Move away from the rotation axis.

• Have the platform spin faster.



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

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

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

1) 1 m/s 2) 2 m/s 3) 3m/s 4) 4m/s1) 1 m/s 2) 2 m/s 3) 3m/s 4) 4m/s

What will be his rotational speed?What will be his rotational speed? 1) 10 RPM 2) 20 RPM 3) 30 RPM 4) 40 RPM1) 10 RPM 2) 20 RPM 3) 30 RPM 4) 40 RPM

Example QuestionExample Question


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

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

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

Answer:Answer: 4 m/s 4 m/s

What will be his rotational speed?What will be his rotational speed?

Answer:Answer: 20 RPM 20 RPM

Example QuestionExample Question


Rotational Inertia Rotational Inertia

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

• This influence is called This influence is called torquetorque..

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

• spinning football• bicycle tires• Frisbee


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

Rotational Inertia Rotational Inertia


Rotational InertiaRotational Inertia


Rotational InertiaRotational Inertia


TorqueTorque

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

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

• wrenches

• see-saws

= 0, no rotational motion !!!

Newton’s First Law for Rotation


Class ProblemClass Problem

• James finds it difficult to muster James finds it difficult to muster enough torque to turn the enough torque to turn the stubborn bolt with the wrench. stubborn bolt with the wrench. He wishes he had a pipe handy He wishes he had a pipe handy to effectively lengthen the to effectively lengthen the wrench handle, but doesn't. He wrench handle, but doesn't. He does, however, have a piece of does, however, have a piece of rope. Will torque be increased if rope. Will torque be increased if he pulls as hard on the rope as he pulls as hard on the rope as shown? shown?



• No, the torque will be the same No, the torque will be the same because the lever-arm distance because the lever-arm distance is the same in both cases. The is the same in both cases. The lever arm is not the distance lever arm is not the distance between axis of turning and the between axis of turning and the point of application of the force, point of application of the force, but the distance from the turning but the distance from the turning axis to the "line of action" of the axis to the "line of action" of the applied force. Note the line of applied force. Note the line of action, and hence the lever arm action, and hence the lever arm is the same in both cases.is the same in both cases.

The pipe that extends the length The pipe that extends the length of the wrench handle puts the of the wrench handle puts the line of action farther from the line of action farther from the turning axis—the rope does not. turning axis—the rope does not.


TorqueTorque


Center of MassCenter of Mass

• The center of mass of an object The center of mass of an object is the average position of mass.is the average position of mass.


Center of MassCenter of Mass

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



• The broom balances at its center of gravity. If you saw the broom into two The broom balances at its center of gravity. If you saw the broom into two parts through the center of gravity and then weigh each part on a scale, which parts through the center of gravity and then weigh each part on a scale, which part will weigh more? part will weigh more?

1) Handle side2) Brush Side3) Both weigh the same



• The short broom part is heavier. It The short broom part is heavier. It balances the long handle just as balances the long handle just as kids of unequal weights can balance kids of unequal weights can balance on a seesaw when the heavier kid on a seesaw when the heavier kid sits closer to the fulcrum. Both the sits closer to the fulcrum. Both the balanced broom and seesaw are balanced broom and seesaw are evidence of equal and opposite evidence of equal and opposite torques—not weights. torques—not weights.



• The 40-kg woman stands at the end of a 4-meter long uniform The 40-kg woman stands at the end of a 4-meter long uniform plank. If the maximum overhang for balance is 1 meter, estimate plank. If the maximum overhang for balance is 1 meter, estimate the mass of the plank.the mass of the plank.

1) 10 kg2) 20 kg3) 40 kg4) 80 kg5) 160 kg



• The mass of the plank is about 40kg. The plank tends to rotate like a seesaw about The mass of the plank is about 40kg. The plank tends to rotate like a seesaw about a pivot point at the end of the building. Her weight multiplied by 1 meter produces a a pivot point at the end of the building. Her weight multiplied by 1 meter produces a torque that tends to rotate the system clockwise. The counterbalancing torque is torque that tends to rotate the system clockwise. The counterbalancing torque is produced by the weight of the plank multiplied by the distance from the pivot point to produced by the weight of the plank multiplied by the distance from the pivot point to the plank's center of gravity. Note that this distance is also 1 meter. So both the the plank's center of gravity. Note that this distance is also 1 meter. So both the woman and the plank weigh the same. Their masses are equal. woman and the plank weigh the same. Their masses are equal.


