Unformatted text preview: Physics for Scientists & Engineers 1
Spring Semester 2011 Lecture 15 Potential Energy February 15, 2011 Physics for Scientists&Engineers 1 1 1 Potential Energy
Definition: Potential energy, U, is the energy stored in the configuration of a system of objects that exert forces on each other.
How are work, kinetic energy, and potential energy related? February 15, 2011 Physics for Scientists&Engineers 1 2 2 Gravitational Potential Energy
Definition of gravitational potential energy Ug = mgy + constant
Change in gravitational potential energy ∆Ug ≡ Ug ( y) − Ug ( y0 ) = mg( y − y0 ) = mgh
Last week we calculated the work done by gravity on a mass to lift it to a height h = y  y0: Wg = − mgh
Compare the two results: ∆Ug = −Wg
February 15, 2011 Physics for Scientists&Engineers 1 3 3 Energy Storage with gravity
We can do work to lift mass up to a higher elevation and leave it there, storing the work in form of potential energy Later we can use that weight and convert this potential energy back by letting the mass do work or gain kinetic energy Hydroelectric Hydroelectric power plants can pump water up to a higher elevation reservoir during times of low power demand and then use this extra potential energy in times of high power demand Not every force can be used in this way to store potential energy in a completely reversible fashion
February 15, 2011 Physics for Scientists&Engineers 1 4 4 Conservative Force Example: Gravity
Lifting: force and displacement in opposite direction Lowering: force and displacement in same direction Total work raising and lowering is zero February 15, 2011 Physics for Scientists&Engineers 1 5 5 Conservative and Nonconservative Forces
Before we can calculate the potential energy from a given force, we need to ask if all kinds of forces can be used to store potential energy We need to consider what happens to the work done by a force when the direction of the path taken by an object is reversed • For gravity, we know what happens • The work done by gravity when an object is lifted has the same magnitude but opposite sign to the work done lowering the object
February 15, 2011 Physics for Scientists&Engineers 1 6 6 Conservative and Nonconservative Forces
The total work done by the force of gravity in lifting an object from some elevation to a different one and then returning it to the same elevation is zero This is the basis for the definition of of a conservative force Definition A conservative force is any force for which the work done over any closed path is zero. A force that does not fulfill this requirement is called a nonconservative force.
February 15, 2011 Physics for Scientists&Engineers 1 7 7 Consequence 1 for Conservative Forces
If we know the work done by a conservative force on an object as the object moves along a path from point A to point B, then we also know the work that the same force does on the object as it moves moves along the path in the reverse direction WB→ A = −WA→ B or WB→ A + WA→ B = 0 February 15, 2011 Physics for Scientists&Engineers 1 8 8 Consequence 2 for Conservative Forces
If we know the work done by a conservative force on an object moving along path 1 from A to B, then we know the work done by the same force on the object when it uses any other path 2 to go go from A to B WA→ B, path 1 = WA→ B,path 2 February 15, 2011 Physics for Scientists&Engineers 1 9 9 NonNonconservative Force Example: Friction
Pushing right: force is opposite to displacement Wf 1 = f • ∆r1 = − f  ( xB − xA )  = − µ k mg  ( xB − x A ) 
Pushing left : force is opposite to displacement (again!) Wf 2 = f • ∆r2 = − f  ( xB − xA )  = − µk mg  ( xB − xA )  February 15, 2011 Physics for Scientists&Engineers 1 10 10 NonNonconservative Force Example: Friction
This result leads us to conclude that the total work done by the friction force on the closed path of A to B and back to A is not zero, but instead Wf = Wf 1 + Wf2 = −2µ k mg  ( xB − x A )  < 0 There appears to be a contradiction between this result and the workkinetic energy theorem • The box starts with zero kinetic energy at a certain position and ends up with zero kinetic energy at the same position
February 15, 2011 Physics for Scientists&Engineers 1 11 11 NonNonconservative Force Example: Friction
This leads us to conclude that the friction force does not do work in the way that a conservative force does work Instead, the friction force converts kinetic energy and/or potential energy into internal excitation of the two objects objects that exert friction on each other The conversion of kinetic and/or potential energy to internal excitation is not reversible • The internal excitation cannot be converted fully back into kinetic and/or potential energy
February 15, 2011 Physics for Scientists&Engineers 1 12 12 Friction as a Nonconservative Force
