Download 09-30--L4c-Work and Potential Energy

Physical Sciences 2: Lecture 4c
October 6, 2015
Work and Potential Energy
•
Highlights from the pre-reading:
In the previous lecture, you showed that the work done by gravity on an object is always
given by: Wby Fgrav = −mgΔh . But the gravitational potential energy is U = mgh. So:
Wby Fgrav = −ΔU grav

In general, for a potential energy U, the work done by its force F is: Wby F = −ΔU .
Given a potential energy U for some interaction, the (vector) components of the force
associated with that interaction are equal to (minus) the partial derivatives of U:
∂U
∂U
Fy = −
Fx = −
∂y
∂x
Using these relationships, along with the others developed earlier, you can calculate the
work done by a force, and the magnitudes of various forces, speeds, and displacements.
It is important to distinguish between work and power: the rate of doing work. For most
machines (including you!), the key limitation is power, not the total amount of work.
The relationship between force and potential energy can be seen in a potential energy
diagram. With these diagrams, you can locate points of stable equilibrium.
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Learning objectives: After this lecture, you will be able to…
1. Calculate the work done by gravity on an object during some process.
2. Distinguish between conservative and non-conservative forces.
3. Derive a relationship between the potential energy of a conservative force and the
(vector) components of that force.
4. Derive a relationship between the work done by a conservative force and the change
in potential energy associated with that force.
5. Derive a relationship between the change in mechanical energy for any system and the
work done by all non-conservative forces on that system.
6. Use any of the relationships derived in Module 4 between work, force, kinetic energy,
and potential energy to calculate the work done by some force on an object, or to
calculate other physical quantities (force, distance, speed, etc.).
7. Calculate the power required to convert energy from one form to another, or the power
required to do a certain amount of work.
8. Draw potential energy diagrams and use these diagrams to find points of stable or
unstable equilibrium, and to determine the overall motion of an object with some total
energy (if mechanical energy is conserved).
9. Approximate the potential energy near a point of stable equilibrium by using a Taylor
series expansion around that point.
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Physical Sciences 2: Lecture 4c
October 6, 2015
Activity 1: Work done by gravity
1.
Recall the basic definition of work:
final
Wby F =
∫
 
F ⋅ dr
initial

The components of the force F are (Fx , Fy ) , and the components of the infinitesimal
displacement dr are (dx, dy) . Rewrite this integral by expanding the dot product inside
the integral in terms of its components:
Wby F =
2.

 
Show that if the force F is constant,
the above expression simplifies to F ⋅ Δr . (Hint:

what are the components of Δr ?)
Wby F =
3.
Choose conventional (x, y) axes and find an expression for the work done by gravity on
an object of mass m. The answer could depend on m, x, y, and/or g:
Wby Fgrav =
4.
What is the relationship between the work done by gravity and the change in the
gravitational potential energy, ΔU grav ? (Hint: what is Ugrav?)
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Bonus! Show that the same relationship found in part
(4) will hold even if you choose non-conventional
coordinates, such as in the diagram shown at right.
y
m
θ
L
x
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Physical Sciences 2: Lecture 4c
October 6, 2015
Am I getting it?
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A box is at rest on a ramp. You push the box up the
ramp (with friction), and stop at the top of the ramp.
You then turn around and push the box back down to
where it started. The box ends at rest.
Consider the work done on the box.
For the first half of this process (pushing the box up the ramp):
1.
The work done by the normal force (of the ramp on the box) is:
a) negative
2.
b) zero
c) positive
The work done by kinetic friction is:
a) negative
4.
c) positive
The work done by gravity is:
a) negative
3.
b) zero
b) zero
c) positive
The work done by you is:
a) negative
b) zero
c) positive
For the entire process (pushing the box up and then back down again):
5.
The work done by the normal force (of the ramp on the box) is:
a) negative
6.
b) zero
c) positive
The work done by kinetic friction is:
a) negative
8.
c) positive
The work done by gravity is:
a) negative
7.
b) zero
b) zero
c) positive
The work done by you is:
a) negative
b) zero
c) positive
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Physical Sciences 2: Lecture 4c
October 6, 2015
Activity 2: Work, Force, and Potential Energy
•
In Activity 1, you found: Wby Fgrav = −ΔU grav . It turns out that this is true in general:
• For every “kind” of potential energy, there is an associated force.
• The work done by that force is equal to minus the change in potential energy.
1.
Suppose there’s an object with kinetic energy K and potential energy U. Assume that the
only force on the object is the force associated with that potential energy, and mechanical
energy is conserved. Show that:
−dU = Fx dx + Fy dy
2.
(Hint: Mechanical energy is conserved: dEmech = 0 . What is dK?)
Using the relationship from question (1) above, show that:
dU
If y is constant, then Fx = −
dx
If x is constant, then Fy = −
3.
The derivatives in question (2) are properly known as “partial derivatives,” which are
written using a funny curly “d.” We should write, for instance:
∂U
“take the derivative with respect to x, treating everything else as constant”
Fx = −
∂x
Suppose the potential energy is given by U = 3x 2 y . Find the components of the force:
Fx = −
4.
dU
dy
∂U
=
∂x
Fy = −
∂U
=
∂y
Using the relationship from question (1) above, show that:
Wby F = −ΔU
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Super duper bonus! If there are two kinds of potential energy U1 andU2, show
 that the
net force is given by the sum of the forces associated with each, i.e. Fnet = F1 + F2 . (Hint:
you’ll need to use one of the relationships derived earlier on this page…)
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Physical Sciences 2: Lecture 4c
October 6, 2015
Activity 3: Putting it together
1.
We must distinguish conservative forces, which are related to some kind of potential
energy, and nonconservative forces, which don’t have any associated potential energy.
List all the kinds of conservative forces we have encountered so far in this course:
List all the kinds of nonconservative forces we have encountered:
2.
Write an equation that represents the following statement, then prove that it is true:
• The change in mechanical energy for a system is equal to the work done by all
nonconservative forces on that system.
3.
A 50-kg skier is at the top of a slope with a height of 20 m and an angle of 20° from the
horizontal. Starting from rest, she skis straight down the hill. At the base of the hill, she
is traveling at a speed of 15.5 m/s. The coefficient of kinetic friction between her skis
and the hill is µk = 0.03; she is affected by both kinetic friction and air drag. Explain
how could you find the work done on the skier by:
• The normal force of the hill
• Gravity
• Kinetic friction
• Air drag
•
Bonus! Carry out all the calculations from question (3) above.
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Physical Sciences 2: Lecture 4c
October 6, 2015
Activity 4: Kinesin, ATP, work, and power
1.
If a system is isolated, show: The work done by all nonconservative forces = –ΔEinternal
2.
The relationship from question (1) above is very useful in thinking about work and
energy. What can you say about the change in internal energy for the following cases:
• A car slows down due to air drag.
• A car accelerates due to the static friction force between the car and the road.
• You stand up from your chair.
3.
Kinesin exerts a force of 7 pN (piconewton = 10–12 N) as it pulls a vesicle over a distance
of 8 nm. Is kinesin doing positive or negative work? What change in internal energy is
required? (Note: Hydrolysis of one molecule of ATP releases 6 × 10–20 J.)
4.
The power required for a given conversion of energy is equal to the amount of energy
converted divided by the time required: P = ΔE Δt . Compared with walking slowly up
the stairs, if I run up the stairs, what is the difference in:
• the amount of work required
• the power required
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Bonus! The instantaneous power of a given
 force is the “rate of work,” dW dt . Show

