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Work, Energy, and Power. Lesson 1: Basic Terminology and Concepts Definition and Mathematics of Work Calculating the Amount of Work Done by Forces Kinetic Energy Potential Energy Mechanical Energy Power Lesson 2 - The Work-Energy Relationship Work-Energy Principle

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Work energy and power
Work, Energy, and Power

Lesson 1: Basic Terminology and Concepts

  • Definition and Mathematics of Work

  • Calculating the Amount of Work Done by Forces

  • Kinetic Energy

  • Potential Energy

  • Mechanical Energy

  • Power

    Lesson 2 - The Work-Energy Relationship

  • Work-Energy Principle

  • Internal vs. External Forces

  • Analysis of Situations Involving External Forces

  • Analysis of Situations in Which Mechanical Energy is Conserved

  • Application and Practice Questions


Definition and Mathematics of Work

force

cause

  • In physics, work is defined as a _________ acting upon an object to ____________ a __________________.

  • In order for a force to qualify as having done work on an object, there must be a displacement and the force must ___________ the displacement

displacement

cause

Work done

Work not done


Let s practice work or no work
Let’s practice – work or no work

  • A student applies a force to a wall and becomes exhausted.

  • A calculator falls off a table and free falls to the ground.

  • A waiter carries a tray full of beverages above his head by one arm across the room

  • A rocket accelerates through space.

no work

work

no work

work


Calculating the amount of work done by forces

F

θ

d

Fy

F

θ

Fx

d

Calculating the Amount of Work Done by Forces

  • F - is the force in Newton, which causes the displacement of the object.

  • d - is the displacement in meters

  • θ = angle between force and displacement

  • W - is work in N∙m or Joule (J). 1 J = 1 N∙m = 1 kg∙m2/s2

  • Work is a scalar quantity

  • Work is independent of time the force acts on the object.

Only the horizontal component of the force (Fcosθ) causes a horizontal displacement.


Example 1
Example 1

  • How much work is done on a vacuum cleaner pulled 3.0 m by a force of 50.0 N at an angle of 30o above the horizontal?.


Example 2
Example 2

  • How much work is done in lifting a 5.0 kg box from the floor to a height of 1.2 m above the floor?

Given: d = h = 1.2 meters; m = 5.0 kg; θ = 0

Unknown: W = ?

W = F∙dcosθ

F = mg = (5.0 kg)(9.81 m/s2) cos0o = 49 N

W = F∙d = (49 N) (1.2 m) = 59 J


Example 3
Example 3

  • A 2.3 kg block rests on a horizontal surface. A constant force of 5.0 N is applied to the block at an angle of 30.o to the horizontal; determine the work done on the block a distance of 2.0 meters along the surface.

  • Given: F = 5.0 N;

    m = 2.3 kg

    d = 2.0 m

    θ= 30o

5.0 N

30o

2.3 kg

  • unknown:

    • W = ? J

  • Solve:

  • W = F∙d∙cosθ

  • W = (5.0 N)(2.0 m)(cos30o) = 8.7 J


Example 4
Example 4

  • Matt pulls block along a horizontal surface at constant velocity. The diagram show the components of the force exerted on the block by Matt. Determine how much work is done against friction.

  • Given: Fx = 8.0 N

    Fy = 6.0 N

    dx = 3.0 m

6.0 N

  • unknown: W = ? J

F

W = Fxdx

W = (8.0 N)(3.0 m) = 24 J

8.0 N

3.0 m


Example 5
Example 5

  • A neighbor pushes a lawnmower four times as far as you do but exert only half the force, which one of you does more work and by how much?

Wyou = Fd

Wneighbor = (½ F)(4d) = 2 Fd = 2 Wyou

The neighbor, twice as much


W f d cos

The sign of work

W = F∙d∙cosθ

Work done – positive, negative or zero work

Positive work

negative work - force acts in the direction opposite the objects motion in order to slow it down.

no work


When F is ┴ to d, W = 0

To Do Work, Forces Must Cause Displacements

W = F∙d∙cosθ = 0


Only the horizontal component of the force (Fcosθ) causes a horizontal displacement.


