Work-Energy Theorem (6.2)

Slide 10-2

Chapter 10: Energy and Work

Forms of Energy

Mechanical Energy = K, Ugravity, Uspring

Ug UsK

Thermal Energy

Eth

Other forms include

Echem Enuclear

Kinetic energy K = energy of motion

Potential energy U = energy of position


Thermal energy Eth = energy associated with temperature

The Basic Energy Model

Energy Transformations are changes of energy within the system from one form to another.

An exchange of energy between the system and environment is called an energy transfer. Two primary energy transfer processes: work and heat (APP2).

Energy Transformations

Kinetic energy K = energy of motion


Thermal energy Eth = energy associated with temperature

System energy E = K + U + Eth + Echem + ...

Energy can be transformed within the system without loss.

Energy is a property of a system.

Some Energy Transformations

Echem Ug K Eth

Echem Eth Us K Ug

Work = ∆ Energy∆ Energy = Work done by Friction = ∆ Heat or Thermal Energy

Question:

If a system is isolated, the total energy of the system

A. increases constantly.B. decreases constantly.C. is constant.D. depends on work into the system.E. depends on work out of the system.

Answer

If a system is isolated, the total energy of the system

A. increases constantly.B. decreases constantly.C. is constant.D. depends on work into the system.E. depends on work out of the system.

Energy Transfers

These change the energy of the system.

Interactions with the environment.

Work is the mechanical transfer of energy to or from a system via pushes and pulls.

The Work-Energy Equation: When work is done on a system, the system’s total energy changes by the amount of work, W, done on it.

W = ∆ E

Consider a rocket with its engines fired, accelerating up toward space… Within a mile of travel (assume the

amount of fuel burned during this time is small compared to the mass of the

rocket), what is the Earth-rocket system’s mechanical energy doing?

Staying constant? Not staying constant? Why?

• The Kinetic Energy is increasing because the rocket is experiencing an increase in speed as it accelerates up into space.

• The rocket’s Gravitational Potential Energy is increasing as it goes higher, away from the surface of the earth, but still within the Earth’s gravitational force field.

• Therefore, the Earth-rocket system’s total Mechanical Energy (combination of K and Ug) is increasing.

• When work is done on a system, the system’s total energy changes by the amount of work, W, done on it. W = ∆ E

Energy Transfers:

Work causes a change in the energy… W = Δ E

W Δ K W Δ Eth

W Δ Us

W Δ Ug

W = F·d = Δ E (Units: N·m = Joule)

where F and d (displacement) are parallel to one another

positive work = increase in Ug negative work = decrease in Ug

W = F·d =Δ KW = (F·cos) d = Δ K

W = F·d =+Δ Ug W = F·d = -Δ Ug

Work… what is the work from the wind causing?

W = ΔK

• Work makes you mad…

• Power makes you mad over time…

• Get it?

• Work makes you m·a·d…

• Power makes you m·a·d/t

Recall…

Mechanical Energy Equations

Rotational kinetic energy of an object rotating with moment of inertia, I, and angular velocity, ω.

Translational Kinetic Energy

Gravitational Potential Energy

Rotational Kinetic Energy

Kinetic Energy

• K = ½ m(v)2 (Translational Kinetic

Energy)

• Looking at this equation, what does KE depend on? Mass and Speed. KE is called energy of motion.

• Work can change KE (work energy theorem.)

W = ΔKE

• F •d = ½ m vf2 – ½ m vi

2

Drivers Ed Question…While driving, if you double your speed, how much more distance

is needed to stop?

Drivers Ed Question…While driving, if you double your speed, how much more

distance is needed to stop?

• W = ΔKE

• F •d = ½ m vf2 – ½ m vi

2

• F •d = ½ m vf2 – ½ m vi

2

• F •d = 0 – ½ m vi2

• d vi2

• (2vi)2 4d

• Doubling vi, or 2x vi, will result in a 22x d or 4x greater distance needing to stop!

What causes the plane’s Kinetic Energy to change?

• A constant net external force F acts over a displacement d and does work on the plane.

• As a result of the work done, the plane’s kinetic energy changes.

