Chapter 8 Potential Energy and Conservation of Energywman/phy111hw/lecture notes... · · 2011-03-11If no friction, K + U does not change. It’s true for all middle positions,

Chapter 8

Potential Energy and Conservation of Energy

Copyright Dr. Weining Man and Pearson

Units of Chapter 8

• Conservative and NonconservativeForces

• Potential Energy and the Work Done by Conservative Forces

• Conservation of Mechanical Energy

• Work Done by Nonconservative Forces

• Potential Energy Curves and Equipotentials



Kinetic Energy

K = ½mv2

Wexternal, Wpush, pull, wind, man, wave, elevator, …

Wkinetic-friction = – fkd

Wmg

Wspring

Kinetic Energy is like your checking account.

What’s special about work done by gravity? From height h to ground, For the red ball, Wmg= mghFor the green ball, Wmg= mg d cosθ = mghFor the blue ball, after a complicated path, Wmg is also = mghWork done by gravity is only determined by the initial and final height.

Copyright Dr. Weining Man

When you lift a box and place it at height h, you did work mgh, where did the energy go?Later the box can drop and gain kinetic energy back.

8-1 Conservative and NonconservativeForces

Conservative force:the work it does is stored in the form of energy

that can be released at a later time

Example of a conservative force: gravity

Example of a nonconservative force: friction

Also: the work done by a conservative force moving an object around a closed path is zero; this is not true for a nonconservative force


8-1 Conservative and Nonconservative Forces

Work done by gravity on a closed path is zero:


up

8-1 Conservative and Nonconservative ForcesThe work done by a conservative force is zero on any closed path:


8-1 Conservative and Nonconservative ForcesWork done by friction on a closed path is not zero:

After that a lot of negative work was doen.



A ball with initial velocity v0, raised ∆h,Work done by gravity: Wmg = mg * ∆h*cos(180) = - mg ∆hGravity force did negative work.

Change of kinetic energy:∆K = Kf –Ki = 0 – ½ m v0

2 = Wmg = - mg ∆hLost Kinetic Energy of amount ½ m v0

2 = mg ∆h

Q: Where did those energy go?

8-2 The Work Done by Conservative Forces


Positive Work done by conservative force reduces Potential Energy and increase Kinetic Energy

Negative Work done by conservative force reduces Kinetic Energy and increase Potential Energy

A: The energy is stored as potential energy.PE is like your saving account. Potential energy gain (mg∆h) during the rising part. We can get that energy back as kinetic E if the ball falls back off. During falling, Kinetic Energy will increase mg∆h. Potential energy will reduce mg∆h.

8-2 The Gravitational potential energy

Falling from height y to ground, Gravitational potential energy reduced from mgyto 0.


U=mgy OR PE =mgyy is the height above “ground”.

You can define any y=0 “ground”.What matters is the Change in height.

PE or U, is due to being under gravitational influence of another mass. (Earth).

8-2 Potential Energy stored in Spring


This is the potential E stored in a deformed spring.

k is spring constant. (not Kinetic Energy)

∆x is the compressed or stretched length of the spring.


Kinetic Energy

K = ½mv2



Wmg

Wspring

Kinetic Energy is like your checking account.


U = mgyPotential Energy

U = ½k(∆x) 2


Kinetic Energy

K = ½mv2


Wmg

Wspring

KE is like your checking account. PE is like your saving account.When you consider the total amount of K+U, no need to worry about transitions between K and U. NO NEED TO calculate W done by mg and spring anymore,When you consider K+U together.


U = ½k(∆x) 2


Kinetic Energy

K = ½mv2

When only mg or spring force (conservative forces) do work, (No friction, no resistance (in air, on ice, not water…),no external forces or energy source does work, (no human, animal, motor, lift, push…) K+U will stay unchanged.

Got Conservation of Mechanical Energy, when Wnc=0

8-3 Conservation of Mechanical EnergyEnergy conservation can make kinematics problems much easier to solve:


When there is no work done by non-conservative forces, Ki + Ui =Kf +Uf; (happily)You don’t need to calculate work here, because work done by gravity and spring forces areincluded in the change of potential energy. ∆K = -∆U= Work of gravity or spring

8-3 Conservation of Mechanical Energy


You can solve v easily if h is known.Q1: If initial height was lower, the final speed to hit the ground would be more or less?Q2: Had the initial height was doubled, the final speed to hit the ground would become Half, Twice, , or 4 times ?