Force vs displacement graph
Force vs. displacement graph

  • The area under a force versus displacement graph is the work done by the force.

Example: a block is pulled along a table with 10. N over a distance of 1.0 m.

W = Fd = (10. N)(1.0 m) = 10. J

work

Force (N)

Displacement (m)

height

base

area


Example
Example

  • A student produced various elongations of a spring by applying a series of forces to the spring. The graph below represents the relationship between the applied force and the elongation of the spring. Determine the work done between 0.0 m to 0.20 m.


The angle in work equation

d

F

The angle in work equation

  • The angle in the equation is the angle between the force and the displacement vectors.

F & d are in the same direction, θ is 0o.


What is in each case
What is θ in each case?


Practices
practices

  • A 20.0 N force is used to push a 2.00 kg cart a distance of 5.00 meters. Determine the amount of work done on the cart by the force.

  • How much work is done in lifting a 5.0 kg box from the floor to a height of 1.2 m above the floor?

  • A 2.3 kg block rests on a horizontal surface. A constant force of 5.0 N is applied to the block at an angle of 30.o to the horizontal; determine the work done on the block a distance of 2.0 meters along the surface.

  • Matt pulls block along a horizontal surface at constant velocity. The diagram show the components of the force exerted on the block by Matt. Determine how much work is done against friction.


Work and energy are related
Work and Energy are related

  • When work is done on a system, that system’s energy equals to the amount of work done on it.

  • Work and energy have the same unit: Joule

  • For example, if you push a cart, you do work on the cart, the cart is going to speed up and its temperature may increase, its energy is increased. If you lift a rock, you do work on the rock and you increase the rock’s energy.

  • There are many forms of energy.

    • Potential energy

    • Kinetic energy

    • Internal energy


Potential energy
Potential energy

  • An object can store energy as the result of its position. Potential energy is the stored energy of position possessed by an object.

  • Two form:

    • Gravitational

    • Elastic


Gravitational potential energy
Gravitational potential energy

  • When you lift an object, you do work against gravity. As a result, its position is higher, and it has more gravitational potential energy.

  • Gravitational potential energy is the energy stored in an object as the result of its vertical position (height)

  • The energy is stored as the result of the gravitational attraction of the Earth for the object.

  • Gravitational depends on

    • m: mass, in kilograms

    • g: acceleration due to gravity = 9.81 m/s2

    • h: height


Gravitational Potential Energy is relative:

To determine the gravitational potential energy of an object, a zero height position must first be assigned. Typically, the ground is considered to be a position of zero height.

But, it doesn’t have to be:

  • It could be relative to the height above the lab table.

  • It could be relative to the bottom of a mountain

  • It could be the lowest position on a roller coaster


Gravitational potential energy equation
Gravitational potential energy Equation

Gravitation potential energy equals to work done against gravity

.

  • change in height, in meters

  • Gravitational attraction between Earth and the object:

  • m: mass, in kilograms

  • g: acceleration due to gravity = 9.81 m/s2


Change in gpe only depends on change in height not path
Change in GPE only depends on change in height, not path

As long as the object starts and ends at the same height, the object has the same change in GPE because gravity does the same amount of work regardless of which path is taken.


Example1
Example

  • The diagram shows points A, B, and C at or near Earth’s surface. As a mass is moved from A to B, 100. joules of work are done against gravity. What is the amount of work done against gravity as an identical mass is moved from A to C?

100 J

As long as the object starts and ends at the same height, the object has the same change in GPE because gravity does the same amount of work regardless of which path is taken.


Unit of energy
Unit of energy

  • The unit of energy is the same as work: Joules

  • 1 joule = 1 (kg)∙(m/s2)∙(m) = 1 Newton ∙ meter

  • 1 joule = 1 (kg)∙(m2/s2)

Work and energy has the same unit


Example2
example

  • How much potential energy is gained by an object with a mass of 2.00 kg that is lifted from the floor to the top of 0.92 m high table?

