TOPIC 5:�Work, Energy & Power

WORK

• Definition of Work:
• When a force causes a displacement of an object
• Components of the force need to be in the direction of the displacement

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Net Work done by a �Constant Net Force

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Work = Force (F) x Displacement (x)

W = Fx

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W = Fx = (Fcosθ)x

** Only the component of the force in the direction of the displacement, contributes to work

Units of Work

Work = Force x Displacement

= Newtons x meters

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Newton x meter 🡪 Joule (J)

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* Joule is named after James Prescott Joule (1818-1889) who made major contributions to the understanding of energy, heat, and electricity

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Work

• Work:
• Scalar quantity
• Can be positive or negative
• Positive work 🡪 Exists when the force & displacement vectors point in the same direction
• Negative work 🡪 Exists when the force & displacement vectors point in opposite directions

Problem

How much work is done on a vacuum cleaner pulled 3 m by a force of 50 N at an angle of 30° above the horizontal?

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W = (Fcosθ)x W = ? F = 50N

d = 3m θ = 30°

W = (50N)(cos30°)(3m)

= 130 J

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ENERGY

Kinetic Energy:

* Energy associated with an object in motion

* Depends on speed and mass

* Scalar quantity

* SI unit for all forms of energy = Joule (J)

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KE = ½ mv²

KE = ½ x mass x (velocity)²

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Kinetic Energy

If a bowling ball and a soccer ball are traveling at the same speed, which do you think has more kinetic energy?

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KE = ½ mv²

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* Both are moving with identical speeds

* Bowling ball has more mass than the soccer ball 🡪 Bowling ball has more kinetic energy

Kinetic Energy Problem

• A 7 kg bowling ball moves at 3 m/s. How fast must a 2.45 g tennis ball move in order to have the same kinetic energy as the bowling ball?

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Velocity of tennis ball = 160 m/s

Work-Kinetic Energy Theorem

Work-kinetic Energy Theorem:

• Net work done on a particle equals the change in its kinetic energy (KE)

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W = ΔKE

PROBLEM

• What is the soccer ball’s speed immediately after being kicked? Its mass is 0.42 kg.

PROBLEM

• What is the soccer ball’s speed immediately after being kicked? Its mass is 0.42 kg.

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W = F ∙ Δx

W = (240 N) (0.20 m) = 48 J

W = ΔKE = 48 J

KE = ½ mv² = 48 J

v² = 2(48 J)/0.42 kg

v = 15 m/s

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Work-Kinetic Energy Theorem

On a frozen pond, a person kicks a 10 kg sled, giving it an initial speed of 2.2 m/s. How far does the sled move if the coefficient of kinetic friction between the sled and the ice is 0.10?

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m = 10 kg v_i = 2.2 m/s v_f = 0 m/s μ_k = 0.10

d = ?

Work-Kinetic Energy Theorem

W_net = F_netdcosθ

* Net work done of the sled is provided by the force of kinetic friction

W_net = F_kdcosθ 🡪 F_k = μ_kN 🡪 N = mg

W_net = μ_kmgdcosθ

* The force of kinetic friction is in the direction opposite of d 🡪 θ = 180°

* Sled comes to rest 🡪 So, final KE = 0

W_net = Δ KE = ½ mv²_f – ½ mv²_i

W_net = -1/2 mv²_i

Work-Kinetic Energy Theorem

Use the work-kinetic energy theorem, and solve for d

W_net = ΔKE

- ½ mv²_i = μ_kmgdcosθ

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d = 2.5 m

POWER

POWER:

* A quantity that measures the rate at which work is done or energy is transformed

* Power = work / time interval

P = W/Δt

(W = Fx 🡪 P = Fx/Δt 🡪 v = x/Δt)

* Power = Force x speed

P = Fv

POWER

SI Unit for Power:

Watt (W) 🡪 Defined as 1 joule per second (J/s)

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Horsepower = Another unit of power

1 hp = 746 watts

POWER PROBLEM

A 193 kg curtain needs to be raised 7.5 m, in as close to 5 s as possible. The power ratings for three motors are listed as 1 kW, 3.5 kW, and 5.5 kW. What motor is best for the job?

POWER PROBLEM

m = 193 kg Δt = 5s d =7.5m

P = ?