StabilityStability

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

• Examples: Examples: • Tower of Pisa

• Touching toes with legs against wall

• Meter stick over the edge


Angular MomentumAngular Momentum

• Another conserved quantity is Another conserved quantity is angular momentumangular momentum, relating to , relating to rotational inertia:rotational inertia:

• Spinning wheel wants to keep on spinning, stationary wheel Spinning wheel wants to keep on spinning, stationary wheel wants to keep still (unless acted upon by an external rotational wants to keep still (unless acted upon by an external rotational force, or torque)force, or torque)

• Newton’s laws for linear (straight-line) motion have direct Newton’s laws for linear (straight-line) motion have direct analogs in rotational motionanalogs in rotational motion


Conservation of Angular MomentumConservation of Angular Momentum

• angular momentum = rotational inertia x rotational velocityangular momentum = rotational inertia x 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 momentum unless acted

upon by an unbalanced external torque.”


Angular MomentumAngular Momentum

• Angular momentum is proportional to rotation speed times rotational Angular momentum is proportional to rotation speed times rotational inertiainertia

• Rotational inertia characterized by (mass)Rotational inertia characterized by (mass)(radius)(radius)22 distribution in distribution in objectobject


Angular Momentum ConservationAngular Momentum Conservation

• Speed up rotation by tucking inSpeed up rotation by tucking in

• Slow down rotation by Slow down rotation by stretching outstretching out

• Seen in diving all the timeSeen in diving all the time

• Figure skaters demonstrate Figure skaters demonstrate impressivelyimpressively

• Effect amplified by moving large Effect amplified by moving large masses to vastly different radiimasses to vastly different radii


Centripetal ForceCentripetal Force

• ……is applied by some object.is applied by some object.• Centripetal means "center seeking".Centripetal means "center seeking".

Centrifugal ForceCentrifugal Force

• ……results from a natural tendency.results from a natural tendency.• Centrifugal means "center fleeing".Centrifugal means "center fleeing".


Direction ofMotion

Centrifugal Force

CentripetalForce

Rotational ForcesRotational Forces


What kind of motions do we What kind of motions do we feelfeel??

• Aside from vibrations, don’t feel constant velocityAside from vibrations, don’t feel constant velocity• But we can But we can feelfeel acceleration acceleration


Rotating Drum RideRotating Drum Ride

• Vertical drum rotates, you’re pressed against wallVertical drum rotates, you’re pressed against wall– Friction force against wall matches gravity– Seem to stick to wall, feel very heavy

The forces real and perceived

Real Forces:

Friction; upCentripetal; inwardsGravity (weight); down

Perceived Forces:

Centrifugal; outwardsGravity (weight); down

Perceived weight; down and out


Simulated GravitySimulated Gravity

Clicker ProblemsClicker Problems


If you replace the wheels and tires of your car with new ones having greater diameters, all of your speedometer readings thereafter will

A) increase B) decrease. C) remain the same.

Suppose the circumference of a bicycle wheel is 2 meters. If it rotates at 1 revolution per second when you are riding the bicycle, then your speed will be

A) 3.14 m/s. B) 2 m/s. C) 6.28 m/s. D) 1 m/s. E) 3 m/s.

Clicker ProblemsClicker Problems


An upright broom is easier to balance when the heavier end is

A) highest, farthest from your hand. B) nearest your hand. C) same either way.

Put a pipe over the end of a wrench when trying to turn a stubborn nut on a bolt, to effectively make the wrench handle twice as long, you'll multiply the torque by

A) four. B) eight. C) two.

Clicker QuestionsClicker Questions


The famous Leaning Tower of Pisa doesn't topple over because its center of gravity is

A) displaced from its center. B) in the same place as its center of mass. C) stabilized by its structure. D) relatively low for such a tall building. E) above a place of support.

The chef at the infamous Fattening Tower of Pizza tosses a spinning disk of uncooked pizza dough into the air. The disk's diameter increases during the flight, while its rotational speed

A) increases. B) decreases. C) remains constant.



To turn a stubborn screw, it is best to use a screwdriver that has a

A) long handle. B) wide handle. C) smooth handle. D) none of these

To kick a football so it won't topple end over end, kick it so the force of impact extends

A) below its center of gravity. B) through its center of gravity. C) above its center of gravity.



A car travels in a circle with constant speed. The net force on the car is

A) directed forward, in the direction of travel. B) zero because the car is not accelerating. C) directed towards the center of the curve. D) none of these

If the polar icecaps melted, the resulting water would spread over the entire earth. This new mass distribution would tend to make the length of a day

A) longer. B) shorter at first, then longer. C) shorter. D) longer at first, then shorter. E) stay the same.

Documents

1Physics 1100 – Spring 2012 Chapter 8 – Rotational Motion Circular MotionCircular Motion Rotational InertiaRotational Inertia TorqueTorque Center of Mass