  • Known:

  • m = 2.00 kg

  • h = 0.92 m

  • g = 9.81 m/s2

Solve:

∆PE = mg∆h

∆PE = (2.00 kg)(9.81m/s2)(0.92 m) = 18 J

  • unknown:

  • PE = ? J


Gpe vs vertical height graph
GPE vs. Vertical Height Graph

The graph of gravitational potential energy vs. vertical height for an object near Earth's surface is a straight line. The slope is the weight of the object.


Elastic potential energy
Elastic potential energy

  • Elastic potential energy is the energy stored in elastic materials as the result of their stretching or compressing when a force is applied.

  • Elastic potential energy can be stored in

    • Rubber bands

    • Bungee cores

    • Springs

    • trampolines


Elastic potential energy in a spring
Elastic potential energy in a spring

  • k: spring constant

  • x: amount of compression or elongation relative to equilibrium position

equilibrium

x

elongation


Elastic potential energy is directly proportional to x 2
Elastic potential energy is directly proportional to x2

Elastic potential energy

elongation


Example3
Example

  • As shown in the diagram, a 0.50-meter-long spring is stretched from its equilibrium position to a length of 1.00 meter by a weight. If 15 joules of energy are stored in the stretched spring, what is the value of the spring constant?

PE = ½ kx2

15 J = ½ k (0.50 m)2

k = 120 N/m


Example4
Example

  • The unstretched spring in the diagram has a length of 0.40 meter and a spring constant k.  A weight is hung from the spring, causing it to stretch to a length of 0.60 meter.  In terms of k, how many joules of elastic potential energy are stored in this stretched spring?

PEs = ½ kx2

PEs = ½ k(0.20 m)2

PEs = (0.020 k) J


Elongation or compression depends on force
Elongation ( or compression) depends on force

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


Hooke s law
Hooke’s Law

  • F in the force needed to displace (by stretching or compressing) a spring x meters from the equilibrium (relaxed) position. The SI unit of F is Newton.

  • k is spring constant. It is a measure of stiffness of the spring. The greater value of k means a stiffer spring because more force is needed to stretch or compress it that spring. The SI units of k are N/m.

  • x the distance difference between the length of stretched/compressed spring and its relaxed (equilibrium) spring.


Example5
example

  • Determine the x in F = kx


The slope of f s vs x

force

elongation

The slope of Fsvs. x

  • Spring force is directly proportional to the elongation of the spring (displacement)

The slope represents spring constant: k = F / x (N/m)


Caution

elongation

force

caution

  • Sometimes, we might see a graph such as this:

The slope represents the inverse of spring constant:

Slope = 1/k


Example6
Example

  • Given the following data table and corresponding graph, calculate the spring constant of this spring.


Example7
Example

  • A 20.-newton weight is attached to a spring, causing it to stretch, as shown in the diagram. What is the spring constant of this spring?


Example8
Example

  • The graph below shows elongation as a function of the applied force for two springs, A and B. Compared to the spring constant for spring A, the spring constant for spring B is

  • smaller

  • larger

  • the same


Work done stretching a spring and its elastic potential energy
Work done stretching a spring and its Elastic Potential Energy

  • Elastic potential energy stored in a spring equals to the work done in stretching it.

Work = Area = ½ (base)(height)

Work = ½ (x)(F)

Work = ½ (x)(k∙x)

Work = ½ k∙x2

PEs = Work = ½ k∙x2


Example9
Example Energy

  • If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?


Example10
Example Energy

  • Determine the potential energy stored in the spring with a spring constant of 25.0 N/m when a force of 2.50 N is applied to it.

Solve:

PEs = ½ k∙x2

To find x, use Fs = kx,

(2.50 N) = (25.0 N/m)(x)

x = 0.100 m

PEs = ½ (25.0 N/m)(0.100 m)2

PEs = 0.125 J

Given:

Fs = 2.50 N

k = 25.0 N/m

Unknown:

PEs = ? J


Example11
Example Energy

  • A 10.-newton force is required to hold a stretched spring 0.20 meter from its rest position. What is the potential energy stored in the stretched spring?


