Chapter 6 - Giancoli - Work and Energy

Work and EnergyWork and Energy

Mr. SharickMr. Sharick

Giancoli – Chapter 6Giancoli – Chapter 6

Lesson 1: 6.1 - Energy Lesson 1: 6.1 - Energy BasicsBasics

Lesson 1: Energy Lesson 1: Energy BasicsBasics

Energy = Motion (in Mechanics)Energy = Motion (in Mechanics) Motion = SpeedMotion = Speed The more speed an object has the more The more speed an object has the more

energy it “carries.”energy it “carries.” Direction doesn’t matter – Energy is a Direction doesn’t matter – Energy is a

scalar, not a vector scalar, not a vector Units for Energy are JoulesUnits for Energy are Joules

1 Joule = 1 N∙m = 1 kg ∙ m1 Joule = 1 N∙m = 1 kg ∙ m22/s/s22 (more on this later) (more on this later)

Lesson 1: Energy Lesson 1: Energy BasicsBasics

Energy can be transferred to and from Energy can be transferred to and from objects (through acceleration)objects (through acceleration) Energy transferred to an object causes the Energy transferred to an object causes the

object to speed upobject to speed up Energy transferred away from an object Energy transferred away from an object

causes the object to slow downcauses the object to slow down

How is this energy transferred?How is this energy transferred? Through forces! Remember, forces cause Through forces! Remember, forces cause

masses to accelerate (F=ma)masses to accelerate (F=ma)

Lesson 1: WORKLesson 1: WORK We define this energy We define this energy

transfer as transfer as workwork.. WorkWork is a scalar is a scalar

quantity as is energy.quantity as is energy. Note the definition of Note the definition of

work in physics is work in physics is different than in different than in common languagecommon language

Lesson 1: WORKLesson 1: WORK Work = Energy (in physics)Work = Energy (in physics) It is energy that a force produces when the It is energy that a force produces when the

force pushes or pulls on the mass.force pushes or pulls on the mass. If the force is going in the same direction If the force is going in the same direction

as the motion it is positive work. (giving as the motion it is positive work. (giving energy to the object)energy to the object)

If the force is going in the opposite If the force is going in the opposite direction as the motion it is negative work.direction as the motion it is negative work.

(taking energy from the object)(taking energy from the object)

Mathematical Definition of Mathematical Definition of Work (Energy) – Dot ProductWork (Energy) – Dot Product

W = the energy that a force transfers to or W = the energy that a force transfers to or from an object as the object MOVES over a from an object as the object MOVES over a distance ∆xdistance ∆x

Work is only done by forces that are in the Work is only done by forces that are in the direction of or opposing the direction of direction of or opposing the direction of motion.motion.

, ( ) cos F x

W F x

W F x θ ∆

≡ • ∆= ∆

v v

1-D Example Problems:1-D Example Problems:

1.) A racecar is sitting on the track when a force of 1500 1.) A racecar is sitting on the track when a force of 1500 N to the East is exerted on it which moves it 10 m to the N to the East is exerted on it which moves it 10 m to the East. What is the angle between the force F and the East. What is the angle between the force F and the displacement Δx? What is the work done by the force?displacement Δx? What is the work done by the force?

1-D Example Problems:1-D Example Problems:

2.) A racecar is moving down the track to the East when a 2.) A racecar is moving down the track to the East when a force of 500 N to the West is exerted on it. The car force of 500 N to the West is exerted on it. The car continues to move 20 m while the force is exerted. What continues to move 20 m while the force is exerted. What is the angle between the force F and the displacement is the angle between the force F and the displacement Δx? What is the work done by the force?Δx? What is the work done by the force?

2-D2-D Dot ProductsDot Products How to take the dot product of any 2 vectors.How to take the dot product of any 2 vectors.

1.1.Draw vectors from a common originDraw vectors from a common origin

2.2.Take angle between vectors. Note: MUST be Take angle between vectors. Note: MUST be between 0between 0oo and 180 and 180oo

3.3.Multiply to get answer (no direction)Multiply to get answer (no direction)

F = 10 N

Δx = 25 mθ = 35°

Ex. Prob. 1: What is the work done by the above force?

Total Work (Net Work) Total Work (Net Work)

The total work done on a particle that moves The total work done on a particle that moves from position xfrom position x11 to x to x2 2 where where

is:is: ( ) ( ) ( )1 2 3

,

...

