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Chapter 2 The Laws of Motion, Part 2

September 14: Seesaws–Rotational motion

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Observations about seesaws

• Equal-weight children balance a seesaw.

• A balanced seesaw rocks back and forth easily.

• Unequal-weight children don’t normally balance the seesaw.

• Moving the heavier child inward restores balance.

• Sitting closer to the pivot speeds up the motion.

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Translational motion and rotational motion

Translational motion: The overall movement of an object from one place to another.Rotational motion: Motion around a fixed point.

Examples of motion:

•A running train

•The spin of the earth

•The moving hands of your watch

•A wind turbine

•A flying stick

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Physics concept: Rotational Inertia

• A body that is rotating tends to remain rotating.• A body that is not rotating tends to remain not

rotating.

Question:

What keeps the merry-go-round rotating?

Analogous question:What keeps a skater moving?

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Physical quantities on rotational motion:

• Angular position – an object’s orientation relative to a certain line.

The SI unit of angular position is radian . 2 radian = 360°.

• Angular velocity – the change in angular position with time. Describes how quickly the object is rotating.

• Torque – A twist or spin.

These are the counterparts of position, velocity and force used in describing translational motion.

time

anglein changespeedangular

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More about angular velocity

•Angular velocity is a vector. It is usually denoted as .

•The direction of angular velocity is defined by the right-hand rule:

Let the fingers of your right hand curl in the way of the rotation, then your thumb is pointing along the angular velocity direction.

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Question:

How much is the angular velocity of the longest hand of my watch?

Answer:

ondradian/sec 105.0second 60

radian 2

time

anglein changespeedangular

π

Question:

What is the direction of this angular velocity?

Answer:

The direction of the angular velocity is pointing into the watch surface according to the right-hand-rule.

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Newton’s first law of rotational motion

A rigid object that’s not wobbling and that is free of outside torques rotates at a constant angular velocity.

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Question:

Andy Wang is trying to harvest a bottle gourd. He is not tall enough to reach the stem. What can he do now to easily pick up the gourd?

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Read: Ch2: 1Homework: Ch2: E2Due: September 25

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September 16: Seesaws–Newton’s second law of rotational motion

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Center of mass of an object

Center of mass: A point in or near an object about which the mass is evenly distributed.

•For a symmetric object the center of mass is located at its geometric center.

•For a less symmetric object the location of the center of mass depends on its mass distribution.

•Center of mass is very useful in describing the motion of an object.

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Examples of center of mass

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Simplification of motion:

A free object naturally spins about its center of mass (e.g., the earth).

The motion of an object can be decomposed into

1)a translational motion of its center of mass, and

2)a rotational motion around its center of mass.

Example: A diver.

Demo: Flying wood board.

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More physical quantities on rotational motion:

• Rotational mass (Moment of inertia) – measure of the rotational inertia of an object. The SI unit of rotational mass is kilogram·meter2.

• Angular acceleration – the change in angular velocity with time. Describes how quickly the angular velocity is changing. The SI unit of angular acceleration is radian/second2.

They are the counterparts of mass and acceleration used in describing translational motion.

time

locityangular vein changeonacceleratiangular

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More about rotational mass:

• Rotational mass of an object measures how difficult to change its angular velocity. The value of rotational mass depends on the ordinary mass of the object and its distribution around the specific rotational axis.

• The contribution of each portion of mass of an object to the rotational mass of the object is proportional to its distance squared from the rotational axis:

dmdmdmI 233

222

211

For the tennis racket, which case has the largest rotational mass?

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More about torque

• The distance from the rotational axis to where the force is exerted is called the lever arm.

• Only the component of force perpendicular to the level arm contributes to torque. Think on how you open a door.

• The direction of torque is determined by the right-hand-rule.

•

Fr

armlever thelar toperpendicu force

armlever torque

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Newton’s second law of rotational motion

The angular acceleration of an object is equal to the net torque exerted on it divided by its rotational mass. The angular acceleration is in the same direction as the net torque.

Inet

onacceleratiangular mass rotationalnet torque

mass rotational

net torqueonacceleratiangular

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Read: Ch2: 1Homework: Ch2: E6,12; P1,6Due: September 25

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September 18: Seesaws–Rotational work

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Balancing a seesaw

A balanced seesaw (with riders) is one that experiences no net torque due to gravity, so that it can rotate smoothly.

When calculating torque, the gravity of an object can be thought as being exerted at the center of gravity of the object.

For smooth rotation the torque caused by the gravity of the seesaw must be zero. The center of gravity of the seesaw must be at the pivot.

Two children with different weights can balance the seesaw by sitting at different distances from the pivot.

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Mechanical advantage of a lever

For a balanced seesaw:(a)If the two weights are equal, the work that the descending child does on the board equals the work that the board does on the rising child.(b)If one child is heavier than the other, the two works involve different forces over different distances, but they are still equal.(c)Therefore the board only transfers energy from one child to the other.

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Question:

The boy exerts a torque on the seesaw board. Does the board also exert a torque on the boy?