Kinetic energy

KE = ½ mv Energy2

Kinetic energy

  • Kinetic energy is the energy of motion.

  • An object which has motion - whether it be vertical or horizontal motion - has kinetic energy.

  • The equation for kinetic energy is:

    • Where KE is kinetic energy, in joules

    • v is the speed of the object, in m/s

    • m is the mass of the object, in kg

  • Kinetic energy is a scalar quantity.


Questions
Questions Energy

  • Which of the following has kinetic energy?

    • a falling sky diver

    • a parked car

    • a shark chasing a fish

    • a calculator sitting on a desk

  • If a bowling ball and a volleyball are traveling at the same speed, do they have the same kinetic energy?

  • Car A and car B are identical and are traveling at the same speed. Car A is going north while car B is going east. Which car has greater kinetic energy?


Speed has more impact on kinetic energy Energy

  • KE is directly proportional to m, so doubling the mass doubles kinetic energy, and tripling the mass makes it three times greater.

  • KE is proportional to v2, so doubling the speed quadruples kinetic energy, and tripling the speed makes it nine times greater.

Kinetic energy

Kinetic energy

speed

mass


Example12
Example Energy

  • A 7.00 kg bowling ball moves at 3.00 m/s. How much kinetic energy does the bowling ball heave? How fast must a 2.45 g table-tennis ball move in order to have the same kinetic energy as the bowling ball? Is this speed reasonable for a table-tennis ball?


Example13
Example Energy

  • An object moving at a constant speed of 25 meters per second possesses 450 joules of kinetic energy. What is the object's mass?

  • Known:

  • KE = 450 J

  • v = 25 m/s

  • Unknown:

  • m = ? kg

Solve:

KE = ½ mv2

450 J = ½ (m)(25 m/s)2

m = 1.4 kg


Example14
Example Energy

  • A cart of mass m traveling at a speed v has kinetic energy KE.  If the mass of the cart is doubled and its speed is halved, the kinetic energy of the cart will be

  • half as great

  • twice as great

  • one-fourth as great

  • four times as great


Example Energy

  • Which graph best represents the relationship between the kinetic energy, KE, and the velocity of an object accelerating in a straight line?

a

b

c

d


Total mechanical energy
Total Mechanical Energy Energy

  • Mechanical energy is the energy that is possessed by an object due to its motion or due to its position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position) or both.

The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy

TME = KE + PE

TME = PEgrav + PEspring + KE

Note: PE includes both gravitational potential energy and elastic potential energy.


Mechanical energy as the ability to do work
Mechanical Energy as the Ability to Do Work Energy

  • Any object that possesses mechanical energy - whether it is in the form of potential energy or kinetic energy - is able to do work.


Internal energy
Internal Energy Energy

  • internal energy: TEMPERATURE/HEAT

Resistance force, such as friction, air resistance, or any force stopping motion, produce internal energy. When resistance force is zero, internal energy does not change.


Classification of Energy Energy

ET = PE + KE + Q


What types of energy are changing scenario 1
What types of energy are changing? Scenario #1 Energy

A car is moving along a flat surface with an increasing speed (while external force is applied). What type(s) of energy are changing?

PEg

PES

KE

Q

X

X

?


What types of energy are changing scenario 2
What types of energy are changing? Scenario #2 Energy

A car is moving up an incline with constant speed (while external forces are applied.) What type(s) of energy are changing?

PEg

PES

KE

Q

X

X

?


What types of energy are changing? Scenario #3 Energy

A car is applying its brakes (external force) and slowing down as it moves down a slope. What type(s) of energy are changing?

PEg

PES

KE

Q

X


What types of energy are changing scenario 4
What types of energy are changing? Scenario #4 Energy

A cart is moving on a flat surface toward a wall and is stopped by a spring on its front. What type(s) of energy are changing?

PEg

PES

KE

Q

X

?


What types of energy are changing? Scenario #5 Energy

A rocket flies upward with an increasing velocity [ignore air resistance]. What type(s) of energy are changing?

