Conservation of EnergyAnd Potential Energy

Law of Conservation of Energy

The system is isolated and allows no exchange with the environment.

No mass can enter or leave

No energy can enter or leave

Energy is constant, or conserved

E = U + K + Eint

= Constant

We only allow U and K to interchange

We ignore Eint (thermal energy)

Law of Conservation of Mechanical Energy

E = U + K= Constant

Law of Conservation of Mechanical Energy

E = U + K = C or E = U + K = 0

for gravity Ug = mghf - mghi K = ½ mvf

2 - ½ mvi2

for springs Us = ½ kxf

2 - ½ kxi2

K = ½ mvf2 - ½ mvi

2

h

Pendulum Energy

½mv12 + mgh1 = ½mv2

2 + mgh2

For any points two points in the pendulum’s swing

Spring Energy

m

m -x

mx

0

½ kx12 + ½ mv1

2

= ½ kx22 + ½

mv22

For any two points in a spring’s oscillation

Conservative Forces

Conservative ForcesThe done on a particle moving b/w any two points is independent of the path takenThe work done on a particle moving through a closed path is zeroEx: gravity

UdxFW f

i

x

x cc W=Ui – Uf = -ΔU

This is only true of conservative forces. This happens because the force is negative (Fg = -mgy or Fx= -kx) and thus the change in energy (work)ends up being –Uf – Ui or –ΔU.

Practice Problem:A 2.0 m pendulum is released from rest when the support

string is at an angle of 25 degrees with the vertical. What is the speed of the bob at the bottom of the string?

Lq Lcosq

h

h = L – Lcosqh = 2-2cosqh = 0.187 m

1.35 m/s = v

Practice Problem: A single conservative force of F = (3i + 5j) N acts on a 4.0 kg particle. Calculate the work done if the particle if the moves from the origin to r = (2i - 3j) m.

a. Does the result depend on path?

b. What is the speed of the particle at r if the speed at the origin was 4.0 m/s?

c. What is the change in potential energy of the system?W = -9Ja) no, Wc is independent of pathb) 3.4 m/sc) 9J

Practice Problem: A bead slides on the loop-the-loop shown and is released from height h = 3.5 R.What is the speed at point A? (assume all energy is conserved)

VA=√(3gR)

Non-conservative forces Wtot = Wnc + Wc = ΔK

Wnc = ΔK – Wc

(Wc = -ΔU)

Wnc = ΔK + ΔU

(Δkfriction= -Fkd – this is the energy lost due to friction, the internal energy that goes into the object (as thermal energy))

Nonconservative ForcesThe work done on a particle b/w any two points is dependent of the path takenCauses a change in mechanical energy (the sum of the kinetic and potential energies)Ex: friction and drag

Practice Problem: A 2,000 kg car starts from rest and coasts down from the top of a 5.00 m long driveway that is sloped at an angel of 20o with the horizontal. If an average friction force of 4,000 N impedes the motion of the car, find the speed of the car at the bottom of the driveway. (remember this is a nonconservative force)

Vf = 3.7 m/s

Practice Problem: A parachutist of mass 50 kg jumps out of a hot air balloon 1,000

meters above the ground and lands on the ground with a speed of 5.00 m/s. How much energy was lost to friction during the descent?

4.9 x 105 J

Force and Potential Energy Before we discuss the relationships between potential energy

and force, lets review a couple of relationships.

Wc = Fx (if force is constant)

Wc = Fdx = - dU = -U (if force varies) Fdx = - dU Fdx = -dU F = -dU/dx

Energy Diagrams: Stable Equilibrium

U

x

Stable Equilibrium:Any displacement from equilibrium results in a force directed back towards x = 0

The positions of stable equilibrium correspond to the points where U(x) is a minimum

Example: A springA ball in a bowl

1st derivative: minimum gives position of stable equilibrium

2nd derivative: would give the spring constant

x and –x give the turning points, a spring will oscillate b/w these points because it can’t exceed ½kx2

-x x

Energy Diagrams: Stable Equilibrium

Us

x

A spring in stable equilibrium:Fs = -dUs/dx = -kxThe force is thus equal to the negative of the slope of the energy curve.

If a force stretches the spring, x is + and the slope is +, thus F is - and brings the spring back to equilibrium.

If a force compresses the spring, x is – and the slope is -, thus F is + and brings the spring back to equilibrium.

Us= ½kx2

Energy Diagrams: Unstable Equilibrium

U

x

Unstable Equilibrium:Any displacement from equilibrium results in an acceleration away from that point

The positions of unstable equilibrium correspond to the points where U(x) is a maximum

Example: A pencil balanced vertically

Energy Diagrams: Neutral Equilibrium

U

x

Neutral Equilibrium:Any displacement from equilibrium results in neither a restorative nor a disruptive force

Example: A ball on a flat table

Practice Problem:The potential energy associated with the force between two

neutral atoms in a molecule is modeled by the Lennard-Jones potential energy function:

Where x is the distance b/w the atoms, and σ and ε are determined experimentally. In this case σ= .263 nm and ε= 1.51 x 10-22 J.

We expect to find the stable equilibrium point where the potential energy of the system is at a minimum, find the equilibrium separation of the two atoms.

X = 2.95 x 10-10 m

Molecular potential energy diagrams

x

UThe potential energy is quite large when the atoms are very close together, at a minimum when they are at their critical separation, and increases again when the atoms move apart.

When U is a minimum, the atoms are in stable equilibrium. This is the most likely separation between the atoms

Graph of the potential energy curve for the molecule at various distances between the atoms