Page 1

Work & Energy

Discussion

Definition

Dot Product

Work of a constant force

Work/kinetic energy theorem

Work of multiple constant forces

Comments

SPH3U:

Introduction to Work and Energy

Work & Energy

One of the most important concepts in physics

Alternative approach to mechanics

Many applications beyond mechanics

Thermodynamics (movement of heat)

Quantum mechanics...

Very useful tools

You will learn new (sometimes much easier) ways to

solve problems

Energy is..

• The ability to do work

• Measured in Joules

• We use energy to do

whatever task we

need to do (move a

car, lift a pencil, etc)

Types of Energy

Energy can come in many forms including:

– Thermal (energy in the form of moving atoms)

– Electrical (energy possessed by charged particles)

– Radiant (energy found in EM waves)

– Nuclear (energy stored in holding the atom together)

– Gravitational (energy stored due to a raised elevation)

Types of Energy cont.

– Kinetic (energy due to motion of objects)

– Elastic (energy stored in compression or stretch)

– Sound (energy in vibrations)

– Chemical (energy stored in molecular bonds)

Energy 101

Energy cannot be created or destroyed, it can

only transform from one form to another.

• Eg. A student turns on the stove to heat a pot

of water

Electrical Radiant Thermal

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Energy and Work

• We know energy is the ability to do work, but

what is work?

• In physics work is the energy transferred to an

object by an applied force over a displacement

Forms of Energy

Kinetic: Energy of motion.

A car on the highway has kinetic energy.

We have to remove this energy to stop it.

The brakes of a car get HOT!

This is an example of turning one form of energy into

another (thermal energy).

21

2K mv

Mass = Energy

Particle Physics:

+ 5,000,000,000 V

e-

- 5,000,000,000 V

e+ (a)

(b)

(c)

E = 1010 eV

M E = MC2

( poof ! )

Energy Conservation

Energy cannot be destroyed or created.

Just changed from one form to another.

We say energy is conserved!

True for any closed system.

i.e. when we put on the brakes, the kinetic energy of the car is turned into heat using friction in the brakes. The total energy of the “car-brakes-road-atmosphere” system is the same.

The energy of the car “alone” is not conserved...

It is reduced by the braking.

Doing “work” on an isolated system will change its “energy”...

Work

• Mechanical work is done on an object when a force displaces an

object.

• Note that this equation only applies when the force is constant and

the force and displacement are in the same direction.

• When the force and displacement are not entirely in the same

direction, the component of the force in the direction of the

displacement is used.

W=FΔd

Definition of Work:

Ingredients: Force (F), displacement (r)

Work, W, of a constant force F

acting through a displacement r is:

W = F r = F r cos = Fr r

F

r Fr

“Dot Product”

The dot product allows us

to multiply two vectors, but

just the components that

are going in the same

direction (usually along

the second vector)

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Definition of Work...

Only the component of F along the displacement

is doing work.

Example: Train on a track.

F

r

F cos

Aside: Dot Product (or Scalar

Product) Definition:

a.b = ab cos

= a[b cos ] = aba

= b[a cos ] = bab

Some properties:

ab = ba q(ab) = (qb)a = b(qa) (q is a scalar) a(b + c) = (ab) + (ac) (c is a vector) The dot product of perpendicular vectors is 0 !!

a

ab b

a

b

ba

Work: Example

(constant force)

A force F = 10 N pushes a box across a frictionless

floor for a distance x = 5 m.

x

F

Work done by F on box is:

WF = Fx = F x (since F is parallel to x)

WF = (10 N) x (5 m) = 50 Joules (J)

Units of Work:

Force x Distance = Work

N-m (Joule) Dyne-cm (erg)

= 10-7 J

BTU = 1054 J

calorie = 4.184 J

foot-lb = 1.356 J

eV = 1.6x10-19 J

cgs other mks

Newton x

[M][L] / [T]2

Meter = Joule

[L] [M][L]2 / [T]2

Example 1

How much mechanical work will be done

pushing a shopping cart 3.5m with a force of

25N in the same direction as the displacement?

.

.

25 3 5

87 5

W Fd

N m

J

Example 2 A curler applies a force of 15.0N on a curling stone and

accelerates the stone from rest to a speed of 8.00m/s

in 3.5s. Assuming the ice surface is level and

frictionless, how much mechanical work is done on the

stone?

15 d

W Fd

N

.

0

2

0 83 5

2

14

fv vt

m m

d

s s s

m

1415

210

W d

N m

F

J

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Example 3

Calculate the work done by a custodian on a vacuum

cleaner if the custodian exerts an applied force of

50.0N on the vacuum hose and the hose makes a 30°

angle with the floor. The vacuum moves 3.0m to the

right on a flat level surface.

cos

.

cos

1

50 3

29 9

30

W F d

Fd

N m

J

Useless work?

• No work: when there is no displacement, no

work is done! Work can also be positive or

negative relative to the motion, as shown in

the next example.

