lecture ch 06

Chapter 6

Work and Energy

Units of Chapter 6

•Work Done by a Constant Force

•Kinetic Energy, and the Work-Energy Principle

•Potential Energy

•Conservative and Nonconservative Forces

•Mechanical Energy and Its Conservation

•Problem Solving Using Conservation of Mechanical Energy

Units of Chapter 6

•Other Forms of Energy; Energy Transformations and the Law of Conservation of Energy

•Energy Conservation with Dissipative Forces: Solving Problems

•Power

So far: Motion analysis with forces.

NOW: An alternative analysis using the concepts of Work & Energy.

Work: Precisely defined in physics.Describes what is accomplished by a force in moving an object through a distance.

6-1 Work Done by a Constant ForceThe work done by a constant force is defined as the distance moved multiplied by the component of the force in the direction of displacement:

(6-1)

W = Fd cosθ

Simple special case: F & d are parallel: θ = 0, cosθ = 1

W = Fd

Example: d = 50 m, F = 30 NW = (30N)(50m) = 1500 N m

Work units: Newton - meter = Joule 1 N m = 1 Joule = 1 J

θ

F

d

6-1 Work Done by a Constant Force

In the SI system, the units of work are joules:

As long as this person does not lift or lower the bag of groceries, he is doing no work on it. The force he exerts has no component in the direction of motion.

W = 0:Could have d = 0 W = 0Could have F d θ = 90º, cosθ = 0

W = 0

6-1 Work Done by a Constant ForceSolving work problems:

1. Draw a free-body diagram.

2. Choose a coordinate system.

3. Apply Newton’s laws to determine any unknown forces.

4. Find the work done by a specific force.

5. To find the net work, either

find the net force and then find the work it does, or

find the work done by each force and add.

W = F|| d = Fd cosθm = 50 kg, Fp = 100 N, Ffr = 50 N, θ = 37º

Wnet = WG + WN +WP + Wfr = 1200 J

Wnet = (Fnet)x x = (FP cos θ – Ffr)x = 1200 J

Example 6-1

Example 6-1ΣF =ma

FH = mgWH = FH(d cosθ) = mgh

FG = mgWG = FG(d cos(180-θ)) = -mgh

Wnet = WH + WG = 0

6-1 Work Done by a Constant Force

Work done by forces that oppose the direction of motion, such as friction, will be negative.

Centripetal forces do no work, as they are always perpendicular to the direction of motion.

6-3 Kinetic Energy, and the Work-Energy Principle

Energy was traditionally defined as the ability to do work. We now know that not all forces are able to do work; however, we are dealing in these chapters with mechanical energy, which does follow this definition.

Energy The ability to do work

Kinetic Energy The energy of motion“Kinetic” Greek word for motion

An object in motion has the ability to do work

Consider an object moving in straight line. Starts at speed v1. Due to presence of a net force Fnet, accelerates (uniform) to speed v2, over distance d.

6-3 Kinetic Energy, and the Work-Energy Principle

Newton’s 2nd Law: Fnet= ma (1)1d motion, constant a

(v2)2 = (v1)2 + 2ad a = [(v2)2 - (v1)2]/(2d) (2) Work done: Wnet = Fnetd (3)

Combine (1), (2), (3):

Wnet = mad = md [(v2)2 - (v1)2]/(2d) OR

Wnet = (½)m(v2)2 – (½)m(v1)2

6-3 Kinetic Energy, and the Work-Energy Principle

6-3 Kinetic Energy, and the Work-Energy Principle

If we write the acceleration in terms of the velocity and the distance, we find that the work done here is

We define the translational kinetic energy:

(6-2)

(6-3)

6-3 Kinetic Energy, and the Work-Energy Principle

This means that the work done is equal to the change in the kinetic energy:

• If the net work is positive, the kinetic energy increases.

• If the net work is negative, the kinetic energy decreases.

(6-4)

Net work on an object = Change in KE.

Wnet = KE The Work-Energy Principle

Note: Wnet = work done by the net (total) force.

Wnet is a scalar.

Wnet can be positive or negative (because KE can be both + & -)

Units are Joules for both work & KE.

6-3 Kinetic Energy, and the Work-Energy Principle

Because work and kinetic energy can be equated, they must have the same units: kinetic energy is measured in joules.

Work done on hammer:Wh = KEh = -Fd = 0 – (½)mh(vh)2

Work done on nail:Wn = KEn = Fd = (½)mn(vn)2 - 0

6-4 Potential Energy

An object can have potential energy by virtue of its surroundings.

Familiar examples of potential energy:

• A wound-up spring

• A stretched elastic band

• An object at some height above the ground

6-4 Potential Energy

In raising a mass m to a height h, the work done by the external force is

We therefore define the gravitational potential energy:

(6-5a)

(6-6)

WG = -mg(y2 –y1) = -(PE2 – PE1) = -PE

Wext = PE

The Work done by an external force to move the object of mass m from point 1 to point 2 (without acceleration) is equal to the change in potential energy between positions 1 and 2

WG = -PE

6-4 Potential Energy

6-4 Potential Energy

This potential energy can become kinetic energy if the object is dropped.

Potential energy is a property of a system as a whole, not just of the object (because it depends on external forces).

If , where do we measure y from?

It turns out not to matter, as long as we are consistent about where we choose y = 0. Only changes in potential energy can be measured.

