chapter 7 work and kinetic energy. reading and review

Chapter 7

Work and Kinetic Energy

Reading and Review

Vertical circular motion

A

B

C vertical (down)

vertical (up)

horizontal

Centripetal acceleration must be

Condition for falling: N=0

at C:

(now apparent weight is in the opposite direction to true weight!)

So, as long as:

at the top, then N>0 and pointing down.

Barrel of Fun

A rider in a “barrel of

fun” finds herself stuck with her back to the wall. Which diagram correctly shows the forces acting on her?

a b c d e

The normal force of the wall on the

rider provides the centripetal force

needed to keep her going around in a

circle. The downward force of gravity

is balanced by the upward frictional

force on her, so she does not slip

vertically.

Barrel of Fun

A rider in a “barrel of

fun” finds herself stuck with her back to the wall. Which diagram correctly shows the forces acting on her?

Follow-up: What happens if the rotation of the ride slows down?

a b c d e

a) centrifugal force is pushing you into the door

b) the door is exerting a leftward force on you

c) both of the above

d) neither of the above

Around the CurveAround the CurveYou are a passenger in a car, not wearing a seat belt. The car makes a sharp left turn, and you find yourself hitting the passenger door. What is the correct description of what is actually happening?

a) centrifugal force is pushing you into the door

b) the door is exerting a leftward force on you

c) both of the above

d) neither of the above

The passenger has the tendency

to continue moving in a straight

line. There is a net centripetal

force, provided by the door, that

forces the passenger into a

circular path.

Around the CurveAround the CurveYou are a passenger in a car, not wearing a seat belt. The car makes a sharp left turn, and you find yourself hitting the passenger door. What is the correct description of what is actually happening?

Atlas holds up the world. Sisyphus pushes his rock

up a hill.

Who does more work?(a)

(b)

Working Hard... or Hardly Working

Atlas holds up the world. Sisyphus pushes his rock

up a hill.

(a)

(b)

Working Hard... or Hardly Working

With no displacement, Atlas does no work

Work Done by a Constant ForceThe definition of work, when the force is parallel to the displacement:

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

Friction and Work IFriction and Work I

a) friction does no work at all

b) friction does negative work

c) 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 (Δr)cos ), because =

180º, then W < 0.

Friction and Work IFriction and Work I

a) friction does no work at all

b) friction does negative work

c) 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?

Can friction ever

do positive work? a) yes

b) no

Friction and Work IIFriction and Work II

Can friction ever

do positive work? a) yes

b) no

Consider the case of a box on the back of a pickup

truck. If the box moves along with the truck, then it is

actually the force of friction that is making the box

move.

Friction and Work IIFriction and Work II

Forces not along displacement

If the force is at an angle to the displacement:

Convenient notation: the dot productThe work can also be written as the dot product of the force and the displacement:

Force and displacement

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

Sum of work by forces = work by sum of forces

If there is more than one force acting on an object, we can find the work done by each force, and also the work done by the net force:

Units of Work

Lifting 0.5 L H2O up 20 cm = 1 J

1 kcal = 1 Cal = 4.186 kJ

In a baseball game, the

catcher stops a 90-mph

pitch. What can you say

about the work done by

the catcher on the ball?

a) catcher has done positive work

b) catcher has done negative work

c) catcher has done zero work

Play Ball!Play Ball!

In a baseball game, the

catcher stops a 90-mph

pitch. What can you say

about the work done by

the catcher on the ball?

a) catcher has done positive work

b) catcher has done negative work

c) catcher has done zero work

The force exerted by the catcher is opposite in direction to the displacement of the ball, so the work is negative. Or using the definition of work (W = F (Δr)cos ), because = 180º, then W < 0. Note that the work done on the ball is negative, and its speed decreases.

Play Ball!Play Ball!

Follow-up: What about the work done by the ball on the catcher?

Tension and WorkTension and Work

a) tension does no work at all

b) tension does negative work

c) tension does positive work

A ball tied to a string is

being whirled around in

a circle. What can you

say about the work

done by tension?

Tension and WorkTension and Work

a) tension does no work at all

b) tension does negative work

c) tension does positive work

A ball tied to a string is

being whirled around in

a circle. What can you

say about the work

done by tension?

v

T

No work is done because the force

acts in a perpendicular direction to

the displacement. Or using the

definition of work (W = F (Δr)cos ),

because = 90º, then W = 0.

Follow-up: Is there a force in the direction of the velocity?

Work by gravity

Fg a

h

A ball of mass m drops a distance h. What is the total work done on the ball by gravity?

N

Fg

A ball of mass m rolls down a ramp of height h at an angle of 45o. What is the total work done on the ball by gravity?

h

a

θ

Fgx = Fg sinθ

h = L sinθ

W = Fd = Fgx x L = (Fg sinθ) (h / sinθ)

W = Fg h = mgh

W = Fd = Fg x h

W = mgh

Path doesn’t matter when asking “how much work did gravity do?”

Only the change in height!

Path independence

• If a force depends on POSITION only then the work done by it on an object moving from

to will NOT depend upon the path.

