Turn in lab from yesterday C Test Takers Go over MC NotesLecture Variable Forces Drag Equation Derivation B Test Takers MOPing on computers pick a problem area Calculating Work a Different Way ID: 649826
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Slide1Slide2Slide3
Plan for Today (AP Physics 2)
Turn in lab from yesterday
C
Test Takers:
Go over MC
Notes/Lecture
Variable Forces
Drag Equation Derivation
B Test Takers:
MOPing
on computers
(
pick a problem area)Slide4
Calculating Work a Different Way
Work is a scalar resulting from the multiplication of two vectors.
We say work is the “dot product” of force and displacement.
W =
F
•
r
dot product representation
W= F r cos
q
useful if given magnitudes and directions of vectors
W = F
x
r
x
+ F
y
r
y
+ F
z
r
z
useful if given unit vectorsSlide5
The “scalar product” of two vectors is called the “dot product”
The “dot product” is one way to multiply two vectors. (The other way is called the “cross product”.)
Applications of the dot product
Work W = F
d
Power P =
F
v
Magnetic Flux
Φ
B
=
B
A
The quantities shown above are biggest when the vectors are completely aligned and there is a zero angle between them.Slide6
Why is work a dot product?
s
W =
F
•
r
W = F r cos
Only the component of force aligned with displacement does work.
F
Slide7
Work and Variable Forces
For constant forces
W = F • r
For variable forces, you can’t move far until the force changes. The force is only constant over an infinitesimal displacement.
dW =
F
• d
r
To calculate work for a larger displacement, you have to take an integral
W =
dW =
F • drSlide8
Variable Forces
If force can vary, what should our new equation for work look like?
W =
This would be by a position dependent force
Slide9
Work and variable force
The area under the curve of a graph of force
vs
displacement gives the work done by the force.
F(x)
x
x
a
x
b
W =
F(x) dx
x
a
x
bSlide10
What if force varies with time?
F = ma
a = dv/
dt
a =
F = m
= mv
=
dx
=
dx =
Slide11
Let’s Integrate that
=
dx =
=
=
=
Look familiar?
It’s the Work-Energy theorem
Work is equal to the change in kinetic energy
And it holds constant whether a force is constant or not
Slide12
What if it’s potential energy
Force of a spring
= -
kx
Hooke’s Law
=
=
=
½ k
Slide13
Spring Potential Energy, U
s
Springs obey Hooke’s Law.
F
s
(x) = -kx
F
s
is restoring force exerted BY the spring.
W
s
=
Fs(x)dx = -k xdxWs is the work done BY the spring.Us = ½ k x2Unlike gravitational potential energy, we know where the zero potential energy point is for a spring.Slide14
Conservative Forces and Potential Energy
=
= Change in potential energy
+
Ui
dU
= -
dx
= -
dU
/dx
Slide15
Force and Potential Energy
In order to discuss the relationships between potential energy and force, we need to review a couple of relationships.
W
c
= F
D
x
(if force is constant)
W
c
= Fdx = - dU = -
D
U
(if force varies) Fdx = - dUFdx = -dU
F = -dU/dxSlide16
Power
P =
dE
/
dt
Average Power = W/t
P =
dW
/
dt
= F *
dr
/
dt = F * vSlide17
Forces Reminders
Be sure to draw
freebody
diagrams
Think about net force and what is going on thereSlide18
Drag and Resistive forces
Drag is a resistive force proportional to the object’s velocity
How can we express this?
= -
bv
v is velocity
b is a constant
Depends on the properties of the medium, shape of the object, size of the object
Slide19
Considering Drag with other forces
Think about the coffee filter lab.
What forces were acting on the coffee filter?
= mg –
bv
= ma
Slide20
Eventually, the coffee filters reached terminal velocitySlide21Slide22
How can we figure out the terminal velocity?
mg - b
= 0
=
Slide23
How can we figure out the velocity at any given point?
mg –
bv
= ma
a =
= g -
Slide24
Assignment
Solve the equation to get an expression for v for all times t (no ds in there)
= g -
Slide25
Sample problem:
Gravitational potential energy for a body a large distance r from the center of the earth is defined as shown below. Derive this equation from the Universal Law of Gravity.
Hint 1:
dW
= F(r)•
dr
Hint 2:
Δ
U = -
W
c
(and gravity is conservative!)
Hint 3:
U
g
is zero at infinite separation of the masses.Slide26
Problem:
The potential energy of a two-particle system separated by a distance r is given by U(r) = A/r, where A is a constant. Find the radial force F that each particle exerts on the other.Slide27
Problem:
A potential energy function for a two-dimensional force is of the form U = 3x
3
y – 7x. Find the force acting at a point (x,y).