lecture 15
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Lecture 15. Chapter 11 Employ the dot product Employ conservative and non-conservative forces Use the concept of power (i.e., energy per time) Chapter 12 Extend the particle model to rigid-bodies U nderstand the equilibrium of an extended object. - PowerPoint PPT PresentationTRANSCRIPT
Physics 207: Lecture 15, Pg 1
Lecture 15Goals:Goals:
• Chapter 11Chapter 11 Employ the dot product Employ conservative and non-conservative forces Use the concept of power (i.e., energy per time)
• Chapter 12Chapter 12 Extend the particle model to rigid-bodies Understand the equilibrium of an extended object. Understand rigid object rotation about a fixed axis. Employ “conservation of angular momentum” concept
Assignment: HW7 due March 10th For Thursday: Read Chapter 12, Sections 7-11
do not concern yourself with the integration process in regards to “center of mass” or “moment of inertia””
Physics 207: Lecture 15, Pg 2
Useful for finding parallel components
A î = Ax
î î = 1 î ĵ = 0
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!
cosBABA
î
A
Ax
Ayĵ
Physics 207: Lecture 15, Pg 4
Work in terms of the dot product
Ingredients: Force ( F ), displacement ( r )
Looks just like a Dot Product!
r
displace
ment
FWork, W, of a constant force F
acts through a displacement r :
If the path is curved at each point
and
rdFdW
rdF
f
i
r
rrdFW
rFrFW
cos
Physics 207: Lecture 15, Pg 6
Energy and WorkWork, W, is the process of energy transfer in which a
force component parallel to the path acts over a distance; individually it effects a change in energy of the “system”.
1.K or Kinetic Energy
2.U or Potential Energy (Conservative)
and if there are losses (e.g., friction, non-conservative)
3. ETh Thermal Energy
Positive W if energy transferred to a system
Physics 207: Lecture 15, Pg 7
A. U K B. U ETh
C. K UD. K ETh
E. There is no transformation because energy is conserved.
A child slides down a playground slide at constant speed. The energy transformation is
Physics 207: Lecture 15, Pg 8
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? State the system (here, just the box) Use a Free Body Diagram Compare force and path
v
Physics 207: Lecture 15, Pg 9
Work and Varying Forces (1D)
Consider a varying force F(x)
Fx
xx
Area = Fx xF is increasingHere W = F · r becomes dW = Fx dx
F = 0°
Start Finish
Work has units of energy and is a scalar!
f
i
x
xx dxxFW )(
F
x
Physics 207: Lecture 15, Pg 10
• How much will the spring compress (i.e. x = xf - xi) to bring the box to a stop (i.e., v = 0 ) if the object is moving initially at a constant velocity (vi) on frictionless surface as shown below with xi = xeq , the equilibrium position of the spring?
x
vi
m
ti
F
spring compressed
spring at an equilibrium position
V=0
t m
f
i
x
xx dxxFW )(box
f
i
x
xeq dxxxkW )(-box
f
ii
xx
xxkW |221
box )( -
K 0 )( - 2212
21
box kxxk ifW
2i2
12
212
21 v0 - mmxk
Example: Hooke’s Law Spring (xi equilibrium)
Physics 207: Lecture 15, Pg 11
Work signs
x
vi
m
ti
F
spring compressed
spring at an equilibrium position
V=0
t m
Notice that the spring force is opposite the displacement
For the mass m, work is negative
For the spring, work is positive
They are opposite, and equal (spring is conservative)
Physics 207: Lecture 15, Pg 12
Conservative Forces & Potential Energy
For any conservative force F we can define a potential energy function U in the following way:
The work done by a conservative force is equal and opposite to the change in the potential energy function.
W = F ·dr ≡ - U
ri
rf Uf
Ui
Physics 207: Lecture 15, Pg 13
Conservative Forces and Potential Energy
So we can also describe work and changes in potential energy (for conservative forces)
U = - W
Recalling (if 1D)
W = Fx x
Combining these two,
U = - Fx x
Letting small quantities go to infinitesimals,
dU = - Fx dx
Or,
Fx = -dU / dx
Physics 207: Lecture 15, Pg 14
ExerciseWork Done by Gravity
An frictionless track is at an angle of 30° with respect to the horizontal. A cart (mass 1 kg) is released from rest. It slides 1 meter downwards along the track bounces and then slides upwards to its original position.
How much total work is done by gravity on the cart when it reaches its original position? (g = 10 m/s2)
1 meter30°
(A) 5 J (B) 10 J (C) 20 J (D) 0 J
h = 1 m sin 30° = 0.5 m
Physics 207: Lecture 15, Pg 19
A Non-Conservative Force
Path 2
Path 1
Since path2 distance >path1 distance the puck will be traveling slower at the end of path 2.
Work done by a non-conservative force irreversibly removes energy out of the “system”.
Here WNC = Efinal - Einitial < 0 and reflects Ethermal
Physics 207: Lecture 15, Pg 20
Work & Power:
Two cars go up a hill, a Corvette and a ordinary Chevy Malibu. Both cars have the same mass.
Assuming identical friction, both engines do the same amount of work to get up the hill.
Are the cars essentially the same ? NO. The Corvette can get up the hill quicker It has a more powerful engine.
Physics 207: Lecture 15, Pg 21
Work & Power:
Power is the rate at which work is done.
InstantaneousPower:
AveragePower:
A person, mass 80.0 kg, runs up 2 floors (8.0 m). If they climb it in 5.0 sec, what is the average power used?
Pavg = F h / t = mgh / t = 80.0 x 9.80 x 8.0 / 5.0 W P = 1250 W
Example:
Units (SI) areWatts (W):
1 W = 1 J / 1st
WP
dtdW
P
Physics 207: Lecture 15, Pg 22
Work & Power:
Power is also,
If force constant, W= F x = F ( v0 t + ½ at2 )
and P = W / t = F (v0 + at)
xxx vFP
txF
tW
P
Physics 207: Lecture 15, Pg 23
Exercise Work & Power
A. Top
B. Middle
C. Bottom
Starting from rest, a car drives up a hill at constant acceleration and then quickly stops at the top.
(Hint: What does constant acceleration imply?) The instantaneous power delivered by the engine during
this drive looks like which of the following,
Z3
timePow
erP
owe r
Pow
e r
time
time
Physics 207: Lecture 15, Pg 24
Chap. 12: Rotational Dynamics
Up until now rotation has been only in terms of circular motion with ac = v2 / R and | aT | = d| v | / dt
Rotation is common in the world around us. Many ideas developed for translational motion are transferable.
Physics 207: Lecture 15, Pg 25
Rotational Variables
Rotation about a fixed axis: Consider a disk rotating about
an axis through its center:
Recall :
(Analogous to the linear case )
/R (rad/s) 2
TangentialvTdt
d
dtdxv
Physics 207: Lecture 15, Pg 26
Rotational Variables...
At a point a distance R away from the axis of rotation, the tangential motion:
x (arc) = R vT (tangential) = R
aT = R
R
v = R
x
rad)in position (angular 2
1
rad/s)in elocity (angular v
)rad/sin accelation(angular constant
200
0
2
tt
t