• Such a force is called a Conservative Force

1r

2r

Motion and energy

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:

Work-Energy Theorem

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

(True for rigid bodies that remain intact)

Lifting a BookLifting a Book

You lift a book with your hand

in such a way that it moves up

at constant speed. While it is

moving, what is the total work

done on the book?

a) mg ∆ r

b) FHAND ∆ r

c) [FHAND + mg] ∆ r

d) zero

e) none of the above

mg

∆r FHAND

v = consta = 0

Lifting a BookLifting a Book

You lift a book with your hand

in such a way that it moves up

at constant speed. While it is

moving, what is the total work

done on the book?

The total work is zero because the net force

acting on the book is zero. The work done

by the hand is positive, and the work done

by gravity is negative. The sum of the two

is zero. Note that the kinetic energy of the

book does not change either!

a) mg ∆ r

b) FHAND ∆ r

c) (FHAND + mg) ∆ r

d) zero


mg

∆r FHAND

v = consta = 0

Follow-up: What would happen if FHAND were greater than mg?

By what factor does the

kinetic energy of a car

change when its speed

is tripled?

a) no change at all

b) factor of 3

c) factor of 6

d) factor of 9

e) factor of 12

Kinetic Energy IKinetic Energy I

By what factor does the

kinetic energy of a car

change when its speed

is tripled?

a) no change at all

b) factor of 3

c) factor of 6

d) factor of 9

e) factor of 12

Because the kinetic energy is mv2, if the speed

increases by a factor of 3, then the KE will increase by a

factor of 9.

Kinetic Energy IKinetic Energy I

Slowing DownSlowing Down

a) 20 m

b) 30 m

c) 40 m

d) 60 m

e) 80 m

If a car traveling 60 km/hr can

brake to a stop within 20 m,

what is its stopping distance if

it is traveling 120 km/hr?

Assume that the braking force

is the same in both cases.

F d = Wnet = ∆KE = 0 –

mv2,

and thus, |F| d = mv2.

Therefore, if the speed doubles,

the stopping distance gets four

times larger.

Slowing DownSlowing Down

a) 20 m

b) 30 m

c) 40 m

d) 60 m

e) 80 m

If a car traveling 60 km/hr can

brake to a stop within 20 m,

what is its stopping distance if

it is traveling 120 km/hr?

Assume that the braking force

is the same in both cases.

Application: ball on a trackhow high must I place the ball so

that it can complete a loop?

Condition: Fcp > mg at top of loop

Fcp = mv2/r = mg v2 = gr

KE = mv2 / 2 = mgr/2Gravity must provide this energy Wg = mg∆h = KE

∆h = r/2 above the top of the loop!

Work Done by a Variable Force

If the force is constant, we can interpret the work done graphically:


If the force takes on several successive constant values:


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


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

Therefore, the work done in stretching the spring is

Application: work by a spring

Hooke’s Law: F = - kx k = (3kg)(9.8 m/s2) / (3.9 cm)k = 760 N/m

Loaded spring: W = kx2/2 = (760 N/m) (0.04m)2/ 2

W = 0.61 J

How fast?: v = d/t = (0.020 m) (0.020 s) = 1 m/s

KE = mv2/2 = (1kg)(1m/s)2 / 2KE = 0.55 J

Kinetic Energy:

Power

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

SI unit: J/s = watt, W

1 horsepower = 1 hp = 746 W

ave

WP

t

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:

Question: what is the total work per unit time done on the object?

F x xP F Fv

t t

a) energy

b) power

c) current

d) voltage


Electric BillElectric Bill

When you pay the electric company

by the kilowatt-hour, what are you

actually paying for?

We have defined: Power = energy / time

So we see that: Energy = power × time

This means that the unit of power × time

(watt-hour) is a unit of energy !!

Electric BillElectric Bill

When you pay the electric company

by the kilowatt-hour, what are you

actually paying for?

a) energy

b) power

c) current

d) voltage


A block rests on a horizontal frictionless surface. A string is attached to the block, and is pulled with a force of 45.0 N at an angle above the horizontal, as shown in the figure. After the block is pulled through a distance of 1.50 m, its speed is 2.60 m/s, and 50.0 J of work has been done on it. (a) What is the angle (b) What is the mass of the block?

The pulley system shown is used to lift a 52 kg crate. Note that one chain connects the upper pulley to the ceiling and a second chain connects the lower pulley to the crate. Assuming the masses of the chains, pulleys, and ropes are negligible, determine (a) the force F required to lift the crate with constant speed, and(b) the tension in two chains

(a) the force F required to lift the crate with constant speed, and(b) the tension in two chains

(a) constant velocity, a=0, so net force =0.

2T - (52kg)(9.8m/s2) = 0

T = 250 NF = -250 Ny

(b) upper pulley doesn’t move:Tch - 2Trope = 0 Tch = 500 N

lower pulley has constant accelerationTch -2Trope =0 Tch = 500 N

Mechanical Advantage!

(a) how much power is applied to the box by the chain?(b) how much power is applied on the rope by the applied force?

What about work?

Trope = 250 NTchain = 500 NF = -250 Ny

(a) P = Fv = 500 N * vbox

(b)P = Fv = 250 N * vhand

hand moves twice as fast

hand moves twice as far

chapter 7 work and kinetic energy. reading and review

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