( )( ) ( )(cos )( )

net

net net x net

W F x F x F x

W F x F xθ

= • ∆ + • ∆ + • ∆ +

= ∆ = ∆

v v vv v v

12 xxx vvv −=∆

2-D Example Problem2-D Example Problem

Calculate the total work done on a mass m as Calculate the total work done on a mass m as it moves from position xit moves from position x11 = 0 m to x = 0 m to x22 = 40 m = 40 m

X2 = 40 X1 = 0

20o

F1 = 5 N

F2 = 6 N

F3 = 2 N

F4 = 2 N 50o

Work Review Work Review QuestionsQuestions

Which of the following statements are true about work?

a. ) Work is a form of energy.

b. ) A Watt is the standard metric unit of work.

c. ) Units of work would be equivalent to a Newton times a meter.

d.) A kg•m2/s2 would be a unit of work.



e.) Work is a time-based quantity; it is dependent upon how fast a force displaces an object.

f.) Superman applies a force on a truck to prevent it from moving down a hill. This is an example of work being done.

g. ) An upward force is applied to a bucket as it is carried 20 m across the yard. This is an example of work being done.



h. ) A force is applied by a chain to a roller coaster car to carry it up the hill of the first drop of the Shockwave ride. This is an example of work being done.

i. ) The force of friction acts upon a softball player as she makes a headfirst dive into third base. This is an example of work being done.

j. ) An eraser is tied to a string; a person holds the string and applies a tension force as the eraser is moved in a circle at constant speed. This is an example of work being done.



k.) A force acts upon an object to push the object along a surface at constant speed. By itself, this force must NOT be doing any work upon the object.

l.) A force acts upon an object at a 90-degree angle to the direction that it is moving. This force is doing negative work upon the object.

m.) An individual force does NOT do positive work upon an object if the object is moving at constant speed.

n.) An object is moving to the right. A force acts leftward upon it. This force is doing negative work.

Lesson 2: Work and Lesson 2: Work and Kinetic Energy TheoremKinetic Energy Theorem

� For a constant net force, the acceleration For a constant net force, the acceleration is constant and we can use:is constant and we can use:

I.I.

II.II.

Solve II. for Solve II. for ∆∆x:x:

2 2

2

2

x x

f i x

F ma (Newton's nd Law)

v v a x (Const. Acceleration Equat.)

∑ =

= + ∆

Fx

vi vf


Substitute Substitute ∆∆x into:x into:

This yields: This yields:

2 2

2f i

x

v vx

a

−∆ =

net xW F x≡ ∆


Substitute Substitute ∆∆x into:x into:

This yields:This yields:

So work has something to do with the So work has something to do with the mass of the object and its speed squared. mass of the object and its speed squared.

2 2

2f i

x

v vx

a

−∆ =

net xW F x≡ ∆

2 2

2 2

( ) ( )2

1 1W

2 2

f inet x

x

net f i

v vW ma

a

mv mv

−=

= −

Kinetic EnergyKinetic Energy

The quantity ½ mvThe quantity ½ mv22 is called Kinetic is called Kinetic Energy (KE)Energy (KE)

Energy due to motion; if an object has Energy due to motion; if an object has speed it has kinetic energy.speed it has kinetic energy.

21

2KE mv=

Einstein said this?Einstein said this?

Einstein’s famous equation is:Einstein’s famous equation is:

Work - Kinetic Energy Work - Kinetic Energy TheoremTheorem

Work done by a constant force is equal Work done by a constant force is equal to the change of kinetic energy of the to the change of kinetic energy of the object.object.

Sign for work designates an increase or Sign for work designates an increase or decrease in energy (Work can be decrease in energy (Work can be positive or negative, but KE is always positive or negative, but KE is always positive; change of KE can be negative)positive; change of KE can be negative)

2 21 1W

2 2done f iKE mv mv= ∆ = −

Example: Finding change Example: Finding change in KE (work)in KE (work)

A 3kg ball is thrown at a wall with A 3kg ball is thrown at a wall with

The ball loses some energy when it strikes the wall. The The ball loses some energy when it strikes the wall. The ball rebounds with a velocity of ball rebounds with a velocity of

a.) Calculate the change in KE of the particle.a.) Calculate the change in KE of the particle.

b.) If the final velocity of the ball isb.) If the final velocity of the ball is

what would be the net work done on the ball? what would be the net work done on the ball?

xsmvi ˆ / 5=

ˆ4.2 / fv m s x= −

ˆ5 / fv m s x=

Lesson 3: Understanding Lesson 3: Understanding Potential EnergyPotential Energy


History of Science: Enrique Joule’s Experiment•What happens as the masses drop?


