Force is applied at the handle The axis of rotation is at the nut or bolt The distance away is the length of the handle. Torque direction is either clockwise

• Force is applied at the handle• The axis of rotation is at the

nut or bolt• The distance away is the

length of the handle.

• Torque direction is either clockwise (cw) or counterclockwise (ccw)

• T=F*L• Torque = force * lever arm.

LEVER ARM

•Always the shortest distance from the rotation axis (axle) to the line of action (applied force).”

WHAT HAPPENS TO THE FORCE IS THE HANDLE LENGTH IS DOUBLED?

TORQUE AT EQUILIBRIUM

• Just like linear forces, torque is at equilibrium if the clockwise torque is equal to the counter clockwise torque.

• The net force is 0

• This means the object will stay at rest (if it was at rest to begin with) or

• It will continue its original motion (if it was in motion before the torque was applied)

• If they are unequal, then there is a net force in either the clockwise or counterclockwise direction.

• This means there is a change in the movement. The object will either speed up or slow down its rotation

What does this setup remind you of?

Units of Chapter 10• Angular Position, Velocity, and Acceleration

• Rotational Kinematics

• Connections Between Linear and Rotational Quantities

•Rolling Motion

• Rotational Kinetic Energy and the Moment of Inertia

• Conservation of Energy

10-1 Angular Position, Velocity, and Acceleration


Degrees and revolutions:


• Arc length s, measured in radians:

• This means

s r

s

r



Calculus: it just means how fast it is turning at the instant that we are talking about. (Just like the speedometer on your car says how fast you are going at the instant)


How long until you make a full circle


Calculus: it just means how fast it is changing speed at the instant that we are talking about. (Just like the speedometer on your car says how fast you are going at the instant)

10-2 Rotational Kinematics

Analogies between linear and rotational kinematics:

An object at rest begins to rotate with a constant angular acceleration. If this object has angular velocity w at time t, what was its angular velocity at the time 1/2 t?

1) 1/2 w

2) 1/4 w

3) 3/4 w

4) 2 w

5) 4 w

Angular Velocity

An object at rest begins to rotate with a constant angular acceleration. If this object has angular velocity w at time t, what was its angular velocity at the time 1/2 t?

1) 1/2 w

2) 1/4 w

3) 3/4 w

4) 2 w

5) 4 w

The angular velocity is w = at (starting from rest), and there is a

linear dependence on time. Therefore, in half the time, the object

has accelerated up to only half the speed.

Angular Velocity

EXAMPLE

• The angular speed of a propeller on a boat increases with constant acceleration from 12 rad/s to 26 rad/s in 2.5 revolutions.

• What is the acceleration of the propeller?

• How long did the change in angular speed take?

10-3 Connections Between Linear and Rotational Quantities

TANGENTIAL SPEED




This merry-go-round has both tangential and centripetal acceleration.

BONNIE AND KLYDE

w

BonnieKlyde

1) Klyde

2) Bonnie

3) both the same

4) linear velocity is zero for

both of them

Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every two seconds. Who has the larger linear (tangential) velocity?

Their linear speeds v will be

different since v = Rw and

Bonnie is located further out

(larger radius R) than Klyde.

w

Bonnie

Klyde

BonnieKlyde V21

V

1) Klyde

2) Bonnie

3) both the same

4) linear velocity is zero for

both of them

BONNIE AND KLYDE

Follow-up: Who has the larger centripetal acceleration?

Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every two seconds. Who has the larger linear (tangential) velocity?

Units of Chapter 7

• Work Done by a Constant Force

• Kinetic Energy and the Work-Energy Theorem

• Work Done by a Variable Force

• Power

7-1 Work Done by a Constant ForceThe definition of work, when the force is in the direction of the displacement:

(7-1)

SI unit: newton-meter (N·m) = joule, J

7-1 Work Done by a Constant Force

If the force is at an angle to the displacement:

(7-3)

The Sign of Work

The work done may be positive, zero, or negative, depending on the angle between the force and the displacement:

TOTAL WORK

• Sum of the works from each force

• Or: Work done by Net Force

Is it possible to do work on an

object that remains at rest?

1) yes

2) no

TO WORK OR NOT TO WORK

Is it possible to do work on an

object that remains at rest?

1) yes

2) no

Work requires that a force acts over a distance. If

an object does not move at all, there is no

displacement, and therefore no work done.

TO WORK OR NOT TO WORK

NORMAL FORCE AND WORK

1) Normal force does no work at all

2) Normal force does negative work

3) Normal force does positive work

A box is being

pulled across a

rough floor at a

constant speed.

What can you say

about the work

done by the

normal force?

f

N

mg

displacement

Pull

The normal force is perpendicular to

the displacement, so the work is

zero. Or using the definition of work

(W = F d cos q ), since = 90o, then

W = 0.

NORMAL FORCE AND WORK

1) Normal force does no work at all

2) Normal force does negative work

3) Normal force does positive work

A box is being

pulled across a

rough floor at a

constant speed.

What can you say

about the work

done by friction?

FRICTION AND WORK

1) friction does no work at all

2) friction does negative work

3) friction does positive work

A box is being

pulled across a

rough floor at a

constant speed.

What can you say

about the work

done by friction?

f

N

mg

displacement

Pull

Friction acts in the opposite direction

to the displacement, so the work is

negative. Or using the definition of

work (W = F d cos q ), since = 180o,

then W < 0.

FRICTION AND WORK

1) friction does no work at all

2) friction does negative work

3) friction does positive work

A box is being

pulled across a

rough floor at a

constant speed.

What can you say

about the work

done by friction?

FORCE AND WORK

1) one force

2) two forces

3) three forces

4) four forces

5) no forces are doing work

A box is being pulled up a

rough incline by a rope

connected to a pulley. How

many forces are doing work

on the box?