Answer:

Yes. The board exerts an equal but oppositely directed torque on the boy. The net torque on the boy is zero, so he rotates smoothly.

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Newton’s third law of rotational motion

For every torque one object exerts on a second object, there is an equal but oppositely directed torque that the second object exerts on the first object (provided that the two objects rotate about the same axis).

Examples:

•Wind turbine and generator.• A child running on a merry-go-round.• Spanner wrench and nut.

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Work done in rotation

work = force · distance along the force direction

work done in a full revolution = force · distance

= force · (level arm · 2 )

= (force · level arm) · 2

= torque · 2

Work done in any amount of rotation:

work = torque · angle (in radian)

W

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Question:

The wind exerts a torque of 500 Newton·meter on a wind turbine. The turbine rotates 2 cycles per second. How much work does the wind do on the turbine per second?

Answer:

work = torque · angle (in radian)

J 628022 mN 500

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Read: Ch2: 1Homework: Ch2: E15Due: September 25

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September 21: Wheels–Friction

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Demo: Pushing a textbook on a table.

Question: Why does the book move slowly even though I push it continuously? (Newton’s second law says that it should accelerate…).

Answer: There exist a frictional force between the book and the table. The frictional force is always along the surfaces and opposing the sliding motion.

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More about frictional force:

1)Its strength depends on how hard the two surfaces are pressed against one another.

2)Its strength depends on how slippery the two surfaces are.

3)Its strength depends on whether or not the two surfaces are moving relative to one another.

4)Its strength does not depend much on the area of contact between the surfaces.5)It adjusts itself in response to the situation.

6)Newton’s third law of motion applies.

7)Friction is ubiquitous. It can help us. It can bother us.

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A microscopic view of friction

Surfaces have microscopic hills and valleys. When two surfaces are relatively moving, the structures collide with each other and produce a horizontal force.

Frictional forces are actually resulted from the electromagnetic forces between the structures.

The magnitude of the frictional force is proportional to the force pressing the two surfaces (normal force):

NF friction

Demo: Pulling textbooks on a table.

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The two types of friction

1.Static friction• Acts to prevent objects from starting to slide.• Ranges from zero to an upper limit.

2.Sliding friction• Acts to stop objects that are already sliding.• Has a fixed magnitude.

(Maximum) static frictional force > sliding frictional force

•In static friction surface features can interpenetrate better.•Frictional force drops when sliding begins.

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Question:

The signal light turns green and you’re in a hurry. Will your car accelerate faster if you1) skid your wheels and “burn rubber” or2) just barely avoid skidding your wheels?Answer:

Just barely avoid skidding your wheels.

Question:

Why is it dangerous (sometimes extremely dangerous) to abruptly step on the brake of your car in a snowy day?

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Friction and energy

1.Static friction• There is no relative motion between the surfaces.• No work is done and so there is no wear.

2.Sliding friction• The two surfaces are relatively moving.• Some work “disappears” and becomes thermal

energy.• The surfaces experience wear.

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Physical quantity: Power

Power is the work done in a unit time. It measures how fast an object is doing work. The SI unit of power is Joule-per-second, called watt (W).

1 kilowatt (kW) = 1000 watt

1 horsepower = 745.7 watt t

WP

time

workpower

Power in translational motion:power = force · distance /time = force · velocity

Power in rotational motion:power = torque · angle /time = torque · angular velocity

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Read: Ch2: 2Homework: Ch2: E22,27,29Due: October 2

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September 23: Wheels–Kinetic energy

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The many forms of energy

1.Kinetic energy: Energy of an object because of motion.

2.Potential energy: Energy stored in objects because of their positions, shapes and structures.• Gravitational potential energy

• Elastic potential energy

• Electric potential energy

• Chemical potential energy

• Magnetic potential energy

• Nuclear potential energy

3. Thermal energy: A disordered mixture of kinetic and potential energy at molecular level.

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Eliminating sliding frictions

Rollers:

•Rollers eliminate sliding friction at roadway.

•Rollers keep moving out from under the object.

•Rollers are not so convenient.

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Wheels:

•Eliminate sliding friction at roadway.•Convenient because they don’t pop out.Question: Who is turning the wheels when the cart is accelerating?Static friction exerts torques on the wheels.Question: Why do I still need to pull the cart even if it is moving at a constant velocity?Wheel hubs have sliding friction with the axels.

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Bearings:

•Eliminate sliding friction in wheel hubs.•Behave like automatically recycling rollers.

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Practical wheels:

•When accelerating, free (non-driving) wheels are turned by a backward static fraction.

•When accelerating, powered (driving) wheels are turned by the engine and a forward static friction.

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Kinetic Energy

1. A translationally moving object has a kinetic energy of:

2. A rotationally moving object has a kinetic energy of:

2

2

2

1

speed · mass · 2

1 energy kinetic

vmK

2

2

2

1

speedangular · mass rotational · 2

1 energy kinetic

IK

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Question:

My car is running on a street in Macomb with a speed of 30 mile/hour. You have a similar car which is running on highway 88 at 60 mile/hour.