6-4 Potential Energy

Potential energy can also be stored in a spring when it is compressed; the figure below shows potential energy yielding kinetic energy.

6-4 Potential Energy

Restoring force

“Spring equation or Hook’s law”

where k is called the spring constant, and needs to be measured for each spring.

(6-8)

The force required to compress or stretch a spring is:

FP = kx

6-4 Potential EnergyThe force increases as the spring is stretched or compressed further (elastic) (not constant). We find that the potential energy of the compressed or stretched spring, measured from its equilibrium position, can be written:

(6-9)

2

2

1)

2

1( kxxkxxFW

Conservative Force The work done by that force depends only on initial & final conditions & not on path taken between the initial & final positions of the mass. A PE CAN be defined for conservative forces

Non-Conservative Force The work done by that force depends on the path taken between the initial & final positions of the mass. A PE CAN’T be defined for non-conservative

forcesThe most common example of a non-conservative force is FRICTION

6-5 Conservative and Nonconservative Forces

6-5 Conservative and Nonconservative Forces

If friction is present, the work done depends not only on the starting and ending points, but also on the path taken. Friction is called a nonconservative force.

6-5 Conservative and Nonconservative Forces

Potential energy can only be defined for conservative forces.

6-5 Conservative and Nonconservative Forces

Therefore, we distinguish between the work done by conservative forces and the work done by nonconservative forces.

We find that the work done by nonconservative forces is equal to the total change in kinetic and potential energies:

Wnet = Wc + Wnc

Wnet = ΔKE = Wc + Wnc

Wnc = ΔKE - Wc = ΔKE – (– ΔPE)

(6-10)

6-6 Mechanical Energy and Its Conservation

If there are no nonconservative forces, the sum of the changes in the kinetic energy and in the potential energy is zero – the kinetic and potential energy changes are equal but opposite in sign.

KE + PE = 0

This allows us to define the total mechanical energy:

And its conservation:

(6-12b)

PE2 = 0 KE2 = (½)mv2

KE1 + PE1 = KE2 + PE2

0 + mgh = (½)mv2 + 0v2 = 2gh

KE + PE = same as at points 1 & 2

The sum remains constant

PE1 = mgh KE1 = 0

6-7 Problem Solving Using Conservation of Mechanical Energy

In the image on the left, the total mechanical energy is:

The energy buckets (right) show how the energy moves from all potential to all kinetic.

6-7 Problem Solving Using Conservation of Mechanical Energy

If there is no friction, the speed of a roller coaster will depend only on its height compared to its starting point.

6-7 Problem Solving Using Conservation of Mechanical Energy

For an elastic force, conservation of energy tells us:

(6-14)

6-8 Other Forms of Energy; Energy Transformations and the Conservation of Energy

Some other forms of energy:

Electric energy, nuclear energy, thermal energy, chemical energy.

Work is done when energy is transferred from one object to another.

Accounting for all forms of energy, we find that the total energy neither increases nor decreases. Energy as a whole is conserved.

6-9 Energy Conservation with Dissipative Processes; Solving Problems

If there is a nonconservative force such as friction, where do the kinetic and potential energies go?

They become heat; the actual temperature rise of the materials involved can be calculated.

We had, in general:WNC = KE + PE

WNC = Work done by non-conservative forcesKE = Change in KEPE = Change in PE (conservative forces)

Friction is a non-conservative force! So, if friction is present, we have (WNC Wfr)

Wfr = Work done by frictionIn moving through a distance d, force of friction

Ffr does work Wfr = - Ffrd

6-9 Energy Conservation with Dissipative Processes; Solving Problems

When friction is present, we have: Wfr = -Ffrd = KE + PE = KE2 – KE1 + PE2 – PE1

– Also now, KE + PE Constant!

– Instead, KE1 + PE1+ Wfr = KE2+ PE2

OR: KE1 + PE1 - Ffrd = KE2+ PE2

• For gravitational PE:

(½)m(v1)2 + mgy1 = (½)m(v2)2 + mgy2 + Ffrd

• For elastic or spring PE: (½)m(v1)2 + (½)k(x1)2 = (½)m(v2)2 + (½)k(x2)2 + Ffrd

6-9 Energy Conservation with Dissipative Processes; Solving Problems

Problem Solving:

1. Draw a picture.

2. Determine the system for which energy will be conserved.

3. Figure out what you are looking for, and decide on the initial and final positions.

4. Choose a logical reference frame.

5. Apply conservation of energy.

6. Solve.

6-10 PowerPower is the rate at which work is done –

The difference between walking and running up these stairs is power – the change in gravitational potential energy is the same.

(6-17)

In the SI system, the units of power are watts:

British units: Horsepower (hp). 1hp = 746 W

6-10 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:

(6-17)

Part a) constant speeda = 0 ∑Fx = 0

F – FR – mg sinθ = 0F = FR + mg sinθ

F = 3100 NP = Fv

= 91 hp

Part b) constant accelerationNow, θ = 0∑Fx = ma

F – FR= mav = v0 + at

a = 0.93 m/s2

F = ma + FR = 2000 NP = Fv

= 82 hp

Example 6-15

Summary of Chapter 6• Work:

•Kinetic energy is energy of motion:

• Potential energy is energy associated with forces that depend on the position or configuration of objects.

•

•The net work done on an object equals the change in its kinetic energy.

• If only conservative forces are acting, mechanical energy is conserved.

• Power is the rate at which work is done.