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Physics 207: Lecture 14, Pg 1
Lecture 14Goals:Goals:
Assignment: Assignment: HW6 due Wednesday, Mar. 11 For Tuesday: Read Chapter 12, Sections 1-3, 5 & 6
do not concern yourself with the integration process in regards to “center of mass” or “moment of inertia””
• Chapter 10 Chapter 10
• Understand spring potential energies & use energy diagramsUnderstand spring potential energies & use energy diagrams
• Chapter 11Chapter 11 Understand the relationship between force, displacement and work Recognize transformations between kinetic, potential, and thermal energies Define work and use the work-kinetic energy theorem Use the concept of power (i.e., energy per time)
Physics 207: Lecture 14, Pg 2
Energy for a Hooke’s Law spring
Associate ½ kx2 with the “potential energy” of the spring
m
2212
212
212
21 ffii mvkxmvkx
xf
xi
f
i
v
vxx
x
xx dvmvdxF
)( eqxxkvdxdv
mdtdx
dxdv
mdtdv
mF xxxx
x
dxxxkdvmvdxF xxx )( eq
Physics 207: Lecture 14, Pg 3
Energy for a Hooke’s Law spring
m
2212
212
212
21 ffii mvkxmvkx
constantK UK fsfisiU
Ideal Hooke’s Law springs are conservative so the mechanical energy is constant
Physics 207: Lecture 14, Pg 4
Energy diagrams In general:
Ene
rgy
K
y
U
Emech
Ene
rgy
K
x
U
Emech
Spring/Mass systemBall falling
Physics 207: Lecture 14, Pg 5
Equilibrium
Example Spring: Fx = 0 => dU / dx = 0 for x=xeq
The spring is in equilibrium position
In general: dU / dx = 0 for ANY function establishes equilibrium
stable equilibrium unstable equilibrium
U U
Physics 207: Lecture 14, Pg 6
Comment on Energy Conservation
We have seen that the total kinetic energy of a system undergoing an inelastic collision is not conserved. Mechanical energy is lost:
Heat (friction)Bending of metal and deformation
Kinetic energy is not conserved by these non-conservative forces occurring during the collision !
Momentum along a specific direction is conserved when there are no external forces acting in this direction. In general, easier to satisfy conservation of momentum
than energy conservation.
Physics 207: Lecture 14, Pg 7
Mechanical Energy Potential Energy (U)
Kinetic Energy (K) If “conservative” forces
(e.g, gravity, spring) then
Emech = constant = K + U
During Uspring+K1+K2 = constant = Emech
Mechanical Energy conserved
Before
During
After
1 2
Physics 207: Lecture 14, Pg 8
Energy (with spring & gravity)
Emech = constant (only conservative forces) At 1: y1 = h ; v1y = 0 At 2: y2 = 0 ; v2y = ? At 3: y3 = -x ; v3 = 0 Em1 = Ug1 + Us1 + K1 = mgh + 0 + 0 Em2 = Ug2 + Us2 + K2 = 0 + 0 + ½ mv2
Em3 = Ug3 + Us3 + K3 = -mgx + ½ kx2 + 0
Given m, g, h & k, how much does the spring compress? Em1 = Em3 = mgh = -mgx + ½ kx2 Solve ½ kx2 – mgx +mgh = 0
1
32
h
0-x
mass: m
Physics 207: Lecture 14, Pg 9
Energy (with spring & gravity)
When is the child’s speed greatest?
(A) At y1 (top of jump)
(B) Between y1 & y2
(C) At y2 (child first contacts spring)
(D) Between y2 & y3
(E) At y3 (maximum spring compression)
1
32
h
0-x
mass: m
Physics 207: Lecture 14, Pg 10
Energy (with spring & gravity)
When is the child’s speed greatest? A: Calculus soln. Find v vs. spring displacement then maximize
(i.e., take derivative and then set to zero) B: Physics: As long as Fgravity > Fspring then speed is increasing
Find where Fgravity- Fspring= 0 -mg = kxVmax or xVmax = -mg / k
So mgh = Ug23 + Us23 + K23 = mg (-mg/k) + ½ k(-mg/k)2 + ½ mv2
2gh = 2(-mg2/k) + mg2/k + v2 2gh + mg2/k = vmax2
1
32
h
0-x
mg kx
Physics 207: Lecture 14, Pg 11
Inelastic Processes If non-conservative” forces (e.g, deformation, friction)
then
Emech is NOT constant
After K1+2 < Emech (before)
Accounting for this loss we introduce Thermal Energy (Eth , new)
where Esys = Emech + Eth = K + U + Eth
Before
During
After
1 2
Physics 207: Lecture 14, Pg 12
Energy & Work
Impulse (Force vs time) gives us momentum transfer Work (Force vs distance) tracks energy transfer Any process which changes the potential or kinetic energy
of a system is said to have done work W on that system
Esys = W
W can be positive or negative depending on the direction of energy transfer
Net work reflects changes in the kinetic energy
Wnet = K
This is called the “Net” Work-Kinetic Energy Theorem
Physics 207: Lecture 14, Pg 13
Circular Motion
I swing a sling shot over my head. The tension in the rope keeps the shot moving at constant speed in a circle.