Friction is an example of a nonconservative force The friction force always acts in a direction opposite to the displacement • The dissipation of energy due to friction is always negative, whether or not the path is closed Work done by a conservative force can be positive or negative, but the dissipation from the friction force is always negative We use the symbol Wf for this dissipated energy to remind ourselves that that we use the same procedures that we use to calculate work for conservative forces The decisive fact is that the friction force always switches direction as a function of the direction of motion and causes dissipation The friction force vector is always antiparallel to the velocity vector • Typical of nonconservative forces such as • Air resistance • Damping force
February 15, 2011 Physics for Scientists&Engineers 1 13 13 Clicker Quiz
A person pushes a box with mass 10.0 kg a distance of 5.00 m across a floor. The coefficient of kinetic friction between the box and the floor is 0.250. The person then picks up the box to a height of 1.00 m, carries the box back to the original starting point, and puts it back down on the floor. How much work has the person done on the box? A. A. 0 J B. 12.5 J W = Fd + 0 C. 98.1 J F = µ mg k D. 123 J W = µk mgd = ( 0.250 )(10.0 kg ) 9.81 m/s 2 ( 5.00 m ) E. 613 J W = 123 J ( ) February 15, 2011 Physics for Scientists&Engineers 1 14 14 Work and Potential Energy
Generalize what we found for gravity For any conservative force, the relationship between potential energy change and work is given by: Since work is in general W = F ( r ' ) • dr ' ∆U = −W ∫
r0 r We find: ∆U = U (r ) − U (r0 ) = − ∫ F ( r ') • dr '
r0 r 1d case: ∆U = U ( x) − U ( x0 ) = − ∫ F ( x ') dx '
x0
Physics for Scientists&Engineers 1 15 x February 15, 2011 15 Two Known Cases
Gravity
y y ∆U g = U g ( y) − Ug ( y0 ) = − (− mg ) dy ' = mg dy ' = mgy − mgy0
y0 y0 Ug ( y ) = mgy + constant Spring force
x x
2 ∆U s = U s ( x ) − Us ( x0 ) = − Fs ( x ') dx ' = − (− kx ') dx ' = 1 kx 2 − 1 kx0 2 2 x0 x0 Us ( x ) = 1 kx 2 + constant 2 February 15, 2011 Physics for Scientists&Engineers 1 16 16 Calculating Force from Potential Energy
We just found out how one can calculate the potential energy from knowing the (conservative!) force by integration Does it also work the other way around? Yes: calculus’s central theorem  derivative inverse operation operation to integral F(x) = − dU ( x ) dx Force is (minus) the derivative of potential energy February 15, 2011 Physics for Scientists&Engineers 1 17 17 Example: Force and Potential Energy (1) (1)
Question: What is the force if the potential energy is given by the formula a U ( x ) = + bx 2 + c x
Answer: We simply have to take the derivative of the potential energy with respect to x: F ( x) = − dU ( x) d a = − + bx 2 + c dx dx x d 1 d2 a = −a − b ( x ) = 2 − 2bx dx x dx x
Physics for Scientists&Engineers 1 18 February 15, 2011 18 Force and Potential Energy
Potential energy is the integral of force over displacement
x ∆U = U ( x ) − U ( x0 ) = − F (x ')dx '
x0 Inverse relationship: force is derivative of potential energy with respect to position F(x) = − dU ( x ) dx February 15, 2011 Physics for Scientists&Engineers 1 19 19 Clicker Quiz
The potential energy U(x) is shown as a function of position x in the figure below. In which region is the magnitude of the force the highest? February 15, 2011 Physics for Scientists&Engineers 1 20 20 Clicker Quiz Solution
The potential energy U(x) is shown as a function of position x in the figure below. In which region is the magnitude of the force the highest? The magnitude of the force is proportional to the magnitude of the slope of the line
February 15, 2011 Physics for Scientists&Engineers 1 21 21 LennardLennardJones Potential
The potential energy associated with the interaction of two atoms in a molecule is given by the LennardJones potential (also known as as the “126 potential”)
U (x ) = 4U0 x0 x
12 x −0 x 6 U0 is a constant and x0 is a constant length
The LennardJones potential is one of the most important concepts in atomic physics and is used for most numerical simulations of molecular systems February 15, 2011 Physics for Scientists&Engineers 1 22 22 Molecular Force
PROBLEM • What is the force resulting from the LennardJones potential? SOLUTION • We take the negative of the derivative of the potential energy with respect to x x0 12 x0 6 dU ( x ) d Fx ( x ) = − = − 4U0 − dx dx x x 1 12 d 1 6d1 61 Fx ( x ) = −4U0 x0 + 4U0 x0 6 = 48U0 x12 13 − 24U0 x0 7 0 dx x12 dx x x x
13 7 24U0 x0 x0 Fx ( x ) = 2 − x0 x x February 15, 2011 Physics for Scientists&Engineers 1 23 23 Molecular Force
PROBLEM • At what value of x does the LennardJones potential have its minimum? SOLUTION • Because we just found that the force is the derivative of the potential energy, we find the point where F (x) = 0 Fx ( x ) x = x min 24U0 x0 = 2 x0 xmin 13 x0 − xmin 7 =0 • This condition can only be true if x x 2 0 = 0 xmin xmin February 15, 2011 13 7 x 20 xmin
Physics for Scientists&Engineers 1 6 =1 24 24 Molecular Force
• This gives us
6 6 2 x0 = xmin or xmin = 21/6 x0 ≈ 1.12 x0 Plot the LennardJones potential for two Ar atoms • x0 = 0.34 nm • U0 = 1.70 1021 J 10 • xmin = 0.38 nm • Note that near the minimum, the force is approximately • Fx(x) ≈ k(x – xmin) • Like a little spring
February 15, 2011 Physics for Scientists&Engineers 1 25 25 ...
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 Spring '08
 Wolf
 Energy, Force, Potential Energy

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