P = W/Δt

= Fx/Δt

= mgx/Δt

= (193kg)(9.8m/s²)(7.5m)/5s

= 280 W 🡪 2.8 kW

** Best motor to use = 3.5 kW motor. The 1 kW motor will not lift the curtain fast enough, and the 5.5 kW motor will lift the curtain too fast

POTENTIAL ENERGY

Potential Energy:

* Stored energy

* Associated with an object that has the potential to move because of its position relative to some other location

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Example:

Balancing rock- Arches National Park, Utah

Delicate Arch- Arches National Park, Utah

GRAVITATIONAL POTENTIAL ENERGY- Definition

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Gravitational potential energy PE_g is the energy an

object of mass m has by virtue of its position relative to the surface of the earth. That position is measured by the height h of the object relative to an arbitrary zero level:

PE_g = mgh

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SI Unit = Joule (J)

Problem

• What is the bucket’s gravitational potential energy?

Problem

• What is the bucket’s gravitational potential energy?

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PE = mgh

PE = (2.00 kg)(9.80 m/s2)(4.00 m)

PE = 78.4 J

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Gravitational Potential Energy

Example: A Gymnast on a Trampoline

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The gymnast leaves the trampoline at an initial height of 1.20 m

and reaches a maximum height of 4.80 m before falling back

down. What was the initial speed of the gymnast?

Gravitational Potential Energy

Elastic Potential Energy

* Energy stored in any compressed or stretched object

• Spring, stretched strings of a tennis racket or guitar, rubber bands, bungee cords, trampolines, an arrow drawn into a bow, etc.

Springs

• When an external force compresses or stretches a spring 🡪 Elastic potential energy is stored in the spring
• The more stretch, the more stored energy

• For certain springs, the amount of force is directly proportional to the amount of stretch or compression (x);
• Constant of proportionality is known as the spring constant (k)

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F_spring = k * x

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Hooke’s Law

• If a spring is not stretched or compressed 🡪 no potential energy is being stored
• Spring is in an Equilibrium position
• Equilibrium position: Position spring naturally assumes when there is no force applied to it
• Zero potential energy position

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Hooke’s Law

• Special equation for springs
• Relates the amount of elastic potential energy to the amount of stretch (or compression) and the spring constant

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PE _elastic = ½kx²

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k = Spring constant (N/m)

Stiffer the spring 🡪 Larger the spring constant

x = Amount of compression relative to the equilibrium position

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Potential Energy Problem

A 70 kg stuntman is attached to a bungee cord with an unstretched length of 15 m. He jumps off the bridge spanning a river from a height of 50m. When he finally stops, the cord has a stretched length of 44 m. Treat the stuntman as a point mass, and disregard the weight of the bungee cord. Assuming the spring constant of the bungee cord is 71.8 N/m, what is the total potential energy relative to the water when the man stops falling?

Potential Energy Problem

* Zero level for gravitational potential energy is chosen to be the surface of the water

* Total potential energy 🡪 sum of the gravitational & elastic potential energy

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PE_total = PE_g + PE_elastic

= mgh + ½ kx²

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* Substitute the values into the equation

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PE_total = 3.43 x 10⁴ J

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Potential Energy

• The energy stored in an object due to its position relative to some zero position
• An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height
• An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position

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Linking Work to Mechanical Energy

• WORK is a force acting upon an object to cause a displacement
• When work is done upon an object, that object gains energy
• Energy acquired by the objects upon which work is done is known as MECHANICAL ENERGY

Mechanical Energy

• Objects have mechanical energy if they are in motion and/or if they are at some position relative to a zero potential energy position

Total Mechanical Energy

*Total Mechanical Energy: The sum of kinetic energy & all forms of potential energy

1. Kinetic Energy (Energy of motion)

KE = ½ mv²

2. Potential Energy (Stored energy of position)

a. Gravitational

PE_g = mgh

b. Elastic

PE_elastic = ½ kx²

Mechanical Energy

CONSERVATION OF MECHANICAL ENERGY:

* In the absence of friction, mechanical energy is conserved, so the amount of mechanical energy remains constant

ME_i = ME_f

Initial mechanical energy = final mechanical energy

(in the absence of friction)

PE_i + KE_i = PE_f + KE_f

mgh_i + ½ mv_i² = mgh_f + ½ mv_f²

Conservation of Energy Problem

Starting from rest, a child zooms down a frictionless slide from an initial height of 3 m. What is her speed at the bottom of the slide? (Assume she has a mass of 25 kg)

Conservation of Energy Problem

h_i = 3m m = 25kg v_i = 0 m/s

h_f = 0m v_f = ?

