Name:___________________ Date:_________

Regents Physics Mr. Morgante

UNIT 3

Work, Energy & Power

2

Work

Work is equal to the force applied to an object that results in the object’s displacement.

Displacement is a vector! So again, direction and sign convention are important.

Eq: W = F x d W = Work (Joules OR N∙m)

Units: J = N x m F = Force (Newtons)

d = displacement (meters)

This J is a unit, it is not impulse!

↓

Joules

Joules is a unit of energy, therefore we must input energy into moving an object to do

work.

If I do work on an object, I have put energy into the object! I push a lawnmower, I put

energy into moving a lawnmower.

Can we derive Kinetic Energy equations from work and Newton’s 2nd

Law!

W = FNet∙d AND FNet = m∙a

W = m∙a∙d where v2 = 2a∙d from kinematics equations therefore v

2 / 2 = a∙d

W = m∙(v2 / 2) W = Joules = energy

KE = Joules = energy

KE = ½ m∙v2

a = ∆v/ t

v = d/t

Typically W = F∙d

Fapp. = 10N

d=10 m

Fapp.=10N

W = F∙d = 10N x 10 m = 100 N∙m = 100 J

3

Kinetic Energy of an Object

“Kinetic” means moving, therefore object has to be moving.

Does it have Kinetic Energy (KE)? Yes it moves, therefore it has KE.

KE = ½ mv2 KE = Kinetic Energy (Joules)

m = mass (kg)

v = Velocity (m/s) [has direction & magnitude]

↓

OR speed (m/s) [has only direction]

Ex. v = 10 m/s

m = 2kg

KE = ½ mv2 = ½ (2kg)(10m/s)

2 = 100 Joules

Different Forms of Energy

Gravitational Potential Energy (PEg)

When work is done against gravity

i.e. lifting a box, climbing stairs, climbing a mountain, etc.

∆PE = mg∆h ∆PE = change in potential energy (Joules)

m = mass (kg)

g = gravity (m/s2)

∆h = change in height OR elevation (m)

mg = weight = (N)

SKETCH

h g

F

m

4

Work Against Friction

Trick Question - Object moves at constant speed of 2 m/s for a distance of 5m. What

work is done on the object? [Ans: NONE b/c constant velocity = no accel.=No Force]

Sketch: FN

Ff Fapplied

Fg=m∙g

Elastic Potential Energy mg = Fg

Hooke’s Law = F = k∙x SKETCH

↓ ↓ x

(N)=(N/m)(m)

k = spring constant (N/m)

When we stretch a spring, we input stored energy (PEs) → Potential Energy of Spring

into the spring. We know this because when the spring is released, it moves and when it

moves it transforms PEs into KE.

Eq. for PEs

PEs = ½ k∙x2 Please note this is NOT F = K∙x

↓ ↓

This is energy This is force

k = spring constant (N/m) **If you use F=k∙x, you can first get k

and then k plug into PEs = ½ k∙x2**

x = displacement (m)

PEs = stored energy (J) or (N∙m)

5

Area Under Graphs and Work & Energy

F

d

Area of triangle = ½ b∙h

Units for energy Joules or N∙m

Slope = spring constant

UNITS ANALYSIS

Y – axis → N

X – axis → m

Therefore looking at units for the area under line

½ Base∙Height OR

½ Force∙distance = F∙d

Units Analysis = N∙ m = J which is energy!

6

7

Pendulum- Period

T=2Π√l/g Time to complete 1 full swing

No friction, pendulum will keep swinging

Period depends on length (l) and gravitational acceleration (g)

Period is independent of mass of the bob

v=0 here so MAX PEg!

h

Equilibrium point Vmax is here so MAX KE!

Period is the time taken by the bob to go from one end of its swing to the opposite end

and then return to the starting point

- Stick different masses on pendulum and determine T

- Set different length and determine T

Total Energy= ∆KE + ∆PE + Internal Energy

We will talk about forms of energy in motors, generators, photocell, battery.

