Mechanics 105

Potential energy of a system The isolated system Conservative and nonconservative forces Conservative forces and potential energy The nonisolated system in steady state Potential energy for gravitational and electrostatic

forces Energy diagrams and stability

Potential energy (chapter seven)Potential energy (chapter seven)

Mechanics 105

Potential energy of a systemPotential energy of a system

Objects and force internal to system External work done on a system that does not

change the kinetic or internal energy – change in potential energy

Gravitational potential energy - the work done by an external force raising an object from ya to yb is:

)( ab yymgrFW

Mechanics 105

Potential energyPotential energy

This work is a transfer of energy to the system. We can define the quantity Ugmgy to be the gravitational potential energy

The work done on the system then gives the change in Ug: W= Ug

The changechange in energy is the key thing – to do problems, you need to first define a reference location (height)

Only depends on height – not on horizontal displacement

Mechanics 105

Isolated systemsIsolated systemsConsider the system of an object only in the earth’s gravitational

field, falling from yb to ya. In free fall, the work done by gravity is mg(yb-ya), which results in

a change in the kinetic energy (work-kinetic energy theorem) K.

This work equals -Ug, the change in the gravitational potential energy of the (earth + object) system. The earth’s kinetic energy will not change, so the change in kinetic energy of the (earth + object) system is just the change of the KE of the object.

This gives the result: K= -Ug, or K+Ug=0

Mechanics 105Isolated systems Isolated systems

We can write this as a continuity equation for the mechanical energy

Emech=K+Ug

Emech=0, or Ki+Ui=Kf+Uf

Mechanics 105

Example – object in free fallExample – object in free fall

Consider earth and object as system Object dropped from height y=h (y=0 is

defined as height at which Ug=0) At height y, what is the speed?

Mechanics 105

Example – object in free fallExample – object in free fall

Initial energy = K+Ug=mgh

Final energy (at any point y) = mgy+½mv2

mgh=mgy+½mv2 v=(2g(h-y))½

yUg=mgyK=½mv2

y=hUg=mghK=0

y=0Ug=0

Mechanics 105

Conceptest Demo

Mechanics 105

ExampleExampleIf m1>m2, How fast will m1 be going when it hits the

floor?Start: K+Ug=m1gh

End: ½m1v2+m2gh

m1gh=½m1v2+m2gh

v=[2gh(m1-m2)/m1]½

m1

m2

Mechanics 105

Conservative and nonconservative forcesConservative and nonconservative forces

Conservative forces- Forces internal to system that cause no

transformation of mechanical to internal energy- Work done is path independent- Work done over closed path = 0- Examples: gravitational, elastic

Mechanics 105

Conservative and nonconservative forcesConservative and nonconservative forcesPotential energy of a spring

Us½kx2

Gravitational potential energy Ugmgh

New formulation of Work-KE thm:K+U+Eint=constant Conservation of energy

Mechanics 105

ExampleExampleMotion on a curved trackChild slides down an irregular frictionless track (total height h) ,

starting from rest. What is the speed at the bottom?Ki+Ui= Kf+Uf

0+mgh=½mv2+0 v=(2gh)½

Mechanics 105

Conservative forces and potential energyConservative forces and potential energy

Since the work done by a conservative force can be written as W=-U

We can express a differential amount of work done as dW=-dU=F·dr

From this we can see that a conservative force can be written as Fx=-dU/dx (F=-U)

e.g. Fg=-dUg/dy=-d(mgy)/dy=-mg

Mechanics 105The nonisolated system in steady state The nonisolated system in steady state

Conservation of energy holds regardless of whether the system is isolated or not. For a nonisolated system, the net energy change can still be zero if the amount of energy entering equals the amount leaving the system.

Mechanics 105

Potential energy for gravitational and electrostatic forcesPotential energy for gravitational and electrostatic forces

Gravitational force between two masses (m1, m2) separated by a distance r

This gives a general form for the gravitational potential energy of:

And for the electrostatic potential energy

12221

12 r̂r

mGmFg

r

mGmU g

21

r

qqkU ee

21

Mechanics 105

ExampleExample

Mechanics 105

ExampleExample

Mechanics 105

Energy diagrams and stabilityEnergy diagrams and stability

Since the potential energy associated with a conservative force can be written Fx=-dU/dx, a plot of U vs. x can tell us something about how a system will behave as a function of position.

For relative minima of U vs. x, there will be no force – we call these points stable equilibria.

For relative maxima of U vs. x, there will also be no force, but for small displacements away from this point, the force will be away from the equilibrium point – we call these points unstable equilibria.

Mechanics 105

Energy diagrams and stabilityEnergy diagrams and stability

Example: mass on a spring – stable equilibrium point at x=0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-20 -10 0 10 20

position (m)

Us (J)

Mechanics 105

Energy diagrams and stabilityEnergy diagrams and stability

Mechanics 105

Mechanics 105

Mechanics 105