PEg

PES

KE

Q

X

X


Power
Power Energy

  • Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation.

  • The standard metric unit of power is the Watt.

All machines are typically described by a power rating. For example, a 60 Watt light bulb indicates 60 J of electrical energy is transferred to light energy every second. A high power car indicates that it can be accelerated very rapidly. Some people are more power-full than others because they are capable of doing the same amount of work in lesstime or more work in the same amount of time


Example15
example Energy

  • Ben Pumpiniron elevates his 80-kg body up the 2.0-meter stairwell in 1.8 seconds. What is his power?

It can be assumed that Ben must apply an (80 kg x 9.81 m/s2) -Newton downward force upon the stairs to elevate his body.



Example16
example Energy

  • Two physics students, Will N. Andable and Ben Pumpiniron, are in the weightlifting room. Will lifts the 100-pound barbell over his head 10 times in one minute; Ben lifts the 100-pound barbell over his head 10 times in 10 seconds. Which student does the most work? ______________ Which student delivers the most power? ______________ Explain your answers.


Example17
example Energy

  • When doing a chin-up, a physics student lifts her 42.0-kg body a distance of 0.25 meters in 2 seconds. What is the power delivered by the student's biceps?


Kilowatt hour is unit for energy
kilowatt-hour is unit for energy Energy

  • Your household's monthly electric bill is often expressed in kilowatt-hours. One kilowatt-hour is the amount of energy delivered by the flow of l kilowatt of electricity for one hour. Use conversion factors to show how many joules of energy you get when you buy 1 kilowatt-hour of electricity.


Lesson 2 the work energy theorem
Lesson 2: The Work-Energy Theorem Energy

  • Work and Energy principle

  • Analysis of Situations Involving external forces

  • Analysis of Situations Involving a closed system

  • Analysis of Situations Involving an ideal system

  • Application and Practice Questions


Work and energy principle
Work and Energy Principle Energy

  • When an external force does work on a system, the work done changes the system's energy.

Workexternal force = ∆ET= Ef- Ei

ET=PE + KE + Q

TME = PE + KE

Workexternal force = (TMEf+ Qf) - (TMEi + Qi)

Workexternal force = ∆PE + ∆KE + ∆Q


What is a system and what are internal external forces
What is a system? And what are internal, external forces? Energy

  • Any sets of objects can be considered A SYSTEM. The definition of system can be arbitrary. However, according to New York State Regents Physics, examples of a system are apple and Earth, two cars in a collision, two cars connected by a spring, rifle and bullets, cannon and cannon ball, spring and its launched toy,car and road.

  • The forces within a system are called internal forces. Example of internal forces include gravity (between apple and Earth), force by the spring between two connected cars, force of friction between car and road.

  • Applied force to a system is an example external force


Problem solving strategies
Problem solving strategies Energy

  • Identify the system in question.

  • Identify external forces (applied force on the system) or external work.

  • Identify initial and final mechanical energy (KE & PE)

  • Determine if there is any resistance force.

  • Apply work-energy theorem

Workexternal force = ∆PE + ∆KE + ∆Q


Example 11
Example 1 Energy

System: barbell

External force = 1000 N

Fresistance= 0

Wexternal force = ∆ET = Ef- Ei


Example 2 Energy

System: cart

External force = 18 N

Fresistance= 0

Wexternal force = ∆ET = Ef- Ei


Example 31
Example 3 Energy

  • A person does 100 joules of work in pulling back the string of a bow. What will be the initial speed of a 0.5-kilogram arrow when it is fired from the bow?

System: bow

External work = 100 J

Fresistance= 0


Energy is conserved in a closed system
ENERGY IS CONSERVED IN A CLOSED SYSTEM Energy

  • A closed system is one in which there are no external forces doing work on the system, no external work being done by the system, an no transfer of energy into or out of the system.

  • Although the energy within a closed system may be transformed from one type to anther, the total amount of energy in a closed system must remain constant – energy is never created or destroyed.

