Potential Energy Potential Energy of Gravity Gravitational ...
Potential Energy
Potential Energy of Gravity
Work: type of energy transfer. Kinetic Energy: energy of motion. Potential Energy: Stored energy by conservative forces.
Wg = (-mg)yf – (-mg)yi = mgyi − mgyf = − ∆mgy Work done by gravity depends on the initial and final y-coordinates only; not Why? on the path! f
mg ∆y i
Demo: Pile Driver
Gravitational Potential Energy Ug ≡ mgy
Potential Energy of Springs Ws = ½kxi2 − ½kxf2 = − ∆ ½kx2 Work done depends on initial and final position.
Then: Wg ≡ Ugi – Ugf = − ∆Ug = mgyi - mgyf NOTE: You must choose where Ug = 0; y = 0.
Elastic potential energy:
Us = ½kx2 Then: Ws = −∆Us
1
Springs
Conservative Forces
E:\SERWAY\CHAP08\SE08_02.PCT
For gravity and for springs, we saw that the work done only depends on the initial and final states, not the actual path traveled.
1. Work done by a conservative force is path independent; it only depends on the initial and final positions. 2. Work done in a closed path is ZERO. 3. A potential energy function can be defined. 4. Wc = −∆U
Nonconservative Forces 4E:\SERWAY\CHAP08\SE08_03.PCT
Q2
Conservation of Mechanical Energy When only conservative forces are acting:
Consider: W = −∆U = Ui – Uf Also: W = ∆K = Kf – Ki So: Ki + Ui = Kf + Uf
Ei = Ef
Mechanical Energy: E = K + U
2
Conservation of Mechanical Energy
For a system of particles: Ei = E f ΣKi + ΣUi = ΣKf + ΣUf
Example: Loop the loop Release the ball from rest at height h. What is speed at the top of the loop?
Ei = Ef Ki + Ui = Kf + Uf
h
R
y=0
0 + mgh = ½mv2 + mg2R
For isolated systems Conservative forces only
v= [2g(h-2R)]1/2 Examples
What is h to barely make it around if vi≠0?
3
Potential Energy
Conservation of Mechanical Energy
For a system of particles: Ei = E f
Gravity: Ug ≡ mgy
ΣKi + ΣUi = ΣKf + ΣUf
Springs: Us = ½kx2 For conservative forces: Wc ≡ − ∆U
For isolated systems Conservative forces only What if non-conservative forces act?
Conservation of Mechanical Energy- revisited
Example vi
Now, W = Wc + Wnc = -∆U + Wnc = Ui – Uf + Wnc
Still:
d
W = ∆K = Kf – Ki
Ei + Wnc = Ef
x
1. What is the maximum compression of the spring?
So: Ki + Ui + Wnc = Kf + Uf
µk≠0
k
d’ m
Can be positive or negative.
2. How far back will the mass move before it stops?
Q1,2
1
Block-Spring Collision • E0 = K0 = 0.5m(vi)2 • E1 = K1 = E0 – fkd = E0 – µk nd = E0 – µk mgd • Maximum compression is for v = 0 ? K2=0J
? E2 = U2 = 0.5kx2 = E1 – µk mgx ? 0.5kx2 – µk mgx + E1 , solve for x > 0!
• Calculate E2 using the above determined x • E3 = E2 - µk mg(x+d´) = 0
? d´ = E2/(µk mg) - x
Potential energy of spring system
Force/Potential Energy Relation Since W = Fx∆x = −∆U dU = −Fxdx Fx = − dU/dx
Spring: U = ½kx2
Fx = −
d 1 2 kx = −kx dx 2
More generally: v ∂U ˆ ∂U ˆ ∂U ˆ F =− i− j− k ∂x ∂y ∂z
Energy Diagrams Example: Mass on a spring. U(x)=½kx 2
Energy
4 E=const ½kA2 slope= −Fx
K= ½mv 2
K+U=E U
-A
x A=amplitude
2
Equilibrium of a system
Unstable Equilibrium
§ X= 0 corresponds to a stable equilibrium,
since the spring force accelerates at every point towards this point
Fs = −
dU S d 1 = − ( kx 2 ) = − kx dx dx 2
§ If U=const ? Fx = 0 ? neutral equlibrium
Applications
Conservation of Energy
4Radioactivity
4The total energy (mechanical energy plus
4Electronic Devices
other energy forms) of an isolated system is constant 4Energy can never be created or destroyed 4But, energy may be transformed from one form into another
4Scanning Tunneling Microscopy U(r) barrier alpha
E
r nucleus forbidden
TUNNELING
3
Potential Energy of Gravity
Work: type of energy transfer. Kinetic Energy: energy of motion. Potential Energy: Stored energy by conservative forces.