that the instantaneous power of a force F on an object with velocity v is: P = F ⋅ v .
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Physical Sciences 2: Lecture 4c
October 6, 2015
Activity 5: Potential energy of a spring
•
Besides gravity, we have seen one other conservative force: the force of a
spring. If a spring stretches along the x-axis, and the equilibrium position is
xeq, the potential energy of the spring is related to the spring constant k as:
2
1
U elastic = k x − xeq
2
Find an expression for the x-component of the force exerted by a spring.
(
1.
)
2.
On the axes below, sketch the potential energy of a spring, and the x-component of the
spring force. (For simplicity, take the equilibrium position to be xeq = 0.)
3.
What do you notice about the point on the potential energy graph where the force is zero?
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Bonus! An object (mass m) hangs vertically from a spring. Set y = 0 when the spring is
not stretched or compressed. Write an expression for the total potential energy of the
system (including spring and gravity forces), and find where the PE is at a minimum.
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xeq
x
Physical Sciences 2: Lecture 4c
October 6, 2015
Activity 6: PE diagrams and stable equilibrium
1.
An object is in equilibrium if the net force on it is zero; it is in stable equilibrium if any
small displacement from equilibrium results in a force that will push it back towards
equilibrium. Assuming that the net force is derived from some potential energy, find the
equilibrium point(s) in each of the following PE diagrams, and identify each equilibrium
point as stable or unstable:
2.
In the third diagram above, if the car starts at rest at point (b), and mechanical energy is
conserved, what will happen? Describe the motion in words…
•
Near any point of stable equilibrium, we can approximate the potential energy by using a
Taylor series expansion around the equilibrium point. If the equilibrium is at x = 0, the
expansion is given by:
⎛ d 2 f ⎞ x2
⎛ df
⎞
f (x) = f (0) + ⎜
x
+
⎜ dx 2 ⎟ 2! +…
⎝ dx x=0 ⎟⎠
⎝
x=0 ⎠
The potential energy of an object is given by:
⎛ 2π x ⎞
U = 1− cos ⎜
⎝ L ⎟⎠
The graph at right shows this function for L = 4 m.
Find an approximate expression for the potential
energy around x = 0.
3.
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Bonus! If xeq is a point of stable equilibrium, show that the potential energy around that
point can always be approximated by: U = U0 + k(x – xeq)2, for some constants U0 and k.
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Physical Sciences 2: Lecture 4c
October 6, 2015
One-Minute Paper
Your name: _________________________________ TF: _____________________________
Names of your group members:
_________________________________
_________________________________
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Please tell us any questions that came up for you today during lecture. Write “nothing”
if no questions(s) came up for you in class from 9:30am–11am.
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What single topic left you most confused after today’s class?
•
Any other comments or reflections on today’s class?
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