History of Science: Enrique Joule’s Experiment•What happens as the masses drop? The original Joule experiment consists of a receptacle filled with water and a mechanism with spinning plates. The kinetic energy of the plates is transformed into heat, because the force of gravity performs work on the weight falling. This gave an experimental confirmation of the equivalence between heat and work, now defined to be exactly 1 calorie for every 4.184 joules and called a "thermochemical calorie".

Potential Energy Kinetic Energy Thermal Energy (Heat)

Lesson 3: Lesson 3: Understanding Understanding Potential EnergyPotential Energy Think about when you ride a roller Think about when you ride a roller

coaster: coaster: On the way up energy is being On the way up energy is being

transferred to you. (negative work transferred to you. (negative work done by gravity)done by gravity)

On the way down, the energy is On the way down, the energy is transferred to kinetic energy (positive transferred to kinetic energy (positive work done by gravity; you speed up work done by gravity; you speed up going down the hill)going down the hill)

Lesson 3: Work done Lesson 3: Work done by gravity (potential by gravity (potential energy)energy)

As you lift a mass to height h, what is the work done by As you lift a mass to height h, what is the work done by gravity?gravity?

yf = h ------------------------

yi =0 m ------------------------


When the mass falls from the same height h, what is the When the mass falls from the same height h, what is the work done by gravity?work done by gravity?

yf = h ------------------------

yi =0 m ------------------------

Gravity does negative work when the object is Gravity does negative work when the object is being lifted and the energy is transferred to the being lifted and the energy is transferred to the object (giving it gravitational potential energy)object (giving it gravitational potential energy)

Change in PE is positive when gravity is doing Change in PE is positive when gravity is doing negative work (ynegative work (yff > y > yii))


done by gravity

f i

PE W

PE mg (y y )

∆ = −

∆ = + −

Potential energy is defined as a function Potential energy is defined as a function of position or height (y) as: of position or height (y) as:

U is the symbol for potential energy U is the symbol for potential energy function in some textbooks and college function in some textbooks and college physics courses. We will stick with using physics courses. We will stick with using PE in this class.PE in this class.

Mathematical Potential Mathematical Potential Energy function for Energy function for GravityGravity

U(y) ≡ mgy

The value for the potential energy at any The value for the potential energy at any given point due to gravity can therefore given point due to gravity can therefore be found by using the equation:be found by using the equation:

Mathematical Potential Mathematical Potential Energy function for Energy function for GravityGravity

gravityPE mgy=U(y)

y

Mechanical EnergyMechanical Energy The sum of the potential and kinetic energy.

ME = KE + PE Mechanical energy in a system can be

conserved if there are ONLY conservative forces doing work.

Initial energy = final energy

MEi = MEf

KEi + PEi = KEf + PEf

Mechanical energy in a system will be lost if there is a non-conservative force doing work. (more on this later).

Conservative ForcesConservative Forces Conservative Forces - Forces for which the work done

only depends on the initial and starting position, but not the path it takes (gravitational; elastic; electric)

Ex: Gravity is a conservative force because the work that gravity does depends only on the change of height of the object, not the path it takes to get to that height.

Non-conservative Non-conservative ForcesForces

Non-conservative Forces – (called dissipative forces) forces for which the work done depends on the path taken by the object. (friction, air resistance, tension in cord, motor or rocket propulsion, push or pull by person)

Ex: The path this box takes as it slides across the floor matters. It takes the least amount of work (energy) to slide it in a straight line from point 1 to 2 because you have to work against a non-conservative force (friction)

EX. A 1kg mass is dropped straight down from a EX. A 1kg mass is dropped straight down from a height of 20 m. For each of the heights below find height of 20 m. For each of the heights below find KE, PE and ME. a.) y = 20 m b.) y = 13 m c.) y = 7 m KE, PE and ME. a.) y = 20 m b.) y = 13 m c.) y = 7 m d.) y = 0 md.) y = 0 m

When there is friction present, it “robs” an object of When there is friction present, it “robs” an object of some of its mechanical energy. The energy is lost some of its mechanical energy. The energy is lost to the environment through heat or sound and to the environment through heat or sound and therefore:therefore:

Since friction always opposes motion, it always Since friction always opposes motion, it always does negative work.does negative work.

