Energy of a System.pdf

PHYS-1408-001

Feb. 28, 2013

Lecture 12

Announcement

Course webpage

http://highenergy.phys.ttu.edu/~slee/1408/

Homework.5 (due by March 7)

Ch.7 1, 6, 11, 14, 28, 31, 37, 41

Ch.8 6, 10, 15, 16, 22, 25, 29, 38

Lab

Monday 11:00 am 12: 50 pm (Sci. 105)

Recitation

Monday 3:00 pm 3:50 pm (Sci. 105)

Quiz 3

3/7 (Next Thursday): Ch7 & 8

Chapter 7

Energy of a System

Kinetic Energy (KE) KE: Energy of a particle due to its motion K = 1/2 mv2

K = Kinetic energy m = mass of the particle v = Speed of the particle

A change in kinetic energy is one possible result of doing work to transfer energy into a system

Calculating the work: 1/2mv22 - 1/2mv12 = Fx = WF

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.

Gravitational 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 at a height y above some reference point:

.

.A person exerts an upward force Fext = mg to lift a brick from y1 to y2 .

WG = FG d = mghcos1800 = -mgh

Gravitational Potential Energy, final

The quantity mgy is identified as the gravitational potential energy, Ug.

! Ug = mgy

Units are joules (J)

Scalar quantity

Work may change the gravitational potential energy of the system.

! Wext = ug

Section 7.6

Potential Energy Example: Potential energy changes for a roller coaster.

A 1000-kg roller-coaster car moves from point 1 to point 2 and then to point 3. (a) What is the gravitational potential energy at points 2 and 3 relative to point 1? That is, take y = 0 at point 1.

Potential Energy Example: Potential energy changes for a roller coaster.

A 1000-kg roller-coaster car moves from point 1 to point 2 and then to point 3. (a) What is the gravitational potential energy at points 2 and 3 relative to point 1? That is, take y = 0 at point 1.

Potential Energy Example: Potential energy changes for a roller coaster.

A 1000-kg roller-coaster car moves from point 1 to point 2 and then to point 3. (b) What is the change in potential energy when the car goes from point 2 to point 3?

Potential Energy Example: Potential energy changes for a roller coaster.

A 1000-kg roller-coaster car moves from point 1 to point 2 and then to point 3. (b) What is the change in potential energy when the car goes from point 2 to point 3?

Potential Energy Example: Potential energy changes for a roller coaster.

A 1000-kg roller-coaster car moves from point 1 to point 2 and then to point 3. (c) Repeat parts (a) and (b), but take the reference point (y = 0) to be at point 3.

8-2 Potential Energy Example 8-1: Potential energy changes for a roller coaster.

A 1000-kg roller-coaster car moves from point 1 to point 2 and then to point 3. (c) Repeat parts (a) and (b), but take the reference point (y = 0) to be at point 3.

Potential Energy General definition of gravitational potential energy:

For any conservative force:

WG = -mgh

Gravitational 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 at a height y above some reference point:

.

.A person exerts an upward force Fext = mg to lift a brick from y1 to y2 .

WG = FG d = mghcos1800 = -mgh

Elastic Potential Energy A spring has potential energy, called elastic potential energy, when it is compressed. The force required to compress or stretch a spring is:

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

A spring (a) can store energy (elastic potential energy) when compressed (b), which can be used to do work when released (c) and (d).

8-2 Potential Energy Then the potential energy is:

8-2 Potential Energy Then the potential energy is:

Elastic Potential Energy, cont.

Section 7.6

This expression is the elastic potential energy: Us = kx2

The elastic potential energy can be thought of as the energy stored in the deformed spring.

The stored potential energy can be converted into kinetic energy. (see next slide)

Energy Bar Chart Example

An energy bar chart is an important graphical representation of information related to the energy of a system.

! The vertical axis represents the amount of energy of a given type in the system.

! The horizontal axis shows the types of energy in the system. In a, there is no energy.

! The spring is relaxed, the block is not moving Section 7.6

Energy Bar Chart Example, cont.

Between b and c, the hand has done work on the system.

! The spring is compressed. ! There is elastic potential energy in the system. ! There is no kinetic energy since the block is held steady.

Section 7.6

Energy Bar Chart Example, final

In d, the block has been released and is moving to the right while still in contact with the spring.

! The elastic PE of the system decreases while the kinetic energy increases. In e, the spring has returned to its relaxed length and the system contains only kinetic energy associated with the moving block.

Section 7.6

Internal Energy

The energy associated with an objects temperature is called its internal energy, Eint.

The friction does work and increases the internal energy of the surface.

When the book stops, all of its kinetic energy has been transformed to internal energy. (we will see this in Ch.8)

The total energy remains the same.

Section 7.7

Conservative and Nonconservative Forces A force is conservative if: the work done by the force on an object moving from one point to another depends only on the initial and final positions of the object, and is independent of the particular path taken. Example: gravity.

Object of mass m: (a) falls a height h vertically.

Conservative and Nonconservative Forces

Another definition of a conservative force:

a force is conservative if the net work done by the force on an object moving around any closed path is zero.

(a) A tiny object moves between points 1 and 2 via two different paths, A and B. (b) The object makes a round trip, via path A from point 1 to point 2 and via path B back to point 1.

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 non-conservative force.

