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Physics 3A: Potential Energy
SvOutPlaceObject^
Yet another form of energy^ Consider lifting a book at constant speed:^ As you lift the book you do work onthe book:^ So your work goes into changing a quantity in the system(book & earth system) of
mgy This represents the " gravitational potential energy
" of the system: Define:^ Gravitational Potential Energy
( U ) to be:
W mgy^ mgy^ (7.1)b^ a^ U mgy (7.2)^ m = mass; y = height of object
∆ ^
^ ^ ^ ^ W F r^ m^ g^ y^ y^ jb^ a^
^ mgy^ mgy^ b^ a
Units of U are:^ Joules (J), just like work and kinetic energy Note:^ that U only depends on vertical change in position: We can choose our origin in the vertical direction anywhere,thus setting the place where U is zero
(usually the surface of the earth) Now lets consider what happens when wedrop the book: Take system as just book:
Physics 3A: Potential Energy )^ (^ ^ * ^ ^ W m g^ r^ m g^ j^ x^ x^ ib^ a^
+ ^ y y j mgy^ mgy^ b a b^ a
W m g rbook ^ ^ W m g j y^ y^ jbook a^ b^ W mgy mgy^ book b a
From Work-Kinetic Energy Theorem:^ this is for book only Relate this to the system (book & earth):^ for mgy side:^ for the kinetic energy side:^ so we have:^
Physics 3A: Potential Energy ^ ^ W^ K^ book^ book^ ^ ^ W^ mgy^ mgy^ book^ b^ a^ K^ mgybook^ (7.5)
mgy^ b a
^ mgy mgy mgy^ mgyb a a^
^ ^ U^ U^ U^ b f^ i^ g ^ K K^ Kbook system^ K U g
This is actually a "continuity equation" or Conservation of Energyequation: So energy in the final state = energy in initial state meansconservation of energy Define this energy as^ Mechanical Energy
Physics 3A: Potential Energy^ ^ ^ K^ U^ g^ ^ ^ ^ K^ K^ U^ U^ f^ i^ f^ i^ *^ *^ ^ K^ U^ K^ U^ f^ f^ i^ i^ *^.^ E^ K^ Umech^
Physics 3A: Potential Energy We now can write out mechanical energy explicitly for book-earthsystem:^ ^ E^ Kmech^ And also the conservation of mechanical energy: When mechanical energy is conserved, in the above, the forceinvolved is called a^ conservative force Can you think of any forces that is not conservative?^ Friction!
*^12 *^ U^ m v^ mgy^2 1 *^1 * 2 2 m v mgy^ m v^ mgy^ ff^ ii 2 2
^
Conservative forces don't transfer mechanical energy intointernal energy^ In addition, the work done by a conservative force doesn'tdepend on the path taken by the objects, it only depends onthe final and initial positions.^ Note: If final position = initial position (i.e.
closed path ), the work done is^ zero. Nonconservative forces do transfer some mechanical energy intointernal energy Prime example is friction So some mechanical energy is transformed into internal energy:
Physics 3A: Potential Energy ^ E f^ xint k^ ^ ^ E^ E^ mech^ int ^ * E K^ Umech
^ *^ ^ K^ U^ E^ int
^
Can rewrite: As: If the change in a quantity is zero, then the quantity is constant:^ This is the most general form of energy conservation for anisolated system
Physics 3A: Potential Energy^ ^ *^ ^ K^ U^ E^ int ^ *^ *^ K U^ E^0 int^ *^ *^ ^ K^ U^ E^ constantint^
Conservative forces don't transfer mechanical energy intointernal energy^ In addition, the work done by a conservative force doesn'tdepend on the path taken by the objects, it only depends onthe final and initial positions.^ Note: If final position = initial position (i.e.
closed path ), the work done is^ zero. Another example of a conservative force:
Spring force Recall work done by spring force moving an object fromx^ to x^ is:i^ f^ Has same form as W = U-Ui^
, so define potential energy off spring force as:
Physics 3A: Potential Energy^112 ^ ^ W^ k x^ k xs^ i^2
2 f 1. 2 U k x s 2
We can choose Uto to be any value we like at a given referencei^ point (like when we choose y=0 to make things simpler)^ Because we are only concerned with differences in U (
∆U)
We can also go the other way, i.e. U => F^ Consider a force (F) acting in the horizontal (x) direction:^ compute the work done over a small displacement dr:
Physics 3A: Potential Energy )^ ^ ^ ^ ^ d W^ F^ d^ r^ F^ dx^ i^ F
dx dUx F dx dU dU x F x dx
So far we have seen^ isolated systems:^ non-isolated systems:^ In both of these, the energy of the system changes^ There can also have non-isolated systems where the
∆E = 0
Physics 3A: Potential Energy^ ^ E^ system but still have energy being transferred in and out of the system. This is a non-isolated system in steady state. The sum of the transfers of energy into the system equalsthe sum of transfers of energy out of the system An example is your home:
^0 ^ E Hsystem^
^
Physics 3A: Potential Energy
^
Lets revisit gravitational potential energy^ We had from before:^ This is only valid for objects close to the earth's surface^ We use our expression for F
and how to go from F to U:g^ Integrate to get U^ :g
unit vector
Physics 3A: Potential Energy^ ^ U^ mgyg^ G M^ m^ E^ F rg^2 r^
Substitute in F^ :g We can choose any reference point for r^ usually choose U^ = 0 where force is zero, i.e. at r = infinity:i^
(for earth) (7.18) (for any pair of (7.19) particles)
Physics 3A: Potential Energy r^ f^ G M^ mE^ ^ ^ U^ gf^2 r^ r^ i
r^ f^ *dr^ * dr U GM mU^ i E i^2 r^ r^ i
r^ f^1 * ^ U GM m U^ gf E i^ r^ r^ i
^ ^11 * ^ U GM m U^ gf E i^ r^ r^ fi^ GM mE U g r f G m m 1 2 U g r f
Physics 3A: Potential Energy This looks like:GM^ mE^ ^ U^ g^ r Must do positive work to increaseseparation U varies as 1/r, not 1/r^2 If there are more than two particles:
*^ * U U U^ U^ total 12 13 23 G m m G m^ m^ G m^ m *^ * 1 21 32 U total r r^1213
3 r 23
,^
Similar U holds for electrostatic force:^ Note: qand q^ can be negative or positive, so force is attractive^1 2 or repulsive. U is negative or positive. So for like charges, potential energy is positive. We have to dopositive work to push them together
Physics 3A: Potential Energy^ U^ e
k^ q^ q^ e^1 2 r
,^
Physics 3A: Potential Energy One final tool we can use to analyze a mechanical system iscalled an^ Energy Diagram This consists of a plot of the potential energy of the systemversus one of the independent variables (i.e. position) uponwhich the potential energy depends Consider the system of a block and spring (attached)with total energy E^ d U^ ^ F^^ d x #^ ^ "
