chapter 6

Copyright 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

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CHAPTER 6Work and Kinetic Energy


Goals for Chapter 6

To understand and calculate work done by a force

To study and apply kinetic energy

To learn and use the work-energy theorem

Calculate work done by a varying force along a curved path

Determine the power in a physical situation by adding time


Introduction Weve studied how Newtons Second Law allows us to

calculate an acceleration from a force but what if the force changes during its application?

We must be able to account for things like an archers bow.

We will use:

Principle of conservation of energy: it cannot be created nor destroyed

Work and energy.


Work, a force through a distance As in the illustration, pushing in the same direction that the object moves

The total work done on particle by all forces acting on it equals the change in kinetic energy (energy of motion, related to particles speed)

More work is done if you push harder (stronger force), or displacement is greater.

Unit is J (Joule) = 1N (newton) 1 m (meter)


Use the parallel component if the force acts at an angle

Force at the angle with respect to displacement


The scalar product

Termed the dot product.

The result is a scalar!


Work general case

Notice that work W is SCALAR quantity.

Constant forceStraight line displacement


Problem 6.8.

A loaded grocery cart is rolling across a parking lot in a strong wind. You apply a constant force F = (30 N)i (40 N)j to the cart as it undergoes a displacement s = (-9.0 m)i (3.0 m)j. How much work does the force you apply do on the grocery cart?


Work: positive, negative and zero

Depending on the angle, work can be positive, negative and even zero.


Q6.1

A. The cable does positive work on the elevator, and the elevator does positive

work on the cable.

v

Motor

Cable

Elevator

An elevator is being lifted at a constant speed by a steel cable attached to an electric motor.

Which statement is correct?

B. The cable does positive work on the elevator, and the elevator does negative work on the cable.

C. The cable does negative work on the elevator, and the elevator does positive work on the cable.

D. The cable does negative work on the elevator, and the elevator does negative work on the cable.


A6.1


work on the cable.

v

Motor

Cable

Elevator

An elevator is being lifted at a constant speed by a steel cable attached to an electric motor.

Which statement is correct?





Q6.2


work on the cable.

v

Motor

Cable

Elevator

An elevator is being lowered at a constant speed by a steel cable attached to an electric

motor. Which statement is correct?





A6.2


work on the cable.

v

Motor

Cable

Elevator

An elevator is being lowered at a constant speed by a steel cable attached to an electric

motor. Which statement is correct?





Three blocks are connected as shown. The ropes and pulleys are of

negligible mass. When released, block C moves downward, block B

moves up the ramp, and block Amoves to the right.

A. positive work on A, B, and C.

B. zero work on A, positive work on B, and negative work on C.

C. zero work on A, negative work on B, and positive work on C.

D. none of these

Q6.8

After each block has moved a distance d, the force of gravity has done


Three blocks are connected as shown. The ropes and pulleys are of

negligible mass. When released, block C moves downward, block B

moves up the ramp, and block Amoves to the right.

A. positive work on A, B, and C.

B. zero work on A, positive work on B, and negative work on C.

C. zero work on A, negative work on B, and positive work on C.

D. none of these

A6.8

After each block has moved a distance d, the force of gravity has done


How can it be such a great workout with no work?

When positive and negative work cancel, the net work is zero even though muscles are exercising.

Negative work: hands on barbell

Positive work: barbell on hands

When one body does negative work on the second body,the second body does an equal amount of positive work on the first body


Total Work How to find it?

2 options

find the net force acting on the body and then find the scalar product of net force and total displacement to find the total work

Find the work by every force acting on the body and sum them all (algebraically) to find the total work


Stepwise solution of work done by several forces 6.2.

A farmer hitches her tractor to a sled loaded with firewood and pulls it a distance 20 m along level ground. The total weight of sled and load is 14,700 N. The tractor exerts a constant 5000-N force at an angle of 36.9 above the horizontal, as shown in Fig. There is a 3500-N friction force opposing the sleds motion. Find the work done by each force acting on the sled and the total work done by all the forces.


The work-energy theorem

Work done on an object can change its motion and energy.


Kinetic energy of object

Lets quantify energy related to objects speed-kinetic energy

Kinetic energyIt is scalar! (unit J)It does not depend on velocity only speed!


We can compare the kinetic energy of different bodies

Changes in the energy of a moving body under the influence of an applied force change differently depending on the direction of application.