Lesson 4: Conservation Lesson 4: Conservation of Energy with Frictionof Energy with Friction

(KEi + PEi) ≠ (KEf + PEf)Specifically:

(KEi + PEi) > (KEf + PEf)and

Wfr = (KEf + PEf) - (KEi + PEi)

Extending the work-energy principle

The net work done by a force on an object was found to be the change in kinetic energy of the object.

Wnet = ΔKE (1)

We write the total (net) work as the sum of the work done by conservative forces (WC) and the work done by non-conservative forces (WNC).

Wnet = WNC + WC (2)


Combining eq. 1 and 2 gives:

ΔKE = WNC + WC (3)

Recalling that the work done by a conservative force can be written as the change in the potential energy (as is the case for gravity)

Wc = -ΔPE (4)


Solving eq. 3 for WNC yields:

WNC = ΔKE + ΔPE

Thus, the work done by the non-conservative forces acting on an object is equal to the total change in kinetic and potential energy.

A bicyclist (combined mass = 90 kg) starts down a hill A bicyclist (combined mass = 90 kg) starts down a hill of height 50 m with a velocity of 10 m/s. He coasts until of height 50 m with a velocity of 10 m/s. He coasts until coming to a stop 55 meters up the next hill. How much coming to a stop 55 meters up the next hill. How much work did friction do on the bicyclist? work did friction do on the bicyclist?

Lesson 4: Example Lesson 4: Example Problem 1Problem 1

Revisiting Conservative Forces

If only conservative forces are acting, then WNC = 0, thus

ΔKE + ΔPE = 0

Or

(KEf – KEi) + (PEf - PEi) = 0

This means that:

KEf + PEf = KEi + PEi

MEf = MEi

And mechanical energy is conserved! This is true when there are only conservative forces doing work!

Lesson 4: Example Lesson 4: Example Problem 2Problem 2

A ball bearing whose mass, m, is 0.0052 A ball bearing whose mass, m, is 0.0052 kg is fired vertically downward from a kg is fired vertically downward from a height, h, of 18m with an initial speed of height, h, of 18m with an initial speed of 14 m/s. It buries itself in sand to a depth, 14 m/s. It buries itself in sand to a depth, d, of 0.21 m. What average upward d, of 0.21 m. What average upward frictional force does the sand exert on the frictional force does the sand exert on the ball as it comes to rest?ball as it comes to rest?

Lesson 5: PowerLesson 5: Power

Power = the rate at which energy is being usedPower = the rate at which energy is being used

Average Power:Average Power:

(average over a time interval)(average over a time interval)

Instantaneous Power: Instantaneous Power:

(at a single instance in time)(at a single instance in time)

WP

t

dWP

dt

=∆

=

Lesson 5: Power Lesson 5: Power equation 2.0equation 2.0

Substituting W into power equation:Substituting W into power equation:

(Constant force)(Constant force)

and W

P W F xt

= = •∆∆

r r

( )

cos

F xP F v t

tP Fv θ

•∆= = •∆

=

r r r r

Power UnitsPower Units

SI: WattsSI: Watts 1 Watt = 1 J/s1 Watt = 1 J/s

Other units:Other units: 1 horsepower = 746 Watts1 horsepower = 746 Watts British horsepower = 550 ft∙lb/sBritish horsepower = 550 ft∙lb/s

Applications:Applications: Electric energy for U.S. houses is measured in Electric energy for U.S. houses is measured in

kilowatt hours (kWh) which is 3.6 x 10kilowatt hours (kWh) which is 3.6 x 1066JJ This is the amount of energy that a 1000W power This is the amount of energy that a 1000W power

source generates in one hour.source generates in one hour.

Power Example: 6.16 – Power Example: 6.16 – Stair Climbing PowerStair Climbing Power

A 70 kg jogger runs up a long flight of stairs in 4.0s. The A 70 kg jogger runs up a long flight of stairs in 4.0s. The vertical height of the stairs is 4.5m. vertical height of the stairs is 4.5m.

a.) Estimate the jogger’s power output in watts and a.) Estimate the jogger’s power output in watts and horsepower. horsepower.

b.) How much energy did this require?b.) How much energy did this require?