• Slide is frictionless 🡪 Mechanical energy is conserved
• Kinetic energy & potential energy = only forms of energy present
• KE = ½ mv² PE_g = mgh
• Final gravitational potential energy = zero (Bottom of the slide) 🡪 PE_gf = 0
• Initial gravitational potential energy 🡪 Top of the slide 🡪 PE_gi = mgh_i 🡪 (25kg)(9.8m/s²)(3m) = 736 J

Conservation of Energy Problem

h_i = 3m m = 25kg v_i = 0 m/s

h_f = 0m v_f = ?

• Initial Kinetic Energy = 0, because child starts at rest
• KE_i = 0
• Final Kinetic Energy
• KE_f = ½ mv² 🡪 ½ (25kg)v²_f
• ME_i = ME_f

PE_i + KE_i = PE_f + Ke_f

736 J + 0 J = 0 J + (1/2)(25kg)(v²_f)

v_f = 7.67 m/s

Mechanical Energy 🡪 Ability to do Work

• An object that possesses mechanical energy is able to do work
• Its mechanical energy enables that object to apply a force to another object in order to cause it to be displaced
• Classic Example 🡪 Massive wrecking ball of a demolition machine

Mechanical Energy is the ability to do work…

• An object that possesses mechanical energy (whether it be kinetic energy or potential energy) has the ability to do work
• That is… its mechanical energy enables that object to apply a force to another object in order to cause it to be displaced

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Mechanical Energy

• Work is a force acting on an object to cause a displacement
• In the process of doing work 🡪 the object which is doing the work exchanges energy with the object upon which the work is done
• When work is done up the object 🡪 that object gains energy

Mechanical Energy

• A weightlifter applies a force to cause a barbell to be displaced
• Barbell now possesses mechanical energy- all in the form of potential energy

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** The energy acquired by the objects upon which work is done is known as mechanical energy

Mechanical Energy is the ability to do work…

Examples on website:

Massive wrecking ball of a demolition machine

The wrecking ball is a massive object which is swung backwards to a high position and allowed to swing forward into a building structure or other object in order to demolish it

Upon hitting the structure, the wrecking ball applies a force to it in order to cause the wall of the structure to be displaced

Mechanical energy = ability to do work

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Work- Energy Theorem

Categorize forces based upon whether or not their presence is capable of changing an object’s total mechanical energy

* Certain types of forces, which when present and when involved in doing work on objects, will change the total mechanical energy of the object

* Other types of forces can never change the total mechanical energy of an object, but rather only transform the energy of an object from PE to KE or vice versa

** Two categories of forces 🡪 Internal & External

Work- Energy Theorem

External Forces:

Applied force, normal force, tension force, friction force and air resistance force

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Internal Forces:

Gravity forces, spring forces, electrical forces and magnetic forces

Work- Energy Theorem

THE BIG CONCEPT!!

* When the only type of force doing net work upon an object is an internal force (gravitational and spring forces)

🡪 Total mechanical energy (KE + PE) of that object remains constant

🡪 Object’s energy simply changes form 🡪 Conservation of Energy

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** Ex) As an object is “forced” from a high elevation to a lower elevation by gravity 🡪 Some of the PE is transformed into KE (Yet, the sum of KE + PE = remains constant)

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Work- Energy Theorem

THE BIG CONCEPT!!

* If only internal forces are doing work 🡪 energy changes forms (KE to PE or vice versa) 🡪 total mechanical energy is therefore conserved

* Internal forces – referred to as conservative forces

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Quick Quiz

Work-Energy Relationship

Analysis of situations in which work is conserved 🡪 only internal forces are involved

TME_i + W_EXT = TME_f

(Initial amount of total mechanical energy (TME_i) plus the work done by external forces (W_EXT) 🡪 equals the final amount of total mechanical energy (TME_f))

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KE_i + PE_i + W_ext = KE_f + PE_f

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KE_i + PE_i = KE_f + Pe_f

_Website

Work- Energy Theorem

THE BIG CONCEPT!!

* Forces are categorized as being either internal or external based upon the ability of that type of force to change an object’s total mechanical energy when it does work upon an object

* Net work done upon an object by an external force 🡪 Changes the total mechanical energy (KE + PE) of the object

🡪 Positive work = object gained energy

🡪 Negative work = object lost energy

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Work- Energy Theorem

THE BIG CONCEPT!!