Power

Power is the rate at which work is done, or the rate at which energy is used transferred.

The SI unit for power is the watt (W).

A power of 1W means that work is being done at the rate of 1 J/s.

Larger units for power are the kilowatt kW (1kW = 1000 W = 103

W) and

the megawatt MW (1 MW = 1000000 W = 106

W).

8

If work is being done by a machine moving at speed v against a constant force, or

resistance, F, then since work doe is force times distance, work done per second is Fv,

which is the same as power.

Example 1

A constant force of 2 kN pulls a crate along a level floor a distance of 10 m in 50s.

What is the power used?

Solution

Alternatively we could have calculated the speed first

and then calculated power

9

Example 2

A hoist operated by an electric motor has a mass of 500 kg. It raises a load of 300 kg

vertically at a steady speed of 0.2 m/s. Frictional resistance can be taken to be constant at

1200 N.

What is the power required?

Solution

From the Previous Power Equation

10

Name_______________________________ Date_________

Regents Physics Mr. Morgante

Energy Notesheet , Part I

Definitions:

1. Energy

2. Work

3. Joule

4. Power

5. Watt

6. Potential Energy

7. Gravitational Potential Energy

Equation Variables/ constants Units Can be used to find Vector/scalar

ΔPE

ΔPE = mgΔh m

g

Δh

W

W = Fd = ΔET F

d

ΔET

P

P = W = Fd =F v W

t t F

d

t

v

A

Ay

θ

Ax

Ay = A sin θ

Ax = A cos θ

11

+y

8. Work and gravitational potential energy examples +x

Case 1 Case 2 Case 3

F F F

no friction μ static μ kinetic

a = _______ a= ___0____ a = ___0____

v= ________ v = ________ v = ________

d = _______ d = ________ d = ________

Ff =_______ Ff= ________ Ff= ________

W = ______ W = ________ W = _______

Summary

______________________ ____________________ __________________

Case 4 Case 5 Case 6 (frictionless)

F

θ

h

μ kinetic

a = _______ h θ

v= ________

d = _______

Ff =_______

W = ______

Summary:

mass mass mass

12

Name_______________________________ Date___________

Regents Physics Mr. Morgante

Energy Notesheet, Part II

9. Force versus displacement graph

“The area under a Force versus displacement graph can be used to find _________”

Calculate the work done in each case:

F F F F

disp disp disp disp

Work =____________ ____________ __________ _________

10. Definitions: Forms of Energy/Devices for converting energy

Internal energy:

Nuclear energy:

Electromagnetic energy:

Photocell

Generator

Motor

Battery

13

Name_______________________________ Date___________

Regents Physics Mr. Morgante

Energy Notesheet, Part III, Elastic Potential Energy

Definitions:

1. Compression

2. Elastic Potential Energy

3. Elongation

4. Hooke’s Law

5. Spring constant

6. Equilibrium position

Equation Variables/ constants Units Can be used to find Vector/scalar

Fs Fs Fs

Fs = kx

k k k

x x x

PEs PEs PEs

PEs = ½ kx2

k k k

x x x

Hooke’s Law:

Sketch the graph of an elastic material that is being elongated according to Fs = kx.

a)What is the slope of this graph?__________

Fs b)What are the units of k?________________

c)What quantity can be computed for the area

under an F versus displacement graph?

x __________________

d)What is the equation for the area of a triangle?_______________________

e) Calculate the work done on a spring if the endpoints of the graph are (0m,0N) & (1.7m,20N)

14

Elastic Potential Energy:

a) Using PEs = ½ kx2 , and k = 15 N/m, complete the table below:

x value 0m 0.25m 0.5m 0.75m 1m 2m

PEs

b) Sketch the graph of an elastic material that is being elongated according to Fs = kx.