Comments:

Time interval not relevant

Run up the stairs quickly or slowly...same Work

Since W = F r

No work is done if:

F = 0 or

r = 0 or

= 90o

Example 4 A shopper pushes a shopping cart on a horizontal

surface with a horizontal applied force of 41.0N for

11.0m. The cart experiences a force of friction of 35.0N.

Calculate the total work done on the cart.

45

41 11

1

shopperW F d

N m

J

385

35 11

frictionW F d

N

J

m

6

451

6

385

Net shopper frictionW W W

J J

J

Work & Kinetic Energy:

A force F = 10 N pushes a box across a frictionless

floor for a distance x = 5 m. The speed of the box is v1

before the push and v2 after the push.

x

F

v1 v2

m

Work & Kinetic Energy...

Since the force F is constant, acceleration a will be

constant. We have shown that for constant a:

v22 - v1

2 = 2a(x2-x1) = 2ax.

multiply by 1/2m: 1/2mv22 - 1/2mv1

2 = max

But F = ma 1/2mv22 - 1/2mv1

2 = Fx

x

F

v1 v2

a m

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Work & Kinetic Energy...

So we find that

1/2mv2

2 - 1/2mv12 = Fx = WF

Define Kinetic Energy K: K = 1/2mv2

K2 - K1 = WF

WF = K (Work/kinetic energy theorem)

x

F a m

v2 v1

Work/Kinetic Energy Theorem:

{Net Work done on object}

=

{change in kinetic energy of object}

netW K

f iK K

2 21 1

2 2f imv mv

Example

A 200g hockey puck initially at rest on ice is pushed by a

hockey stick by a constant force of 6.0N. What is the

hockey puck’s speed after it has moved 2m?

WW KE

.

1

10 95

1

m

s

s

m

. . .

2

21 16 0 2 0 2 0 2 0

2 2 2f

mN m kg v kg

2 21 1

2 2Net f iF d mv mv

22

2120f

mv

s

. 212 0 1 fJ kg v

Work & Energy Question

Two blocks have masses m1 and m2, where m1 > m2. They

are sliding on a frictionless floor and have the same kinetic

energy when they encounter a long rough stretch (i.e. m > 0)

which slows them down to a stop.

Which one will go farther before stopping?

(a) m1 (b) m2 (c) they will go the same distance

m1

m2

Solution

The work-energy theorem says that for any object WNET = K

In this example the only force that does work is friction (since

both N and mg are perpendicular to the block’s motion).

m f

N

mg

Solution

The work-energy theorem says that for any object WNET = K

In this example the only force that does work is friction (since

both N and mg are perpendicular to the blocks motion).

The net work done to stop the box is - fD = -mmgD.

m

D

This work “removes” the kinetic energy that the box had:

WNET = K2 - K1 = 0 - K1

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Solution

The net work done to stop a box is - fD = -mmgD.

This work “removes” the kinetic energy that the box had:

WNET = K2 - K1 = 0 - K1

This is the same for both boxes (same starting kinetic energy).

mm2gD2 mm1gD1 m2D2 m1D1

m1

D1

m2

D2

Since m1 > m2 we can see that D2 > D1

A Simple Application:

Work done by gravity on a falling object

What is the speed of an object after falling a distance H, assuming it

starts at rest?

Wg = F r = mg r cos(0) = mgH

Wg = mgH

Work/Kinetic Energy Theorem:

Wg = mgH = 1/2mv2

r mg

H

j

v0 = 0

v 2v gH

POWER

Simply put, power is the rate at which work gets done (or

energy gets transferred).

Suppose you and I each do 1000J of work, but I do the

work in 2 minutes while you do it in 1 minute. We both did

the same amount of work, but you did it more quickly (you

were more powerful)

WorkPower

time

WP

t

J

s

Watt (W)

• Power = change in energy time

Measured in Watts (J/s or W) or in horsepower

Ex. 1 horsepower = 746 W or 0.75 kW Vehicle horsepower can be calculated using its RPM & Torque (in the manual).

Watt determined that the average horse could do 33, 000 foot-pounds of work per

minute (Ex. Move 1000 lbs of coal 33 ft every minute).

The equation for power is: P = ΔE = W Δt Δt

Power is a measure of: the change in energy

over time. aka. Energy used or Work done over time

Kahn Academy: Speed of Weight-Lifter:

https://www.youtube.com/watch?v=RpbxIG5HTf4

Power

Power is also needed for acceleration and for moving against the force of

gravity.

The average power can be written in terms of the force and the average

velocity:

av av

W FdP Fv

t t

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Understanding

A mover pushes a large crate (m= 75 kg) from one side of a truck to the

other side ( a distance of 6 m), exerting a steady push of 300 N. If she

moves the crate in 20 s, what is the power output during this move?

WP

t

Fd

t

300 6

20

90

N m

s

W

Understanding

What must the power output of an elevator motor be such that it

can lift a total mass of 1000 kg, while maintaining a constant

speed of 8.0 m/s?

WP

t

t

Fd vF vmg

1000 9.8 8.0

78,000

78

N mkg

kg s

W

kW