Power Example – 6.59Power Example – 6.59

If a car generates 18 hp (1 hp= 746W) when traveling at a If a car generates 18 hp (1 hp= 746W) when traveling at a steady 90 km/hr (25 m/s), what must be the average steady 90 km/hr (25 m/s), what must be the average force exerted on the car due to friction and air resistance?force exerted on the car due to friction and air resistance?

Lesson 6: SpringsLesson 6: Springs

Lesson 6: Hooke’s LawLesson 6: Hooke’s Law

The minus sign means that the force exerted by the The minus sign means that the force exerted by the spring is always opposite the stretching or compressing spring is always opposite the stretching or compressing motion.motion.

The spring force is always a restoring force. That is, it The spring force is always a restoring force. That is, it wants to restore the mass to the equilibrium position.wants to restore the mass to the equilibrium position.

The proportionality constant k is called the The proportionality constant k is called the spring spring constantconstant and is different for different springs. (Large k and is different for different springs. (Large k means it takes more force to compress or stretch)means it takes more force to compress or stretch)

Since the force is not constant, but is linear (as seen) Since the force is not constant, but is linear (as seen) we will use the average force to find the work done we will use the average force to find the work done between 2 points xbetween 2 points x11 and x and x22..

Lesson 6: Work done Lesson 6: Work done by a spring (derive w/ by a spring (derive w/ algebra)algebra)


Since the force is not constant, but is linear (as seen) Since the force is not constant, but is linear (as seen) we will use the average force to find the work done we will use the average force to find the work done between 2 points xbetween 2 points x11 and x and x22..

1 2

1( )2avg x xF F F= +


The work done is therefore:The work done is therefore:

avgW F x= ∆



1 2 2 1

1( )( )2avg x xW F x F F x x= ∆ = + −



1 2 2 1

1 2 2 1

1( )( )2

1( )( )2

avg x xW F x F F x x

W kx kx x x

= ∆ = + −

= − − −



1 2 2 1

1 2 2 1

2 21 2 2 2 1 1

1( )( )2

1( )( )21( )2


W kx kx x x

W kx x kx kx x kx

= ∆ = + −

= − − −

= − − + +



1 2 2 1

1 2 2 1

2 21 2 2 2 1 1

2 22 1

1( )( )2

1( )( )21( )21( )

2


W kx kx x x

W kx x kx kx x kx

W k x x

= ∆ = + −

= − − −

= − − + +

= − −


Recall that work done by a conservative Recall that work done by a conservative force is:force is:

W=-W=-ΔΔPEPE

So:So:

PEPEspringspring= ½ kx= ½ kx22

Lesson 6: Comparing Lesson 6: Comparing PE of spring vs. gravityPE of spring vs. gravity

Lesson 6: Spring Force Lesson 6: Spring Force problemproblem

A 2 kg block is connected to a spring with a spring A 2 kg block is connected to a spring with a spring constant k=400N/m as it oscillates on a frictionless constant k=400N/m as it oscillates on a frictionless horizontal plane. At x=0, the spring is neither stretched horizontal plane. At x=0, the spring is neither stretched nor compressed. nor compressed.

a.) Determine the work done by the net force on the block a.) Determine the work done by the net force on the block as it moves from xas it moves from x11 = 0.1m to x = 0.1m to x22 = 0.2m. = 0.2m.

b.) If the speed of the block is 3 m/s when it is at xb.) If the speed of the block is 3 m/s when it is at x 11=0.1m, =0.1m,

how fast is it moving as it passes the point xhow fast is it moving as it passes the point x22 = 0.2m? = 0.2m?

Lesson 6: Work and KE Lesson 6: Work and KE for springsfor springs

A 2kg block is moving with a constant speed of 10 m/s on the A 2kg block is moving with a constant speed of 10 m/s on the horizontal, frictionless plane until it hits the end of the spring. The horizontal, frictionless plane until it hits the end of the spring. The spring constant is 200N/m. spring constant is 200N/m.

a.) How far is the spring compressed before the block comes to rest a.) How far is the spring compressed before the block comes to rest and reverses its motion?and reverses its motion?

b.) What is the speed of the block as the spring subsequently comes b.) What is the speed of the block as the spring subsequently comes back to its original length?back to its original length?