Displacement, d

W = ∆K

Which requires more work, a car changing its velocity from

5 m/s to 10 m/s or the same car

changing its velocity from 50 m/s to 55 m/s?

Work changes the energy of an object…

Kinetic Energy • Kinetic Energy can be solely translational

K… K = ½ m v2

• Kinetic Energy can be solely rotational K…

• Or Kinetic Energy can be a combination of both… ball rolling down a ramp.

Rotational Kinetic Energy

I = Moment of Inertia, M = Mass, R = radius

Gravitational PE• Ug = GPE = magΔh

• What does GPE depend on?

• Mass, acceleration due to gravity & height.

• GPE is called energy of location or position.

• ΔGPE does not care about that path taken, just the change in height.

• W= ΔGPE = magΔh

Work done by the force of gravity• Gravity exerts a force

mg (g = ag) on the basketball. Work is done by the gravitational force as the basketball falls from a height of ho to a height of hf (relative to the earth’s surface).

• d = ho - hf

• Wg = Fg d = mag (ho-hf)

• Wg = ΔUg = magΔh

• An object can move along different paths in going from an initial height ho to a final height of hf. In each case, the work done by the gravitational force is the same, since the change in vertical distance is the same.

• d = ho - hf

• Wg = Fg d = mg (ho-hf)

• Wg = ΔUg = magΔh

• Note: ΔUg does not depend on path taken.

Gravitational Potential Energy

• An object possessing energy by virtue of its position relative to earth is said to have gravitational potential energy.

• The hammer has the potential to do the work of driving the pile into the ground.

• Ug = PEg = magh

• Wg = mag (Δh) = ΔUg

Work-Energy Theorem:

W = ΔK + ΔUg = ΔE

• The total energy of a system changes by the amount of work done on it.

• When a net force performs work on an object, the result could be a change in the kinetic energy of the object and/or a change in the potential energy.


W = ∆ E

Conservation of Energy• Energy cannot be created nor destroyed, but

energy can change from one form into another.

• ΔE = ΔUg + Δ KE = W

• For an isolated system, W = 0, or the total energy of an isolated system remains constant (is conserved).

• ΔE = Δ Ug + Δ KE = W = 0 (isolated system)

• Conservation of Energy… Efinal = Einitial

A sledder, starting from rest, slides down a 10-m high hill. At the

bottom of the hill is a long horizontal patch of rough snow. The

hill is nearly frictionless, but the coefficient of friction, µk between

the sled and the rough snow at the bottom is 0.30.

How far will the sled slide along the rough patch?

Did you get… d= 33 m

Question

If you raise an object to a greater height, you are increasing

A. kinetic energy. B. heat. C. potential energy.D. chemical energy.E. thermal energy.

Answer

If you raise an object to a greater height, you are increasing

A. kinetic energy. B. heat. C. potential energy.D. chemical energy.E. thermal energy.

Checking Understanding

A skier is moving down a slope at a constant speed. What energy transformation is taking place?

A. K Ug

B. Ug Eth

C. Us Ug

D. Ug K

E. K Eth

A skier is moving down a slope at a constant speed. What energy transformation is taking place?

A. K Ug

B. Ug Eth

C. Us Ug

D. Ug K

E. K Eth

Answer

Checking Understanding

A child is on a playground swing, motionless at the highest point of his arc. As he swings back down to the lowest point of his motion, what energy transformation is taking place?

A. K Ug

B. Ug Eth

C. Us Ug

D. Ug K

E. K Eth

Answer

A child is on a playground swing, motionless at the highest point of his arc. As he swings back down to the lowest point of his motion, what energy transformation is taking place?

A. K Ug

B. Ug Eth

C. Us Ug

D. Ug K

E. K Eth

A heavy ball is hanging from a 4.5m cable,

and it is released from a height of 16.5 m off the ground and then falls to its lowest point

which is 15 m off the ground. What is the

speed of the ball at its lowest position?

Did you get…

•Vf= 5.4 m/s


W = ∆ E

The Law of Conservation of Energy

Summary

Work-Energy Theorem (6.2)

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