Wext= ∆ET = ETf– ETi= 0


Example 12
Example 1 Energy

system: mitt and ball

Resistance force = 6000 N

Fexternal = 0 N, closed system

∆Q = -Wresistance

Ef= Ei


Example 21
Example 2 Energy

Closed system: car and ground

Fresitance = 6000 N

∆Q = -Wresistance


Example Energy

  • A shopping cart full of groceries is sitting at the top of a 2.0-m hill. The cart begins to roll until it hits a stump at the bottom of the hill. Upon impact, a 0.25-kg can of peaches flies horizontally out of the shopping cart and hits a parked car with an average force of 500 N. How deep a dent is made in the car (i.e., over what distance does the 500 N force act upon the can of peaches before bringing it to a stop)?

Closed system: can of peaches and parked car

Fresitance = 500 N

∆Q = -Wresistance


Conservative vs non conservative forces
Conservative vs. non-conservative Forces Energy

  • When work done against a force is independent of the path taken, the force is said to be a CONSERVATIVE FORCE.

  • When work done against a force is dependent of the path taken, the force is said to be a NON-CONSERVATIVE FORCE.


Ideal mechanical system
Ideal Mechanical System Energy

  • An ideal mechanical system is a closed system in which no friction or other non-conservative force acts. The only forces doing work are gravity, or force of a spring.

  • In an ideal mechanical system total mechanical energy remains the same:

Wext= ∆ET = 0


Ideal mechanical system Energy

example #1 - Skiing

  • In skiing, the only force doing work is gravity. Normal force is perpendicular to the displacement. The system is ideal, TME is constant. Energy only transforms from one form to another.


Ideal mechanical system Energy

example #2 - Pendulum

  • In a pendulum, the only force doing work is gravity. Tension is perpendicular to displacement. The system is ideal, TME is constant. Energy only transforms from one form to another,


Example Energy

  • As the 2.0-kg pendulum bob in the above diagram swings to and fro, its height and speed change. Use energy equations and the above data to determine the blanks in the above diagram.

0

0.306

0.153

0.306

1.73

2.45


Example18
Example Energy

  • As the pendulum swings from position A to position C as shown in the diagram, what is the relationship of kinetic energy to potential energy? [Neglect friction.]

  • The kinetic energy decreases more than the potential energy increases.

  • The kinetic energy increases more than the potential energy decreases.

  • The kinetic energy decrease is equal to the potential energy increase.

  • The kinetic energy increase is equal to the potential energy decrease.


Example19
example Energy

  • A pendulum is pulled to the side and released from rest. Sketch a graph best represents the relationship between the gravitational potential energy of the pendulum and its displacement from its point of release.

PE

pos


Example20
Example Energy

  • In the diagram, an ideal pendulum released from point A swings freely through point B. Compared to the pendulum's kinetic energy at A, its potential energy at B is

  • half as great

  • twice as great

  • the same

  • four times as great


Ideal mechanical system Energy

example #3 – Roller Coaster

  • In Roller Coster, the only force doing work is gravity. Normal force is perpendicular to the displacement. The system is ideal, TME is constant. Energy only transforms from one form to another.


Ideal mechanical system Energy

example #4 – Free Fall or projectile

  • In Free Fall, the only force doing work is gravity. The system is ideal, TME is constant. Energy only transforms from one form to another.


Energy graph of a free falling object
Energy graph of a free falling object Energy

The graph shows as a ball is dropped, how its energy is transformed.

  • The total mechanical energy remains _____________.

  • GPE decreases as KE increases

constant


Example21
example Energy

  • A 3.0-kilogram object is placed on a frictionless track at point A and released from rest. (Assume the gravitational potential energy of the system to be zero at point C.) Calculate the kinetic energy of the object at point B.

KEi + PEi = KEf + PEf

0 + mg(3.0m) = KEf + mg(1.0m)

KEf = mg∆h = (3.0 kg)(9.81 m/s2)(3.0 m – 1.0 m) = 59 J

(KE gained at B is potential lost from A to B

Since only force is gravity, TME is constant


Example22
example Energy

  • A 250.-kilogram car is initially at rest at point A on a roller coaster track. The car carries a 75-kilogram passenger and is 20. meters above the ground at point A. [Neglect friction.] Compare the total mechanical energy of the car and passenger at points A, B, and C.