PEs

X

c) Using PEs = ½ kx2

Solve for k solve for x

____________________ ________________________

d) Prove that PEs is equivalent to Work is equivalent to ΔPE using units:

_______________ ________________ _______________

15

Name_______________________________ Date___________

Regents Physics Mr. Morgante

Energy Notesheet, Part IV, Kinetic Energy

Definitions:

1. Kinetic energy

2. Potential energy

3. Momentum

Equation Variables/ constants Units Can be used to find Vector/scalar

KE KE KE

KE = ½ mv2

m m m

v v v

ΔPE ΔPE ΔPE

ΔPE = mgΔh

m m m

g g g

Δh Δh Δh

p = mv p p p

m m m

v v v

Algebra review:

1. If KE = ½ mv2, solve for: m = ______________, v = ________________

2. If the velocity of an object is doubled, the KE is ______________________

3. If the velocity of an object is halved, the KE is ______________________

Graph review: Sketch the basic shapes of the graphs below(mass remains constant):

KE p

velocity velocity

16

Unit Review:

a)show how Work = KE with units

Practice Problem:

A 2-kg object falls from rest from a 490 m cliff. Ignore air resistance. Use g = 9.8 m/s2

Time (s) 0 1 2 3 4 5 6 7 8 9 10

Velocity

Displacement

PEg

KE

Momentum

(Not to scale)

+x

+y

17

Name_______________________________ Date_________

Regents Physics Mr. Morgante

Energy Notesheet, Part V, Conservation of Energy

Definitions:

4. Conservative force

5. Nonconservative force

6. Closed system

7. Law of conservation of energy:

8. Mechanical energy

9. Ideal mechanical system

10. Simple pendulum

11. Nonideal mechanical system

12. Total energy

Equation Variables/ constants Units Can be used to find Vector/scalar

ET ET ET

ET=PE+KE+Q

PE PE PE

KE KE KE

Q Q Q

ΔPE ΔPE ΔPE

ΔPE = mgΔh

m m m

g g g

Δh Δh Δh

KE = ½ mv2 KE KE KE

m m m

v v v

18

Algebra review:

1. Given: KE = ½ mv2, ΔPE = mgΔh, W = Fd, PEs = ½ kx

2

a) If ½ mv2

= mgΔh , solve for v: b) If ½ mv2

= mgΔh , solve for Δh:

a)_____________ b)_____________

c) If mgΔh = ½ kx2 , solve for k: d) If mgΔh = ½ kx

2, solve for x:

c)_______________ d)_____________

e) If mgΔh = ½ kx2 , solve for Δh: f ) If mgΔh = ½ kx

2 , solve for m:

e)_______________ f)_____________

g) If ½ mv2 = ½ kx

2 , solve for v: h ) If ½ mv

2 = ½ kx

2 , solve for m:

g)______________ h)____________

i) If ½ mv2 = ½ kx

2 , solve for k: j) If ½ mv

2 = ½ kx

2 , solve for x:

i)______________ j)_____________

19

Conservation of Energy Systems:

Case 1 Case 2

Object in free-fall above ground to h = 0 Object projected vertically (+y) from ground to

v i = 0, h top of arc

+y

+y

h = 0 h = 0

PE i = ________ KE i = ________ PE i = ________ KE i = _______

PE f = ________ KE f = ________ PE f = ________ KE f = _______

What Energy transformation occurs? What energy transformation occurs?

______________________________ ________________________________

Case 3 Case 4

Object on inclined plane Object launched vertically from spring

v i = 0 System:

h

h = 0

PE i = ________ KE i = ________ PE i = ________ KE i = _______

PE f = ________ KE f = ________ PE f = ________ KE f = _______

What Energy transformation occurs? What energy transformation occurs?

______________________________ ________________________________

20

Name:__________________ Date:__________

Regents Physics Mr. Morgante

Work Worksheet

1. What is the spring constant of a spring of negligible mass which gained 8 joules

of potential energy as a result of being compressed 0.4 meters?

