chapter 6: work, energy and power - national maglabmagnet.fsu.edu/~shill/teaching/2048...
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Chapter 6: Work, Energy and Power Thursday February 12th
Reading: up to page 97 in the text book (Ch. 6)
• Discuss Mini Exam II • Review: Work and Kinetic Energy • Conservative and non-conservative forces • Work and Potential Energy • Conservation of Energy • Calculus method for determining work • Power • As usual – iclicker, examples and demonstrations
Mini Exam III next Thursday • Will cover LONCAPA #7-10 (Newton’s laws and energy cons.)
Work - Definition
• There are only two relevant variables in one dimension: the force, Fx, and the displacement, Δx.
Work W is the energy transferred to or from an object by means of a force acting on the object. Energy transferred to the object is positive work, and energy transferred from the object is negative work.
W = FxΔxDefinition: [Units: N.m or Joule (J)]
Fx is the component of the force in the direction of the object’s motion, and Δx is its displacement.
Kinetic Energy - Definition
Fx Frictionless surface
Δx
K = 12 mv 2
12 mv f
2 − 12 mvi
2 = 12 m× 2axΔx
ΔK = K f − Ki = maxΔx = FxΔx =W
v f2 = v i
2+2axΔx
Work-Kinetic Energy Theorem
ΔK = K f − Ki =Wnet
change in the kinetic net work done on
energy of a particle the particle
⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
K f = Ki +Wnet
kinetic energy after kinetic energy the net
the net work is done before the net work work done
⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Fx = F cosφW = Fd cosφW =
!F ⋅!d
More on Work To calculate the work done on an object by a force during a displacement, we use only the force component along the object's displacement. The force component perpendicular to the displacement does zero work
• Caution: for all the equations we have derived so far, the force must be constant, and the object must be rigid.
• I will discuss variable forces later.
Work – More Examples
k M
K=½mv2
Frictionless surface
M
K=0 k
Δx
Fsp.
Frictionless surface
Work – More Examples
Wspr. = −FΔx = ΔK
k M
K=½mv2
Δx
Fsp.
Frictionless surface
Kinetic energy is completely recovered: a ‘conservative’ force
Work – More Examples
Wspr. = +FΔx = ΔK
F
Mg
h
Wlift = FhWg = −Mgh
These two examples are similar – they both involve conservative forces Examples: electrostatic (spring/elastic) forces, gravitational forces.
Work – More Examples
Wn.c. = Fh = +MghWcons. =Wg = −MghWnet =Wn.c. +Wcons. = 0
If F = Mg,Wnet = 0Then ΔK = 0
F
Mg
h
It turns out that one can define a ‘Potential Energy’, U, for ALL conservative forces as follows:
Work & Potential Energy
ΔU =U f −Ui
= −Wcons.
ΔUg = Mgh=Wn.c.
F
Mg
ΔU =U f −Ui
= −Wcons.
ΔUg = Mgh=Wn.c.
The Potential Energy change, ΔU, does not care how the height change was achieved.
h
It turns out that one can define a ‘Potential Energy’, U, for ALL conservative forces as follows:
Work & Potential Energy
Conservation of Energy
ΔK = K f − Ki =Wnet
=Wcons. +Wn.c.
Work-Kinetic Energy theorem
• We can now replace any work due to conservative forces by potential energy terms, i.e.,
ΔK = −ΔU +Wn.c.
ΔEmech = ΔK + ΔU =Wn.c.
Or
• Here, Emech is the total mechanical energy of a system, equal to the sum of the kinetic and potential energy of the system.
• If work is performed on the system by an external, non-conservative force, then Emech increases.
Conservation of Energy Special case: a completely isolated system, solely under the influence of conservative forces:
ΔEmech = ΔK + ΔU = 0
⇒ K f − Ki( ) + U f −Ui( ) = 0
Or K f +U f = Ki +Ui
If there is an outside influence from a non-conservative force, then:
K f +U f = Ki +Ui( ) +Wn.c.
Power • Power is defined as the "rate at which work is done." • If an amount of work W is done in a time interval Δt by a force, the average power due to the force during the time interval is defined as
Pavg =
WΔt
• Instantaneous power is defined as P = dW
dt• The SI unit for power is the Watt (W).
1 watt = 1 W = 1 J/s = 0.738 ft · lb/s 1 horsepower = 1 hp = 550 ft · lb/s = 746 W
1 kilowatt-hour = 1 kW · h = (103 W)(3600 s) = 3.60 MJ
General (calculus) method for calculating Work
ΔU = −Wc
= − Fc1Δx + Fc2Δx + ....FcjΔx + ...( )= − FcjΔx
j∑
ΔU = −Wc = − Fc (x)dx
xi
x f∫
Take the limit Δx → 0
Elastic/Spring Potential Energy
ΔU = − (−kx)dxxi
x f∫ = k x dxxi
x f∫= 1
2 k x2⎡⎣ ⎤⎦xi
x f = 12 kx f
2 − 12 kxi
2
ΔU = −Wc = − Fc (x)dx
xi
x f∫F
xi xf
Fc = -kx Δx
Fc = kx
U = − (−kx)dx0
x
∫ = k x dx0
x
∫= 1
2 k x2⎡⎣ ⎤⎦0
x= 1
2 kx2
0
x
-kxf