  • The total mechanical energy is less at point C than it is at points A or B.

  • The total mechanical energy is greatest at point A.

  • The total mechanical energy is the same at all three points.

  • The total mechanical energy is greatest at point B.


Example23
example Energy

  • The diagram represents a 0.20-kilogram sphere moving to the right along a section of a frictionless surface. The speed of the sphere at point A is 3.0 meters per second.

  • Approximately how much kinetic energy does the sphere gain as it goes from point A to point B?

Since only force is gravity, TME is constant

KEi + PEi = KEf + PEf

KEf - KEi = PEi - PEf

The KE gained is PE lost

∆KE = ∆PE = mg∆h

KE = (0.20 kg)(9.81 m/s2)(1.00 m) = 2.0 J


Example24
example Energy

  • A 1.0 kg mass falls freely for 20. meters near the surface of Earth. What is the total KE gained by the object during its free fall?

Since only force is gravity, TME is constant

KEi + PEi = KEf + PEf

KEf - KEi = PEi - PEf

∆KE (gained) = GPE (lost)

∆KE = (1.0 kg)(9.81 m/s2)(20.m-0) = 2.0x102 J


Ideal mechanical system Energy

example #5 – Pop Up toy

  • In Pop Up toy, the only force doing work is gravity and force on a spring. The system is ideal, TME is constant. Energy only transforms from one form to another.

Initially, at the bottom: TMEi = ½ kx2

Finally, at the top: TMEf = mgh

TMEf = TMEi

½ kx2 = mgh


Work and energy thorem
Work and Energy Thorem Energy

  • When external force does work on a system, its total energy is changed.

When external force is zero, the system is closed, its total energy is conserved.

In a closed system, if the only force doing work are conservative forces (no friction), the total mechanical energy is conserved.


Example25

B Energy

0.50 m

Example

  • A box with a mass of 0.04 kg starts from rest at point A and travels 5.00 meters along a uniform track until coming to rest at point B, as shown in the picture. Determine the magnitude of the frictional force acting on the box. (assume the frictional force is constant.)

A

0.80 m


Example26
example Energy

  • Base your answer to the question on the information and diagram. A 250.-kilogram car is initially at rest at point A on a roller coaster track. The car carries a 75-kilogram passenger and is 20. meters above the ground at point A. [Neglect friction.] Calculate the speed of the car and passenger at point B.

20. m/s


Example27
example Energy

  • The diagram shows a 0.1-kilogram apple attached to a branch of a tree 2 meters above a spring on the ground below. The apple falls and hits the spring, compressing it 0.1 meter from its rest position. If all of the gravitational potential energy of the apple on the tree is transferred to the spring when it is compressed, what is the spring constant of this spring?

Since only internal forces is doing work, TME is constant

KEi + PEi = KEf + PEf

0 + mgh = 0 + ½ kx2

mgh = ½ kx2

(0.1 kg)(9.81m/s2)(2m) = ½∙k (0.1m)2

k = 400 N/m


Example28
Example Energy

In the diagram below, 450. joules of work is done raising a 72-newton weight a vertical distance of 5.0 meters. How much work is done to overcome friction as the weight is raised?

KEi + PEi + Wext = KEf + PEf

There are two external forces: applied force and friction force

The applied force did 450 J of work:

0 + 0 + 450 J + Wf = 0 + (72 N)(5.0m)

450 J + Wf = (72 N)(5.0 m)

Wf = -90 J

90 J of work is done to overcome friction


Practices1
Practices Energy

A block weighing 15 newtons is pulled to the top of an incline that is 0.20 meter above the ground. If 3.5 joules of work are needed to pull the block the full length of the incline, how much work is done against friction?

A block weighing 40. newtons is released from rest on an incline 8.0 meters above the horizontal. as shown in the diagram below. If 50. joules of heat is generated as the block slides down the incline, what is the maximum kinetic energy of the block at the bottom of the incline?