(1 ) 100N/m (2 ) 50N/m

(3 ) 0.3N/m (4 ) 40N/m

2. Work is done when a force _______

(1 ) acts vertically on a cart that can only move horizontally

(2 ) exerted by one team in a tug of war when there is no movement

(3 ) is exerted while pulling a wagon up a hill

(4 ) of gravitational attraction acts on a person standing on the surface of the Earth

3. A spring of negligible mass with a spring constant of 200 newtons per meter is

stretched 0.2 meters. How much potential energy is stored in the spring?

(1 ) 40 J (2 ) 20 J

(3 ) 8 J (4 ) 4 J

4. An object gains 10 joules of potential energy as it is lifted vertically 2.0 meters. If

a second object with one-half the mass is lifted vertically 2.0 meters, the potential

evergy gained by the second object will be

(1 ) 10. J (2 ) 20. J

(3 ) 5.0 J (4 ) 2.5 J

5. A cart of mass M on a frictionless track starts from rest at the top of a hill having

height h1, as shown in the diagram below. What is the kinetic energy of the cart

when it reaches the top of the next hill, having height h2?

(1 ) mgh1 (2 ) mg(h1-h2)

(3 ) mg(h2-h3) (4 ) 0

21

6. A force is applied to a block, causing it to accelerate along a horizontal,

frictionless surface. The energy gained be the block is equal to the

(1 ) Work done on the block (2 ) power applied to the block

(3 ) impulse applied to the block (4 ) momentum given to the block

7. A 1.0 x103 -kilogram car is moving at a constant speed of 4.0 meters per second.

What is the kinetic energy of the car?

(1 ) 1.6 x 103 J (2 ) 2.0 x 10

4 J

(3 ) 8.0 x 103 J (4 ) 4.0 x 10

3 J

8. When a spring is stretched 0.200 meter from its equilibrium position, it possesses

a potential energy of 10.0 joules. What is the spring constant for this spring?

(1 ) 100. N/m (2 ) 125 N/m

(3 ) 250. N/m (4 ) 500. N/m

9. A constant force of 2.0 newtons is used to push a 3.0-kilogram mass 4.0 meters

across the floor. How much work is done on the mass?

(1 ) 6.0J (2 ) 8.0J

(3 ) 12J (4 ) 24J

10. A student rides a bicycle up a 30° hill at a constant speed of 6.00 meters per

second. The combined mass of the student and bicycle is 70.0 kilograms. What is

the kinetic energy of the student-bicycle system during this ride?

(1 ) 210. J (2 ) 420. J

(3 ) 1,260 J (4 ) 2,520 J

11. If the distance a spring is stretched is doubled, the potential energy is:

(1) four times as great (2) one fourth as great

(3) twice as great (4) the same

12. A 5-kg cart is rolling along on the ground when an additional 5-kg mass is placed

on the cart. The KE of the cart is now:

(1) four times as great (2) one fourth as great

(3) twice as great (4) the same

13. In raising an object vertically at a constant speed of 2.0 meters per second, 10

watts of power is developed. The weight of the object is

(1) 5.0 N (3) 40. N

(2) 20. N (4) 50. N 14. An object moving at a constant speed of 25 meters per second possesses 450 joules of

kinetic energy. What is the object’s mass?

(1) 0.72 kg (3) 18 kg

(2) 1.4 kg (4) 36 kg

22

Name:_________________ Date:__________ Regents Physics Mr. Morgante

Work and Power Worksheet Priscilla and Larry begin to climb the stairs at the end of the science wing in WHS to travel to physics class. The vertical rise of the stairs is 4.0 meters. Priscilla’s mass is 45.0 kg, while Larry has a mass of 60.0 kg. Priscilla makes the climb in 2.0 seconds, while Larry takes 30.0 seconds to climb the same distance. 1. Calculate Larry’s weight (metric) 2. Calculate the work done by Priscilla in climbing the stairs? 3. Calculate Priscilla’s power rating. Larry normally takes 30.0 seconds to climb the stairs, on a particular day he is the recipient of verbal help from a teacher. With the help he manages to climb the stairs in 2.0 seconds. 4. Compare the work he did climbing the stairs on a normal day to this special day. 5. Compare his usual power rating to his power rating on that special day.