Work-energy Theorem

The work done by the net force on a particle equals the change in the particles kinetic energy:

Work = Final KE Initial KE

Work-energy theorem


The work-energy theorem - analysis

KE increases KE decreases KE stays the same


Problem 6.73. You and your bicycle have a combined mass 80.0 kg. When you reach

the base of a bridge, you are traveling along the road at 5m/s. At the top of the bridge, you have climbed a vertical distance of 5.2 m and have slowed to 1.5 m/s. You can ignore work done on by friction and any inefficiency in the bike or your legs.

A) What is the total work done on you and your bicycle when you go from the base to the top of the bridge?

B) How much work have you done with the force you apply to your pedals?


Two iceboats (one of mass m, one of mass 2m) hold a race on a frictionless,

horizontal, frozen lake. Both iceboats start at rest, and the wind exerts the

same constant force on both iceboats.

A. The iceboat of mass m: it has twice as much KE as the other.

B. The iceboat of mass m: it has 4 times as much KE as the other.

C. The iceboat of mass 2m: it has twice as much KE as the other.

D. The iceboat of mass 2m: it has 4 times as much KE as the other.

E. They both cross the finish line with the same kinetic energy.

Q6.3

Which iceboat crosses the finish line with more kinetic energy (KE)?


Two iceboats (one of mass m, one of mass 2m) hold a race on a frictionless,

horizontal, frozen lake. Both iceboats start at rest, and the wind exerts the

same constant force on both iceboats.

A. the iceboat of mass m: it has twice as much KE as the other

B. the iceboat of mass m: it has 4 times as much KE as the other

C. the iceboat of mass 2m: it has twice as much KE as the other

D. the iceboat of mass 2m: it has 4 times as much KE as the other

E. they both cross the finish line with the same kinetic energy

A6.3

Which iceboat crosses the finish line with more kinetic energy (KE)?


A tractor driving at a constant speed pulls

a sled loaded with firewood. There is

friction between the sled and the road.

A. positive.

B. negative.

C. zero.

D. not enough information given to decide

Q6.4

The total work done on the sled after it has moved a distance d is


A tractor driving at a constant speed pulls

a sled loaded with firewood. There is

friction between the sled and the road.

A. positive.

B. negative.

C. zero.


A6.4

The total work done on the sled after it has moved a distance d is


A nonzero net force acts on an object. Which of the following quantities could be constant?

A. the objects kinetic energy

B. the objects velocity

C. both of the above

D. none of the above

Q6.5


A nonzero net force acts on an object. Which of the following quantities could be constant?

A6.5

A. the objects kinetic energy

B. the objects velocity

C. both of the above

D. none of the above


How fast?What is the final speed of the sled? 6.3.


Forces on a hammerheadExample 6.4 In a pile driver, a steel hammerhead with mass 200 kg is lifted 3.00m above the top of a

vertical I-beam being driven into the ground. The hammer is then dropped, driving the I-beam 7.4 cm farther into the ground. The vertical rails that guide the hammerhead exert a constant 60 N friction force on the hammerhead. Use the work-energy theorem to find:

A) the speed of the hammerhead just as it heats the I-beam

B) the average force the hammerhead exerts on the I-beam. Ignore the effects of the air.


Work and energy with varying forces So far we considered only work done by a

constant force and resulting in straight-line motion

But both the force can be varying (spring) and trajectory can be curved

The first case: varying force, straight line motion

Perhaps the best example is driving a car, alternating your attention between the gas and the brake.

The effect is a variable positive or negative force of various magnitude along a straight line.

Area underthe curve


The stretch of a spring and the force that caused it

The force applied to an ideal spring will be proportional to its stretch Hookes law

Fx = kx k force (spring)

constant (N/m)

The graph of force on the y axis versus stretch on the x axis will yield a slope of k, the spring constant.


Problem 6.39. At a waterpark, sleds with riders are sent along a slippery,

horizontal surface by the release of a large compressed spring. The spring with force constant k = 40 N/cm and negligible mass rests on the frictionless horizontal surface. One end is in contact with a stationary wall. A sled and rider with total mass 70.0 kg are pushed against the other end, compressing the spring 0.375 m. The sled is then released with zero velocity. What is the sleds speed when the spring

A) returns to its uncompressed length?

