Section 5-3: Conservation of energy

Objectives1.Identify situations in which conservation of

mechanical energy is valid.2.Recognize the forms that conserved energy

can take.3.Solve problems using conservation of

mechanical energy.

Mechanical 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.

TME = KE + PEg + PEs

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

Mechanical Energy as the Ability to Do Work

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

Conservative vs. non - conservative Forces• There are a variety of ways to categorize all the types

of forces. 1. Contact force: Forces that arise from the physical

contact of two objects.2. Field force exist between objects, even in the

absence of physical contact between the objects.• We can also categorize forces based upon whether or

not their presence is capable of changing an object's total mechanical energy.

1. Conservative force can never change the total mechanical energy of an object

2. Non-conservative forces will change the total mechanical energy of the object

• The conservative forces include the gravity forces, spring force, magnetic force, electrical force.

• We will simply say all the other forces are non-conservative forces, such as applied force, normal force, tension force, friction force, and air resistance force.

conservative forces

Non-conservative forces

Fgrav

Fspring

Fapp

Ffrict

Ften

FNorm

Work energy theoremWnet = ∆KE

Wgrav + Wspring + Wother = ∆KEWgrav = work done by Gravity

Wgrav = mg(hi – hf)Wspring = work done by spring

Wspring = ½ k(xi2 – xf

2)

mghi – mghf + ½ kxi2 – ½kxf

2 + Wother = ½ kvf2 – ½kvi

2

mghi + ½ kxi2 + ½kvi

2 + Wother = ½ kvf2 + mghf + ½kxf

2

PEgi + PEsi + KEi + Wother = PEgf +

PEsf + KEf

• When net work is done upon an object by an non-conservative force, the total mechanical energy (KE + PE) of that object is changed. – If the work is positive, then the object will gain energy. – If the work is negative, such as friction doing work, then the

object will lose energy, the object will gain heat (internal energy).– The gain or loss in energy can be in the form of potential energy,

kinetic energy, or both. – The work done will be equal to the change in mechanical energy

of the object.

PEgi + PEsi + KEi + Wother = PEgf + PEsf + KEf

TMEi + Wother = TMEf

Wother = TMEf - TMEi

• When the only type of force doing net work upon an object is conservative force (Wother = 0), the total mechanical energy (KE + PE) of that object remains constant. TMEf = TMEi. In such cases, the object's energy changes form.

• For example, as an object is "forced" from a high elevation to a lower elevation by gravity, some of the potential energy of that object is transformed into kinetic energy. Yet, the sum of the kinetic and potential energies remain constant.

When Wother= 0 TMEi = TMEf

The Example of Pendulum Motion• Consider a pendulum bob swinging to and fro on the end of a

string. There are only two forces acting upon the pendulum bob. Gravity (an internal force) acts downward and the tensional force (an external force) pulls upwards towards the pivot point. The external force does not do work since at all times it is directed at a 90-degree angle to the motion. The only force doing work is gravity, which is a conservative force.

KEi + PEi + Wext = KEf + PEf

Wext = 0

KEi + PEi = KEf + PEf

The pendulum: Wother = 0• The sum of the kinetic and potential energies in system is

called the total mechanical energy.• In the case of a pendulum, the total mechanical energy (KE +

PE) is constant: at the highest point, all the energy is potential energy, at the lowest point, all the energy is kinetic 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.

0.306 0.1531.73

02.45

0.306

Roller coaster – friction is ignored, Wother = 0• A roller coaster operates on the principle of energy

transformation. Work is initially done on a roller coaster car to lift the car to the first and highest hill. The roller coaster car has a large quantity of potential energy and virtually no kinetic energy as it begins the trip down the first hill. As the car descents hills and loops, it potential energy is transformed into kinetic energy; as the car ascends hills and loops, its kinetic energy is transformed into potential energy. The total mechanical energy of the car is conserved when friction is ignored.

The skier• Transformation of energy from the potential to the kinetic

also occurs for a ski jumper. As a ski jumper glides down the hill towards the jump ramp and off the jump ramp towards the ground, potential energy is transformed into kinetic energy. If friction can be ignored, the total mechanical energy is conserved.

A free falling object• If a stationary object having mass m is located a vertical

distance h above Earth’s surface, the object has initial PE = mgh and KE = 0. As object falls, its PE decreases and KE increases. The total mechanical energy is conserved.

Energy conversion of a free falling object

• The total mechanical energy remains _____________. • GPE decreases as KE increases

constant

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

Example 1• 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?

In this situation, the force doing the work is gravity, which is an internal force.

KEi + PEi = KEf + PEf

KEf = 1080 J

Example 2• 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.]