Name:_______________ Date:__________

Regents Physics Mr. Morgante

23

Work, Power, KE, and PE Problems

Work and Power

A) What work is done by a girl who pushes a box along a floor with a force of 52.0 N

for a distance of 11.0 m?

B) A boy raises a 20.0 Kg rock 2.3 meters: -What is the force that the boy uses to raise the rock?

-Calculate the amount of work that he does?

C) A student is pulling on a wagon handle with a force of 40.0 N. The handle is at a 30-degree angle with the horizontal. The wagon moves 8 meters in 10 seconds. Find the

work done by the student and the power exerted by the student. Kinetic Energy

A) What is the kinetic energy of an object who’s mass is 5.0 Kg and is moving a 4.0

m/s? If the object was accelerated from rest for a distance of 10.0 m, what was the

force applied to it?

(OVER) B) A force of 10 N is applied to a body on a practically frictionless table over a distance

of 8.0 m, what is the Kinetic energy it imparts to the body? If the body starts at rest

24

and has a mass of 4.0 kg, what velocity does the force impart?

C) When the brake is applied to a car having a mass of 1000 Kg, its speed is reduced

from 30 m/s to 20 m/s. How much work does the brake do on the car? If the brake is applied for a distance of 25 m, what force does it exert on the car?

Potential Energy

A) A 5.0 Kg rock is located on a ledge 10 meters above the ground, calculate the

potential energy of the rock relative to the ground.

Name:_______________ Date:________

Regents Physics Mr. Morgante

25

Work and Energy Graph Worksheet

1. Calculate the work done on the cart by the force shown in the diagram below.

30N

60o

25 m

2. The force on an object varies as shown in the graph below.

Force (N) vs. Distance (m)

0

10

20

30

40

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Distance (m)

Fo

rce (

N)

Series1

What is the work done on the object in the first 1.1 meters?

3. How many joules of energy are produced by a 60 watt light bulb in 2 minutes?

(OVER)

26

4. The coefficient of friction between a 5.0 kg mass and a desktop is 0.20. An

unbalanced force of 50.0 N acts horizontally on the mass to move it along the

desktop as shown below.

5.0 kg

50 N

+X

a. What is the weight of the mass?

b. What is the size and direction of the frictional force acting on the mass?

c. What is the value and direction of the unbalanced force acting on the mass?

d. What is the acceleration experienced by the mass?

e. The unbalanced force is reduced to zero by lowering the outside push to

9.81N after the mass has reached a speed of 6 m/s. At what power is work

being done on the mass if this is true?

27

Name:_______________ Date:________

Regents Physics Mr. Morgante

More Work, KE, PE in Springs & Pendulum Problems

1. A spring is stretched for a distance of 1m by 10N force. What is the potential energy stored in the spring (Think F=kx)?

2. A spring with a spring constant of 3 N/m is stretched a distance of 4m. What is

the PEs?

3. A 30-kg box is pulled at constant velocity of 4 m/s across a rough surface.

15 N

θ = 25º

+x

a) Calculate the horizontal component of the pulling force, Fx. (Show mag. & dir.)

b) Calculate the friction force, Ff Sketch the vector on the diagram(Show mag. &

dir.)

c) Calculate the work against friction in 90 seconds (Show mag. & dir.):

d) Calculate the work done by friction in 90 seconds (Show mag. & dir.):

(OVER)

28

4. Answer the following questions based on the diagram below:

Pt. B

h

Pt. A

mass of pendulum ball = 2 kg h = 2 m length of string = 4 m

a) What is the PEmax?

b) What is the maximum velocity of the ball when it reaches Pt. A after being

released from Pt. B?

c) What is the KEmax?

5. Find the potential energy stored in the spring below. The mass of the object is 40

dg and the distance the spring is elongated 2m. Show all of your work.

Spring

m

h=2m

m

a) What is the max. velocity and KEmax of the mass?

b) What is the max PEg of the mass?