Wg = (-mg)yf – (-mg)yi = mgyi − mgyf = − ∆mgy Work done by gravity depends on the initial and final y-coordinates only; not Why? on the path! f
mg ∆y i
Demo: Pile Driver
Gravitational Potential Energy Ug ≡ mgy
Potential Energy of Springs Ws = ½kxi2 − ½kxf2 = − ∆ ½kx2 Work done depends on initial and final position.
Then: Wg ≡ Ugi – Ugf = − ∆Ug = mgyi - mgyf NOTE: You must choose where Ug = 0; y = 0.
Elastic potential energy:
Us = ½kx2 Then: Ws = −∆Us
1
Springs
Conservative Forces
E:\SERWAY\CHAP08\SE08_02.PCT
For gravity and for springs, we saw that the work done only depends on the initial and final states, not the actual path traveled.
1. Work done by a conservative force is path independent; it only depends on the initial and final positions. 2. Work done in a closed path is ZERO. 3. A potential energy function can be defined. 4. Wc = −∆U
Nonconservative Forces 4E:\SERWAY\CHAP08\SE08_03.PCT
Q2
Conservation of Mechanical Energy When only conservative forces are acting:
Consider: W = −∆U = Ui – Uf Also: W = ∆K = Kf – Ki So: Ki + Ui = Kf + Uf
Ei = Ef
Mechanical Energy: E = K + U
2
Conservation of Mechanical Energy
For a system of particles: Ei = E f ΣKi + ΣUi = ΣKf + ΣUf
Example: Loop the loop Release the ball from rest at height h. What is speed at the top of the loop?
Ei = Ef Ki + Ui = Kf + Uf
h
R
y=0
0 + mgh = ½mv2 + mg2R
For isolated systems Conservative forces only
v= [2g(h-2R)]1/2 Examples
What is h to barely make it around if vi≠0?
3
Potential Energy
Conservation of Mechanical Energy
For a system of particles: Ei = E f
Gravity: Ug ≡ mgy
ΣKi + ΣUi = ΣKf + ΣUf
Springs: Us = ½kx2 For conservative forces: Wc ≡ − ∆U
For isolated systems Conservative forces only What if non-conservative forces act?
Conservation of Mechanical Energy- revisited
Example vi
Now, W = Wc + Wnc = -∆U + Wnc = Ui – Uf + Wnc
Still:
d
W = ∆K = Kf – Ki
Ei + Wnc = Ef
x
1. What is the maximum compression of the spring?
So: Ki + Ui + Wnc = Kf + Uf
µk≠0
k
d’ m
Can be positive or negative.
2. How far back will the mass move before it stops?
Q1,2
1
Block-Spring Collision • E0 = K0 = 0.5m(vi)2 • E1 = K1 = E0 – fkd = E0 – µk nd = E0 – µk mgd • Maximum compression is for v = 0 ? K2=0J
? E2 = U2 = 0.5kx2 = E1 – µk mgx ? 0.5kx2 – µk mgx + E1 , solve for x > 0!
• Calculate E2 using the above determined x • E3 = E2 - µk mg(x+d´) = 0
? d´ = E2/(µk mg) - x
Potential energy of spring system
Force/Potential Energy Relation Since W = Fx∆x = −∆U dU = −Fxdx Fx = − dU/dx
Spring: U = ½kx2
Fx = −
d 1 2 kx = −kx dx 2
More generally: v ∂U ˆ ∂U ˆ ∂U ˆ F =− i− j− k ∂x ∂y ∂z
Energy Diagrams Example: Mass on a spring. U(x)=½kx 2
Energy
4 E=const ½kA2 slope= −Fx
K= ½mv 2
K+U=E U
-A
x A=amplitude
2
Equilibrium of a system
Unstable Equilibrium
§ X= 0 corresponds to a stable equilibrium,
since the spring force accelerates at every point towards this point
Fs = −
dU S d 1 = − ( kx 2 ) = − kx dx dx 2
§ If U=const ? Fx = 0 ? neutral equlibrium
Applications
Conservation of Energy
4Radioactivity
4The total energy (mechanical energy plus
4Electronic Devices
other energy forms) of an isolated system is constant 4Energy can never be created or destroyed 4But, energy may be transformed from one form into another
4Scanning Tunneling Microscopy U(r) barrier alpha
E
r nucleus forbidden
TUNNELING
3