Similarly, if an object was thrown UPWARD with v0, maximum height can be easily found.

Figure 8-9

When no work is done bynon-conservative forces, Ki + Ui =Kf +Uf;When we look for changeof KE or PE, only initial and final height matters.

½ mv02+mgh0= ½ mvf

2+mghfNo matter which direction velocities are. If no friction, K + U does not change. It’s true for all middle positions, not only the start and the end. You have ½ mv0

2+mgh0= ½ mvf2+mghf

=½mv12+mgh1= ½ mv2

2+mgh2, or any middle position at 1,2,3,…. regardlessvelocity directions.

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At maximum height, ½ mv0

2+mgh0=½mvtop2+mghmax

Is vtop = 0 ?

For A object launched at angle θ and V0, through the entire motion, for all points in between.

½ mv02+mgh0= ½ mvf

2+mghf=½mv1

2+mgh1= ½ mv22+mgh2

Sure, vtop_y = 0, But vtop= vtop_x = v0cosθ; ax=0.

But for a pendulum, or an object sliding up alone a incline, track etc, total v IS zero at maximum height.

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Example, A truck slide down for distance Lfrom rest, on an incline. (θ)If there is no friction,After distance L what will be its final speed?

Lh

θ

Ki + Ui =Kf +Uf;

0+ mgh0 = ½ mvf2 + 0

h0 =L sinθ

mg L sinθ = ½ mvf2

g L sinθ = ½ vf2

vf2 = 2g L sinθ

Don’t memorize conclusions like v2 =2ghIt only works without friction and when initial or final v=0Learn the physics, UNDERSTAND it,UNDERSTAND when to use the sweet


Total Energy conservation:Energy cannot be created or destroyed. It only change forms.

When Energy seems to be created/ destroyed, itactually is being converted from/to other forms.

When friction does work, it reduces kinetic energy and convert that into Heat.


U = ½k(∆x) 2


Kinetic Energy

K = ½mv2



When nonconservative forces do work, the total mechanical energy is not conserved: we need to calculate work done by Non-Conservative forces. (Wnc= sum of all work except for mg and spring’s work)

8-4 Work Done by Nonconservative Forces

In this example, the nonconservative force is water resistance:

By comparing Ef and Ei

You can find Wnc, which is otherwise hard to calculate since water resistance may not be constant force.


New Problem solving strategy : 1.Ananlyze all forces and decide whether Non-Conservative force does work or not. 2. Only compute total work down by NC forces 3. Set equation

Yeah! No need to calculate ALL work any more, when I understand conservative and non-conservative forces.

m=1000kg, v0=0, µk=0.2φ=30ο, d=0.5m,Find vf

N=mgcosφfk=µkN=µkmgcosφfk=0.2∗9800∗cos30=1697(Ν)

WNC = Wfk = – fk d = – 1697*0.5= – 849 J ; Kf + Uf = Ki + Ui +WNC

½mvf2 + 0 = 0 + mgh – fk d ; h=d sin φ

½mvf2 = mg d sin30 – fk d =1601J vf=1.79m/s

Summary of Chapter 8

• Conservative forces conserve mechanical energy. They do work to convert energy between Kinetic Energy and Potential Energy.

• Conservative force does zero work on any closed path

• Work done by a conservative force is independent of path

• Conservative forces: gravity, spring force.

• Nonconservative forces convert mechanical energy into other forms.


Summary of Chapter 8• Work done by nonconservative force on closed path is not zero, and depends on the path

• Nonconservative forces: friction, air resistance, pull, push, most other forces…

• Energy in the form of potential energy can be converted to kinetic or other forms

• Work done by a conservative force is the negative of the change in the potential energy

• Gravity: U = mgy

• Spring: U = ½ k(∆x)2


Summary of Chapter 8

• Mechanical energy is the sum of the kinetic and potential energies; it is conserved only in systems with purely conservative forces to do work.

• Nonconservative forces change a system’s mechanical energy

• Work done by nonconservative forces equals change in a system’s mechanical energy (K+U)

•Gravitational Potential energy at distance r from earth center:

U= GMearth m/rCopyright Dr. Weining Man and Pearson



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Chapter 8 Potential Energy and Conservation of Energywman/phy111hw/lecture notes... · · 2011-03-11If no friction, K + U does not change. It’s true for all middle positions,