FORCE AND WORK

N

f

T

mg

displacementAny force not perpendicular

to the motion will do work:

N does no work

T does positive work

f does negative work

mg does negative work

1) one force

2) two forces

3) three forces

4) four forces

5) no forces are doing work

A box is being pulled up a

rough incline by a rope

connected to a pulley. How

many forces are doing work

on the box?

7-2 Kinetic Energy and the Work-Energy Theorem

When positive work is done on an object, its speed increases; when negative work is done, its speed decreases.

Kinetic Energy

After algebraic manipulations of the equations of motion, we find:

Therefore, we define the kinetic energy:

(7-6)

The Work-Energy Theorem

Work-Energy Theorem: The total work done on an object is equal to its change in kinetic energy.

(7-7)

A child on a skateboard is moving

at a speed of 2 m/s. After a force

acts on the child, her speed is 3

m/s. What can you say about the

work done by the external force on

the child?

1) positive work was done

2) negative work was done

3) zero work was done

CONCEPTEST 7.7 WORK AND KE

A child on a skateboard is moving

at a speed of 2 m/s. After a force

acts on the child, her speed is 3

m/s. What can you say about the

work done by the external force on

the child?

1) positive work was done

2) negative work was done

3) zero work was done

The kinetic energy of the child increased because her speed

increased. This increase in KE was the result of positive work

being done. Or, from the definition of work, since W = DKE =

KEf – KEi and we know that KEf > KEi in this case, then the

work W must be positive.

CONCEPTEST 7.7 WORK AND KE

Follow-up: What does it mean for negative work to be done on the child?

CONCEPTEST 7.8B SPEEDING UP I

1) 0 30 mph

2) 30 60 mph

3) both the same

A car starts from rest and

accelerates to 30 mph. Later, it

gets on a highway and accelerates

to 60 mph. Which takes more

energy, the 030 mph, or the 3060

mph?

The change in KE (1/2 mv2 ) involves the velocity squared.

So in the first case, we have: 1/2 m (302 - 02) = 1/2 m (900)

In the second case, we have: 1/2 m (602 - 302) = 1/2 m (2700)

Thus, the bigger energy change occurs in the second case.

CONCEPTEST 7.8B SPEEDING UP I

1) 0 30 mph

2) 30 60 mph

3) both the same

A car starts from rest and

accelerates to 30 mph. Later, it

gets on a highway and accelerates

to 60 mph. Which takes more

energy, the 030 mph, or the 3060

mph?

EXAMPLE

• A 1300 kg car coasts on a horizontal road with a velocity of 18 m/s, E. After crossing an unpaved, sandy stretch of road 30.0 m long, it’s velocity decreases to 15 m/s, E.

• Was the net work done on the car positive, negative, or zero?• Find the net work done on the car.• What is the magnitude and direction of the average net force on the car in the

sandy section?

7-3 Work Done by a Variable ForceIf the force is constant, we can interpret the work done graphically:

Multiple Rectangles

If the force takes on several successive constant values:

Varying Force

We can then approximate a continuously varying force by a succession of constant values.

Work done by a spring

The force needed to stretch a spring an amount x is F = kx.

Therefore, the work done in stretching the spring is

(7-8)

EXAMPLE

• A 1.2 kg block is held against a spring of force constant 1.0x104 N/m, compressing it a distance of 0.15 m. How fast is the block moving after it has been released and the spring pushes it away (in other words, just after the spring stops pushing on it)?

7-4 Power

Power is a measure of the rate at which work is done:

(7-10)

SI unit: J/s = watt, W

1 horsepower = 1 hp = 746 W

7-4 Power

PAYING YOUR ELECTRIC BILL

Engine #1 produces twice the power

of engine #2. Can we conclude that

engine #1 does twice as much work

as engine #2?

1) yes

2) no

CONCEPTEST 7.11C POWER

Engine #1 produces twice the power

of engine #2. Can we conclude that

engine #1 does twice as much work

as engine #2?

1) yes

2) no

No!! We cannot conclude anything about how much work

each engine does. Given the power output, the work will

depend upon how much time is used. For example,

engine #1 may do the same amount of work as engine #2,

but in half the time.

CONCEPTEST 7.11C POWER

CONCEPTEST 7.12B ENERGY CONSUMPTION

Which contributes more to

the cost of your electric bill

each month, a 1500-Watt

hair dryer or a 600-Watt

microwave oven?

1) hair dryer

2) microwave oven

3) both contribute equally

4) depends upon what you cook in the oven

5) depends upon how long each one is on

1500 W

600 W

We already saw that what you actually pay for is

energy. To find the energy consumption of an

appliance, you must know more than just the

power rating — you have to know how long it

was running.

CONCEPTEST 7.12B ENERGY CONSUMPTION

Which contributes more to

the cost of your electric bill

each month, a 1500-Watt

hair dryer or a 600-Watt

microwave oven?

1) hair dryer

2) microwave oven

3) both contribute equally

4) depends upon what you cook in the oven

5) depends upon how long each one is on

1500 W

600 W

7-4 Power

If an object is moving at a constant speed in the face of friction, gravity, air resistance, and so forth, the power exerted by the driving force can be written:

(7-13)

Summary of Chapter 7• If the force is constant and parallel to the displacement, work is force times distance

• If the force is not parallel to the displacement,

• The total work is the work done by the net force:

Summary of Chapter 7

• SI unit of work: the joule, J

• Total work is equal to the change in kinetic energy:

where

Summary of Chapter 7

• Work done by a spring force:

• Power is the rate at which work is done:

• SI unit of power: the watt, W

Documents

Force is applied at the handle The axis of rotation is at the nut or bolt The distance away is the length of the handle. Torque direction is either clockwise