How much work is done after the ball makes one full revolution?
v(A) W > 0
(B) W = 0
(C) W < 0
(D) need more info
Physics 207: Lecture 14, Pg 14
Examples of “Net” Work (Wnet)
Examples of No “Net” Work
K = Wnet
Pushing a box on a rough floor at constant speed Driving at constant speed in a horizontal circle Holding a book at constant height
This last statement reflects what we call the “system”
( Dropping a book is more complicated because it involves changes in U and K, U is transferred to K )
K = Wnet
Pushing a box on a smooth floor with a constant force; there is an increase in the kinetic energy
Physics 207: Lecture 14, Pg 15
Changes in K with a constant F
xf
xi
f
i
v
vxx
x
xx dvmvdxF
If F is constant
xf
xi
f
i
v
vxx
x
xx dvmvdxF
KmvmvxFxxF xixfxifx 2212
21)(
Physics 207: Lecture 14, Pg 16
Net Work: 1-D Example (constant force)
Net Work is F x = 10 x 5 N m = 50 J= 10 x 5 N m = 50 J 1 Nm ≡ 1 Joule and this is a unit of energy Work reflects energy transfer
x
A force F = 10 N pushes a box across a frictionless floor for a distance x = 5 m.
F = 0° Start Finish
Physics 207: Lecture 14, Pg 17
Units:
N-m (Joule) Dyne-cm (erg)
= 10-7 J
BTU = 1054 J
calorie = 4.184 J
foot-lb = 1.356 J
eV = 1.6x10-19 J
cgs Othermks
Force x Distance = Work
Newton x
[M][L] / [T]2
Meter = Joule
[L] [M][L]2 / [T]2
Physics 207: Lecture 14, Pg 18
Net Work: 1-D 2nd Example (constant force)
Net Work is F x = -10 x 5 N m = -50 J= -10 x 5 N m = -50 J
Work reflects energy transfer
x
F
A force F = 10 N is opposite the motion of a box across a frictionless floor for a distance x = 5 m.
= 180° Start Finish
Physics 207: Lecture 14, Pg 19
Work in 3D….
2212
21)( zizfzifz mvmvzFzzF
x, y and z with constant F:
2212
21)( yiyfyify mvmvyFyyF
2212
21)( xixfxifx mvmvxFxxF
2222
2212
21
with zyx
ifzyx
vvvv
KmvmvzFyFxF
Physics 207: Lecture 14, Pg 20
Work: “2-D” Example (constant force)
(Net) Work is Fx x = F cos(= F cos(-45°) x = 50 x 0.71 Nm = 35 J = 50 x 0.71 Nm = 35 J Work reflects energy transfer
x
F
A force F = 10 N pushes a box across a frictionless floor for a distance x = 5 m and y = 0 m
= -45°
Start Finish
Fx
Physics 207: Lecture 14, Pg 21
Useful for performing projections.
A î = Ax
î î = 1 î j = 0
î
A
Ax
Ay
A B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz )
Calculation can be made in terms of components.
Calculation also in terms of magnitudes and relative angles.
Scalar Product (or Dot Product)
A B ≡ | A | | B | cos
You choose the way that works best for you!
A · B ≡ |A| |B| cos()
Physics 207: Lecture 14, Pg 22
Scalar Product (or Dot Product)
Compare:
A B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz )
with A as force F, B as displacement r
and apply the Work-Kinetic Energy theorem
Notice:
F r = (Fx )(x) + (Fy )(z ) + (Fz )(z)
Fx x +Fy y + Fz z = K
So here
F r = K = Wnet
More generally a Force acting over a Distance does Work
Physics 207: Lecture 14, Pg 23
Definition of Work, The basics
Ingredients: Force ( F ), displacement ( r )
“Scalar or Dot Product”
r
displace
ment
FWork, W, of a constant force F
acts through a displacement r :
W = F · r (Work is a scalar)(Work is a scalar)
If we know the angle the force makes with the path, the dot product gives us F cos and r
If the path is curved at each point
and rdFdW
rdF
f
i
r
rrdFW
Physics 207: Lecture 14, Pg 24
Remember that a real trajectory implies forces acting on an object
Only tangential forces yield work! The distance over which FTang is applied: Work
a
vpathand time
a
a
a = 0
Two possible options:
Change in the magnitude of v
Change in the direction of v
a = 0
a = 0
aaa= +
aradialatanga= +
FradiallFtangF= +
Physics 207: Lecture 14, Pg 26
ExerciseWork in the presence of friction and non-contact forces
A. 2
B. 3
C. 4
D. 5
A box is pulled up a rough ( > 0) incline by a rope-pulley-weight arrangement as shown below. How many forces (including non-contact ones) are doing work on the box ? Of these which are positive and which are negative? Use a Free Body Diagram Compare force and path
v
Physics 207: Lecture 14, Pg 28
Lecture 14
Assignment: Assignment: HW6 due Wednesday, March 11HW6 due Wednesday, March 11 For Tuesday: Read Chapter 12, Sections 1-3, 5 & 6For Tuesday: Read Chapter 12, Sections 1-3, 5 & 6
do do notnot concern yourself with the integration process concern yourself with the integration process