A crate is pushed at constant speed across a rough floor from position 1 to position 2 via two paths, one straight and one curved. The pushing force FP is always in the direction of motion. (The friction force opposes the motion.) Hence, the work it does is W = FPd, so if d is greater (as for the curved path), then W is greater. The work done does not depend only on points 1 and 2; it also depends on the path taken.

Conservative and Nonconservative Forces

Potential energy can only be defined for conservative forces.

Conservative Forces and Potential Energy

Define a potential energy function, U, such that the work done by a conservative force equals the decrease in the potential energy of the system.

The work done by such a force, F, is

! U is negative when F and x are in the same direction."

f

i

x

xxW F dx Uint = =

Section 7.8

We can invert this equation to find U(x) if we know F(x):

Conservative Forces and Potential Energy cross check!!!

Look at the case of a deformed spring:

! This is Hookes Law and confirms the equation for U

212

ss

dU dF kx kxdx dx

! "= = = $ %

& 'Chapter 8

Conservation of Energy

Three youngsters enjoy the transformation of potential energy to kinetic energy on a waterslide. !We can analyze processes such as these with the techniques developed in this chapter. !

Energy Review

Kinetic Energy

! Associated with movement of members of a system

Potential Energy

! Determined by the configuration of the system ! Gravitational & Elastic Potential Energies have been studied

Internal Energy

! Related to the temperature of the system

BTW, whats the system?

Introduction

Types of Systems

Non-isolated systems

! Energy can cross the system boundary in a variety of ways. (see next slide) ! Total energy of the system changes

Isolated systems

! Energy does not cross the boundary of the system ! Total energy of the system is constant

Conservation of energy

! Can be used if no non-conservative forces act within the isolated system

Introduction

Examples of Ways to Transfer Energy Into or out of a system

Section 8.1

Work ! Mechanical Wave !

Heat !

Matter Transfer !

Electrical Transmission !

Electromagnetic Radiation!

Conservation of Energy

Energy is conserved

! This means that energy cannot be created nor destroyed. ! If the total amount of energy in a system changes, it can only be due to the

fact that energy has crossed the boundary of the system by some method of energy transfer.

Section 8.1

Conservation of Energy, cont.

Mathematically, Esystem = #! Esystem = total energy of the system ! T = energy transferred across the system boundary

! Established symbols: Twork = W and Theat = Q

The primarily mathematical representation of the energy version of the analysis model of the non-isolated system is given by the full expansion of the above equation.

! K + U + Eint = W + Q + TMW + TMT + TET + TER ! TMW transfer by mechanical waves ! TMT by matter transfer ! TET by electrical transmission ! TER by electromagnetic transmission"

Section 8.1

The Work-Kinetic Energy theorem is a special case of Conservation of Energy

Isolated System

For an isolated system, Emech = 0

! Remember Emech = K + U ! This is conservation of energy for an isolated system with no non-

conservative forces acting.

If non-conservative forces are acting, some energy is transformed into internal energy.

Conservation of Energy becomes Esystem = 0

! Esystem is all kinetic, potential, and internal energies ! This is the most general statement of the isolated system model.

Section 8.2

Isolated System, cont.

The changes in energy can be written out and rearranged.

Kf + Uf = Ki + Ui ??? -- How? See next slide!

Section 8.2

Conservation of Mechanical Energy

We define the sum of kinetic and potential energies as mechanical energy in the system: Emech = K + Ug

Conservation of Mechanical Energy for an isolated system: Kf + Uf = Ki+ Ui

Look at work done by the book as it falls from some height to a lower height

From work-kinetic energy theorem, Won book = Kbook

Also, Won book = Fr = mgyb mgya = Kbook mgyb mgya = -(Uf - Ui) = -Ug

So, K = -Ug => K + Ug = 0

Conservation of Energy, Example 1 (Drop a Ball)

Initial conditions: Ei = Ki + Ui = mgh

The configuration for zero potential energy is the ground

Conservation rules applied at some point y above the ground gives Kf + Uf = Ki + Ugi 1/2 mvf2 + mgy = 0 + mgh

Conservation of Energy, Example 2 (Pendulum)

As the pendulum swings, there is a continuous change between PE and KE

@ A, the energy is PE

@ B, all of the PE @ A is transformed into KE

Let 0 PE be @ B

@ C, the KE has been transformed back into PE

Conservation of Energy, Example 3 (Spring Gun)

Choose point A as the initial point and C as the final point

EA = EC KA + UgA + UsA = KC + UgC + UsC 0 + 0 + 1/2 kx2 = 0 + mgh + 0 1/2 kx2 = mgh k = 2mgh/x2 where h = (xC - xB)

Example Spring Gun, final

The energy of the gun-projectile-Earth system is initially zero.

The popgun is loaded by means of an external agent doing work on the system to push the spring downward.

After the popgun is loaded, elastic potential energy is stored in the spring and the gravitational potential energy is lower because the projectile is below the zero height.

As the projectile passes through the zero height, all the energy of the system is kinetic.

At the maximum height, all the energy is gravitational potential.

Section 8.2

Problem Solving Using Conservation of Mechanical Energy

Example: Falling rock.

If the original height of the rock is y1 = h = 3.0 m, calculate the rocks speed when it has fallen to 1.0 m above the ground.

Problem Solving Using Conservation of Mechanical Energy

Example: Falling rock.

If the original height of the rock is y1 = h = 3.0 m, calculate the rocks speed when it has fallen to 1.0 m above the ground.