A 20.-kilogram object strikes the ground with 1960 joules of kinetic energy after falling freely from rest.  How far above the ground was the object when it was released?

A 55.0-kilogram diver falls freely from a diving platform that is 3.00 meters above the surface of the water in a pool. When she is 1.00 meter above the water, what are her gravitational potential energy and kinetic energy with respect to the water's surface?


  • A person does 64 joules of work in pulling back the string of a bow. What will be the initial speed of a 0.5-kilogram arrow when it is fired from the bow?

  • A spring in a toy car is compressed a distance, x. When released, the spring returns to its original length, transferring its energy to the car. Consequently, the car having mass m moves with speed v. Derive the spring constant, k, of the car’s spring in terms of m, x, and v. [Assume an ideal mechanical system with no loss of energy.]

  • A child, starting from rest at the top of a playground slide, reaches a speed of 7.0 meters per second at the bottom of the slide. What is the vertical height of the slide? [Neglect friction.]


  • Which of the following statements are true about work? Include all that apply.

    • Work is a form of energy.

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

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

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

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

    • 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.

    • 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.


Lab 15 power
Lab 15 – Power Include all that apply.

Purpose: To determine my power requirement for climbing a staircase - both by walking and by running. Compare the two results.

Material: stop watch, ruler.

Data section: should contain colomns of measured and calculated data for both walking and running up a flight of stairs. The rows and columns should be labeled; units should be identified. Work should be shown for each calculation; the work should be labeled and easy to follow.

Conclusion/discussion of results: The Conclusion should (as always) answer the questions posed in the Purpose.


Data table for individual
Data table for individual Include all that apply.


Data table for each group
Data table for each group Include all that apply.


Lab 16 energy of a tossed ball
Lab 16 - Include all that apply.Energy of a Tossed Ball

OBJECTIVES

  • Measure the change in the kinetic and potential energies as a ball moves in free fall.

  • See how the total energy of the ball changes during free fall.

    MATERIALS


Preliminary questions
PRELIMINARY QUESTIONS Include all that apply.

  • For each question, consider the free-fall portion of the motion of a ball tossed straight upward, starting just as the ball is released to just before it is caught. Assume that there is very little air resistance.

    1. What form or forms of energy does the ball have while momentarily at rest at the top of the path?

    2. What form or forms of energy does the ball have while in motion near the bottom of the path?

    3. Sketch a graph of postion vs. time for the ball.

    4. Sketch a graph of velocity vs. time for the ball.

    5. Sketch a graph of kinetic energy vs. time for the ball.

    6. Sketch a graph of potential energy vs. time for the ball.

    7. Sketch a graph of total energy vs. time for the ball.

    8. If there are no frictional forces acting on the ball, how is the change in the ball’s potential energy related to the change in kinetic energy?


Data table
DATA TABLE Include all that apply.


Analysis
ANALYSIS Include all that apply.

  • Inspect kinetic energy vs. time graph for the toss of the ball.

  • Inspect potential energy vs. time graph for the free-fall flight of the ball.

  • Inspect Total energy vs. time graph for the free-fall flight of the ball.

  • Your conclusion from this lab

    • How does the kinetic and potential energy change?

    • How does the total energy change?


Lab 14 hooke s law 1
Lab 14 – Hooke’s Law (1) Include all that apply.

  • Purpose: To determine the spring constant of a given spring.

  • Material: spring, masses, meter stick.

  • Procedure: Hook different masses on the spring, record the force Fs (mg) and corresponding elongation x. Plot the graph of Force vs. elongation

  • Data section: should contain colomns: force applied, elongation.

    • Data measured directly from the experiment. The units of measurements in a data table should be specified in column heading only.

  • Data analysis: Graph force vs. elongation on graph paper, answer following questions:

    • What does the slope mean in Force vs. elongation graph?

    • Determine the spring constant


  • Force vs. elongation Include all that apply.

    Force (N)

    Elongation (m)


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