How much more kinetic energy does your car have compared to mine?

Answer:

Because the kinetic energy is proportional to the square of speed, your car has four times kinetic energy compared to mine.

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Read: Ch2: 2No homework

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September 25: Bumper Cars–Linear momentum

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Momentum: A basic physical quantity

• Momentum (linear momentum) measures the translational motion of an object. Specifically

• The SI unit of momentum is kilogram·meter/second.

• Momentum is a vector quantity. It is in the direction of the velocity. On a certain direction, it can be positive or negative.

vp

mvelocitymassmomentum

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Question:

My car is running on a street in Macomb with a speed of 30 mile/hour. You have a similar car which is running on highway 88 at 60 mile/hour.

How much more momentum does your car have compared to mine?

Answer:

Because the magnitude of momentum is proportional to the speed, your car has two times momentum compared to mine.

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Impulse

• Momentum is transferred by impulse. Specifically

• The SI unit of impulse is newton·second.

• Impulse is a vector quantity. It is in the direction of the force.

t

Fimpulse

timeforceimpulse

• It comes from Newton’s second law of motion:

It is therefore correct!

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The impulse-momentum relationship

• The change in the momentum of an object equals the net impulse exerted on the object:

t

Fpreceived impulse momentumin Change

tmt

mm

Fvpv

aF

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Question: My car has a mass of 2500 kg and it is running at a velocity of 4 m/s. It is out of control because the brake is defect. How much impulse is needed to stop my car?

Answer:

sN 10,000

m/skg 10,000

m/s 4kg 2500

momentumin changeneeded impulse

vm

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Question: I am sliding on ice. My mass is 75 kg and my velocity is 2 m/s. How much is my momentum?

Answer: m/s.kg 150m/s 2kg 75velocitymassmomentum

Question: How much impulse should I receive so that I can stop?

Answer: s.N 150or m/s,kg 150

Question: If I push the ice forward with a force of 10 N, how long time does it take me to stop?

Answer:

s. 15N 10

sN 150

force

impulsetime

timeforceimpulse

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Read: Ch2: 3Homework: Ch2: P9,10,11Due: October 2

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September 28: Bumper Cars–Momentum conservation

Review: Momentum, impulse and their relations

• Momentum measures the translational motion of an object:

• Impulse transfers momentum:

• Impulse-momentum relationship: The change in the momentum of an object equals the net impulse received by the object:

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t

Fpreceived impulse momentumin Change

vp

mvelocitymassmomentum

t

Fimpulse

timeforceimpulse

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Demo: Tablecloth revisited (impulse-momentum view)

• Goal: Let v of the glasses be as small as possible.

• Principle: Change in the momentum of the glasses equals to the impulse received:

• Known: F (sliding frictional force) is small and fixed for given glasses and cloth. The mass of the glasses m is also fixed.

• Solution: Keep t as short as possible.

m

ttm

FvFv

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Law of Conservation of Momentum(A fundamental law in physics)

When there is no external force, the momentum of a system remains unchanged.

Movie: Space experiment: momentum conservationDiscussion: Newton’s third law of motion

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Demo: Momentum conservation in Collision (Newton’s pendulum).

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Examples of momentum conservation

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Question:

Your are caught by a monster and put onto a frozen river covered by slippery ice which cannot provide any frictional force. The monster knows a little physics and believes that without a frictional force you cannot let the ice help you to slide, which is unfortunately true.

How can you manage to escape?

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Read: Ch2: 3Homework: Ch2: P13Due: October 9

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September 30: Bumper Cars–Angular momentum

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Angular momentum: A basic physical quantity

• Angular momentum measures the rotational motion of an object. Specifically

• The SI unit of angular momentum is kilogram·meter2/second.

• Angular momentum is a vector quantity. It is in the direction of the angular velocity.

ωL

Ilocityangular vemass rotationalmomentumangular

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Angular impulse

• Angular momentum is transferred by angular impulse. Specifically

• The SI unit of angular impulse is newton·meter·second.

• Angular impulse is a vector quantity. It is in the direction of the torque.

t

τ

timetorqueimpulseangular

• It comes from Newton’s second law of rotational motion:

It is therefore correct!

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The angular impulse-angular momentum relationship

• The change in the angular momentum of an object equals the net angular impulse exerted on the object:

t

τLreceived impulseangular momentumangular in Change

tIt

II

τωLω

ατ

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Law of Conservation of Angular Momentum(A fundamental law in physics)

When there is no external torque, the angular momentum of a system remains unchanged.

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Question:

You are riding on the edge of a spinning playground merry-go-round. If you pull yourself to the center of the merry-go-round, what will happen to its rotation?

Answer:

The total angular momentum is conserved. The rotational mass is reduced when you go to the center of the merry-go-round. Therefore it will spin faster.

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Examples of angular momentum conservation

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Demo: Angular momentum conservation

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Read: Ch2: 3Homework: Ch2: E38;P14 Due: October 9