* Gain or loss in energy can be in the form of

🡪 PE, KE, or both

Under such circumstances, the work which is done is equal to the change in mechanical energy of the object

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** External forces 🡪 capable of changing the total mechanical energy of an object (Nonconservative forces)

Work-Energy Relationship

Analysis of situations involving external forces

TME_i + W_EXT = TME_f

(Initial amount of total mechanical energy (TME_i) plus the work done by external forces (W_EXT) 🡪 equals the final amount of total mechanical energy (TME_f))

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KE_i + PE_i + W_ext = KE_f + PE_f

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Practice Problems

DEFINITION OF A CONSERVATIVE FORCE

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Version 1 A force is conservative when the work it does on a moving object is independent of the path between the object’s initial and final positions.

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Version 2 A force is conservative when it does no work on an object moving around a closed path, starting and finishing at the same point.

Conservative Versus Nonconservative Forces

Conservative Versus Nonconservative Forces

Version 1 A force is conservative when the work it does on a moving object is independent of the path between the object’s initial and final positions.

Conservative Versus Nonconservative Forces

Version 2 A force is conservative when it does no work

on an object moving around a closed path, starting and

finishing at the same point.

Conservative Versus Nonconservative Forces

An example of a nonconservative force is the kinetic

frictional force.

The work done by the kinetic frictional force is always negative. Thus, it is impossible for the work it does on an object that moves around a closed path to be zero.

The concept of potential energy is not defined for a

nonconservative force.

Conservative Versus Nonconservative Forces

In normal situations both conservative and nonconservative

forces act simultaneously on an object, so the work done by

the net external force can be written as

Conservative Versus Nonconservative Forces

THE WORK-ENERGY THEOREM

The Conservation of Mechanical Energy

If the net work on an object by nonconservative forces

is zero, then its energy does not change:

The Conservation of Mechanical Energy

THE PRINCIPLE OF CONSERVATION OF

MECHANICAL ENERGY

The total mechanical energy (E = KE + PE) of an object

remains constant as the object moves, provided that the net

work done by external nonconservative forces is zero.

The Conservation of Mechanical Energy

The Conservation of Mechanical Energy

Example A Daredevil Motorcyclist

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A motorcyclist is trying to leap across the canyon by driving

horizontally off a cliff at 38.0 m/s. Ignoring air resistance, find

the speed with which the cycle strikes the ground on the other

side.

The Conservation of Mechanical Energy

The Conservation of Mechanical Energy

Nonconservative Forces and the Work-Energy Theorem

THE WORK-ENERGY THEOREM

Nonconservative Forces and the Work-Energy Theorem

Example Fireworks

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Assuming that the nonconservative force

generated by the burning propellant does

425 J of work, what is the final speed

of the rocket. Ignore air resistance. The mass

of the rocket is 0.2kg.

Nonconservative Forces and the Work-Energy Theorem

POWER

POWER:

* A quantity that measures the rate at which work is done or energy is transformed

* Power = work / time interval

P = W/Δt

W = Fd 🡪P = Fd/Δt 🡪 v = d/Δt

* Power = Force x speed

P = Fv

POWER

SI Unit for Power:

Watt (W) 🡪 Defined as 1 joule per second (J/s)

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Horsepower = Another unit of power

1 hp = 746 watts

POWER PROBLEM

A 193 kg curtain needs to be raised 7.5 m, in as close to 5 s as possible. The power ratings for three motors are listed as 1 kW, 3.5 kW, and 5.5 kW. What motor is best for the job?

POWER PROBLEM

m = 193 kg Δt = 5s d =7.5m

P = ?

P = W/Δt

= Fd/Δt

= mgd/Δt

= (193kg)(9.8m/s²)(7.5m)/5s

= 280 W 🡪 2.8 kW

** Best motor to use = 3.5 kW motor. The 1 kW motor will not lift the curtain fast enough, and the 5.5 kW motor will lift the curtain too fast

THE PRINCIPLE OF CONSERVATION OF ENERGY

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Energy can neither be created nor destroyed, but can

only be converted from one form to another.

* Disclaimer: This powerpoint presentation is a compilation of various works.

Question

A cart is loaded with a brick and pulled at constant speed along an inclined plane to the height of a seat-top. If the mass of the loaded cart is 3.0 kg and the height of the seat top is 0.45 meters, then what is the potential energy of the loaded cart at the height of the seat-top?

PE = m*g*h

PE = (3 kg ) * (9.8 m/s/s) * (0.45m) PE = 13.2 J

Question

If a force of 14.7 N is used to drag the loaded cart (from previous question) along the incline for a distance of 0.90 meters, then how much work is done on the loaded cart?

W = F * d * cos Theta

W = 14.7 N * 0.9 m * cos (0 degrees)

W = 13.2 J 

(Note: The angle between F and d is 0 degrees because the F and d are in the same direction)