0 + ½ kx2 = ½ mv2 + 0

½ kx2 = ½ mv2

k = mv2/x2

KEi + PEi = KEf + PEf Since only force is elastic, TME is constant

Example 3• 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?

0 + mgh = 0 + ½ kx2

mgh = ½ kx2

(0.1 kg)(9.81m/s2)(2m) = ½∙k (0.1m)2

k = 400 N/m

KEi + PEi = KEf + PEf

Since only internal forces is doing work, TME is constant

Sample Problem 5E• Starting from rest, a child zooms down a

frictionless slide from an initial height of 3.00 m. What is her speed at the bottom of the slide? Assume she has a mass of 25.0 kg.

7.67 m/s

Class work

• Page 185 – practice 5E

1. 20.7 m/s2. 9.9 m/s; 14.0 m/s3. 14.1 m/s4. 0.25 m5. 0.18 m

Mechanical Energy is not conserved when Wother ≠ 0

• When non-conservative forces do work– TME changes,

– Wother = TMEf – TMEi

• When non-conservative forces is friction, heat is generated.

TMEi + Wother = TMEf

Example 1

Example 2

Example 3

Example 4• A block weighing 15 N is pulled to the top of an incline that is

0.20 meter above the ground, as shown below. If 5.0 joules of work are needed to pull the block the full length of the incline, how much work is done against friction?

KEi + PEi + Wext = KEf + PEf

The external forces are applied force and friction force0 + 0 + Wapp + Wf = 0 + (15 N)(0.20 m)5.0 J + Wf = 3.0JWf = -2.0 J 2.0 J of work is done to overcome friction

Example 5In 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?

There are two external forces: applied force and friction forceThe applied force did 450 J of work:0 + 450 J + Wf = (72 N)(5.0m)

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

Wf = -90 J

90 J of work is done to overcome friction

TMEi + Wext = TMEf

Example 6• 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 B

0.50 m

Given: hA = 0.80 mhB= 0.50 md = 5.00 mm=0.04 kg

Unknown: Ff = ? N

TMEi + Wother = TMEf 0+mg(0.80m)+Wf=0+ mg(0.50m)Wf= -0.12 J0.12 J of work is done to

overcome frictionWf = Ff∙d = 0.12 J Ff∙(5.00m) = 0.12 JFf = 2.4 x 10-2 N

Example 7• 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?

KEi + PEi + Wext = KEf + PEf

The external force is friction force only0 + (40.N)(8.0m) + Wf = KEf + 0320 J – 50 J = KEf

KEf = 270 J

Example 8• 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?

16 m/s

example• Which of the following statements are true about work? Include

all that apply.1. Work is a form of energy. 2. Units of work would be equivalent to a Newton times a meter. 3. A kg•m2/s2 would be a unit of work. 4. Work is a time-based quantity; it is dependent upon how fast a

force displaces an object. 5. Superman applies a force on a truck to prevent it from moving

down a hill. This is an example of work being done. 6. 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. 7. 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.

example• Determine the meaning of slope in each graph

elongation

forc

e

force

elon

gatio

n

mass

wei

ght

height

Gra

vita

tiona

l po

tent

ial

ener

gy

A B

DC

K 1/K

gmg

Lab 17 - Energy of a Tossed BallOBJECTIVES1. Measure the change in the kinetic and potential energies as

a ball moves in free fall.2. See how the total energy of the ball changes during free fall.

MATERIALS

computer ballVernier computer interface Logger ProVernier Motion Detector wire basket

PRELIMINARY QUESTIONS• 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 position 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 TABLEMass of the ball (kg)

Position Time Height Velocity PE KE TE(s) (m) (m/s) (J) (J) (J)

After releaseTop of path

Before catch

ANALYSIS1. Inspect kinetic energy vs. time graph for the

toss of the ball. 2. Inspect potential energy vs. time graph for

the free-fall flight of the ball. 3. Inspect Total energy vs. time graph for the

free-fall flight of the ball. 4. Your conclusion from this lab

1. How does the kinetic and potential energy change?

2. How does the total energy change?

Lab 17 - Energy of a Tossed BallOBJECTIVES1. Measure the change in the kinetic and potential energies as

a ball moves in free fall.2. See how the total energy of the ball changes during free fall.

MATERIALS

computer ballVernier computer interface Logger ProVernier Motion Detector wire basket

PRELIMINARY QUESTIONS• 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 position 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 TABLEMass of the ball (kg)

Position Time Height Velocity PE KE TE(s) (m) (m/s) (J) (J) (J)

After releaseTop of path

Before catch

ANALYSIS

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

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

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

4. Your conclusion from this lab1. How does the kinetic and potential energy

change?2. How does the total energy change?