B) is still compressed 0.20 m?


Stepping on a scaleExample 6.6 Whether you like the result or not, stepping on a scale is an excellent example of

applied force and the work being done to compress that spring.

A woman weighing 600 N steps on a bathroom scale containing a stiff spring. In equilibrium the spring is compressed 1 cm under her weight. Find the force constant of the spring and the total work done on it during the compression.


Motion with a varying force 6.7. An air-track glider of mass 0.1

kg is attached to the end of a horizontal air track by a spring with force constant 20 N/m. Initially the spring is unstretched and the glider is moving at 1.5 m/s to the right. Find the maximum distance d that the glider moves to the right

A) if the air-track is turned on so that there is no friction

B) if the air is turned off so that there is kinetic friction with coefficients k = 0.47.


Work-energy Theorem for Motion Along a Curve Force varies in direction and magnitude

Displacement lies along the curved path

dW = F cos dl = Fk dl = F dlW =

R P2P1F cosdl = R P2

P1Fkdl =

R P2P1F dl

Wtot = K2 K1 = K True always


Motion on a curved path Example 6.8. If you watch a child on a swing set, you can also consider the motion of a

particle along a curved path.

At a family picnic you are appointed to push your cousin in a swing. His weight is w, the length of the chains is R, and you push the boy until the chains make an angle 0 with the vertical. To do this, you exert a varying horizontal force F that starts at zero and gradually increases just enough so that the boy and the swing move very slowly and remain very nearly in equilibrium. What is the total work done on the boy by all forces? What is the work done by the tension T in the chains? What is the work you do by exerting the force F? (Neglect the weight of the chains and seat).


Watt about power? How quickly is work done? Once work is calculated, dividing by the time that passed

determines power. Power is the time rate at which work is done.

Average power is

Instantaneous power

The pun is credit to James Watt. (You will see that scientists of that era often were privileged to leave their names on the topic of their efforts.) Unit is watt (W).

Also note the popular culture power unit of horsepower: 1hp = 550ft lb/s = 746 W

The energy you use may be noted from the meter the electric company probably installed to measure your consumption of energy in kilowatt-hours: 1 kW h = 3.6 MJ

P = FkvP =

F v

Pav =Fkst = Fk

st = Fkv

t 0


An object is initially at rest. A net force (which always points in the same direction) is applied to the object so that the power of the net force is constant. As the object gains speed,

A. the magnitude of the net force remains constant.

B. the magnitude of the net force increases.

C. the magnitude of the net force decreases.


Q6.10


An object is initially at rest. A net force (which always points in the same direction) is applied to the object so that the power of the net force is constant. As the object gains speed,

A. the magnitude of the net force remains constant.

B. the magnitude of the net force increases.

C. the magnitude of the net force decreases.


A6.10


Force and power you depend uponExample 6.10

Each of the two jet engines in a Boing 767 airliner develops a thrust (a forward force on the airplane) of 197,000 N. When the airplane is flying at 250 m/s, what horsepower does each engine develop?


An example you might do if the elevator is out - 6.11. A 50 kg marathon runner runs up the stairs to the top of 443 m tall Sears

Tower in Chicago. To lift herself to the top in 15 min, what must be her average power output in watts? In kilowatts? In horsepower?

Its interesting how a lighter stair climber and heavier stair climber can expend the same power by using different amounts of time.

Slide Number 1Slide Number 2Goals for Chapter 6IntroductionWork, a force through a distanceUse the parallel component if the force acts at an angleSlide Number 7Work general caseProblem 6.8.Work: positive, negative and zeroSlide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16How can it be such a great workout with no work?Total Work How to find it?Stepwise solution of work done by several forces 6.2.The work-energy theoremKinetic energy of objectWe can compare the kinetic energy of different bodiesWork-energy TheoremSlide Number 24Problem 6.73.Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31How fast?What is the final speed of the sled? 6.3.Forces on a hammerheadExample 6.4Work and energy with varying forcesThe stretch of a spring and the force that caused itProblem 6.39.Slide Number 37Motion with a varying force 6.7.Work-energy Theorem for Motion Along a CurveMotion on a curved path Example 6.8.Watt about power? How quickly is work done?Slide Number 42Slide Number 43Force and power you depend uponExample 6.10An example you might do if the elevator is out - 6.11.

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