29

NAME________________________________ DATE________

Regents Physics Mr. Morgante

Nonideal Mechanical Systems

Objective: Investigators will analyze nonideal mechanical system and develop

solutions based on evidence.

Definition:

Nonideal mechanical system:________________________________________________

Sketch: vi = 0 m/s

m = 40 kg

Point A

L = 3 m

40O h = 0

1. A 40-kg box initially at rest slides down a 3 m long inclined plane that is elevated 40º

to the horizontal. At point A, the bottom of the plane, the velocity of the block is found to

be 4 m/s.

a) Calculate the gravitational potential energy of the box at the top of the inc. plane

a)________________

b) What is the grav. potential energy of the box at the bottom of the plane?

b)________________

c) What is the kinetic energy of the box at the bottom of the plane?

c)________________

d) Has ET been conserved in this system? Yes No

Explain: __________________________________________________________

e) If ET = PE + KE + Q, What is the probable value of Q in this system?

(over) e)________________

30

f) What condition probably caused the increase in Q? f)________________

g) How much work was done by friction in this system? g)________________

h) Calculate the average friction force acting on the box:

Magnitude Direction

h)_______ ________

i) Sketch the box on the plane and show friction force direction, Normal force exerted on

the box by the plane, Fg on box:

Calculate Fg

Magnitude Direction

_______ ________

j) Neglecting friction on the inclined plane, calculate the theoretical speed of the box

at the bottom of the plane:

j)______________

k) What is the magnitude of the normal force acting on the box?

k)_______________

l) Additional comments

31

NAME________________________________ DATE________

Regents Physics Mr. Morgante

Law of Conservation of Energy Practice

1. A 3.0-kg mass free-falls from rest a distance of 10m to the ground (h=0)

Use g = 10 m/s2

; neglect air resistance; Complete the table below

Time

Dist.

from

ground

(m)

10 9 8 7 6 5 4 3 2 1 0

PEg

KE

Velocity

2. Using the strip of graph paper provided, Label x-axis 0 to 10m, y-axis Energy

Using RED pencil plot PEg, BLUE pencil KE, GREEN pencil ET

Attach graph here

3. Explain how this exercise helps illustrate the Law of Conservation of Energy

32

Name:_______________________ Date:__________

Regents Physics Mr. Morgante

Energy Stored in Head-On Collisions

The graph below shows the kinetic energy of a moving cart vs. time as it collides with

the spring bumper of a fixed cart. The mass of the moving cart is 1.0 kg.

KE (J) vs. Time (s)

012345678

0 1 2 3 4 5 6 7

Time (S)

KE

(J)

KE (J)

1. Determine the time at which the spring reaches its maximum compression.

2. The graph shows that the kinetic energy of the cart on rebounding from the spring

bumper is less than before the collision. Explain a possible cause.

3. The cart compresses the spring and then rebounds. Why is the graph not drawn as

the graph shown below?

(OVER)

33

KE (J) vs. Time (s)

-3-2-1012345678

0 1 2 3 4 5 6 7 8

Time (s)

KE

(J

)Series1

4. If the KE of the cart at t=1.0 seconds is 8 J calculate the speed of the cart.

The graph below relates the force exerted on a spring to its compression.

Force (N) vs. Compression (m)

0

1

2

3

4

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Compression (m)

Fo

rce (

N)

Series1

5. Use the graph to find potential energy stored in the spring when it is compressed

by 0.04 m.

34

Name:__________________________ Date:_________

Regents Physics Graphs and Work Worksheet Mr. Morgante

Find the work done in each of the cases pictured below. Show All Work!!!!!!

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30

Displacement (m), Left

Fo

rce

(N

)

Series1

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6

Displacement (m), North

Fo

rce

(N

)

Series1

02468

1012141618

1 2 3 4 5 6 7

Fo

rce

(N

)

Displacement (m), West

Series1

0

2

4

6

8

10

12

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Spring Elongation (m)

Fo

rce (

N)

Series1

Z:\Physics\Regents Physics\Class Material\Unit 3 Work & Energy 1-11-10.doc