Download L4a--09-22--Energy

Physical Sciences 2: Lecture 4a
September 22, 2016
Pre-reading for Lecture 4a: Energy
The most important concept about energy is that energy is conserved: for an isolated
system, it can be transformed from one form into another, but the total amount must
remain the same. The most important kinds of energy that we will encounter are:
- If an object’s center of mass is moving at a speed v, it has
kinetic energy associated with the motion of its center of mass:
- If an object is at some height h above the ground (or some other
reference point), it has gravitational potential energy
associated with its height:
- If a spring has a spring constant k, and it is stretched (or
compressed) by a distance x from its equilibrium length, the
spring has elastic potential energy given by:
- If an object of mass m with a specific heat C changes its
temperature by an amount ΔT, then its internal energy (thermal
energy) changes by an amount:
K CM =
1 2
mv
2
U grav = mgh
U elastic =
1 2
kx
2
ΔEthermal = mCΔΤ
- The sum of the kinetic and potential energy for an object is referred to its mechanical
energy. Other forms of energy include light, sound, chemical energy, thermal energy,
nuclear energy, etc. We will call these other forms internal energy, Einternal.
- The total energy of any object (or system) is given by Etotal = K + U + Einternal, where K
represents the total kinetic energy and U represents the total potential energy for all
objects involved. If a system is isolated, it must have ΔEtotal = 0.
- Often, we can assume that the internal energy does not change, ΔEinternal = 0. This
requires, for instance, no kinetic friction or drag forces, which raise the temperature.
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Learning objectives: After this lecture, you will be able to…
1. Define some of the the different forms of energy: kinetic, potential, and internal.
2. Use energy conservation to solve problems in which energy is converted from one
form into another.
3. Use mechanical energy to solve mechanics problems involving speed, height,
displacement, etc., in situations where the change in internal energy is negligible.
4. Use energy conservation to derive the equation of motion for an object falling
vertically, a mass on a spring, or other one-dimensional problems.
5. Calculate the dot product of two vectors.
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Physical Sciences 2: Lecture 4a
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September 22, 2016
The following excerpt is from the Feynman Lectures on Physics; it’s a very nice way of
thinking about energy from one of our greatest 20th-century physicists…
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Physical Sciences 2: Lecture 4a
September 22, 2016
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Physical Sciences 2: Lecture 4a
September 22, 2016
Energy
•
Highlights from the pre-reading:
The most important concept about energy is that energy is conserved: for an isolated
system, it can be transformed from one form into another, but the total amount must
remain the same. The most important kinds of energy that we will encounter are:
- If an object’s center of mass is moving at a speed v, it has
kinetic energy associated with the motion of its center of mass:
- If an object is at some height h above the ground (or some other
reference point), it has gravitational potential energy
associated with its height:
- If a spring has a spring constant k, and it is stretched (or
compressed) by a distance x from its equilibrium length, the
spring has elastic potential energy given by:
- If an object of mass m with a specific heat C changes its
temperature by an amount ΔT, then its internal energy (thermal
energy) changes by an amount:
K CM =
1 2
mv
2
U grav = mgh
U elastic =
1 2
kx
2
ΔEthermal = mCΔΤ
- The sum of the kinetic and potential energy for an object is referred to its mechanical
energy. Other forms of energy include light, sound, chemical energy, thermal energy,
nuclear energy, etc. We will call these other forms internal energy, Einternal.
- The total energy of any object (or system) is given by Etotal = K + U + Einternal, where K
represents the total kinetic energy and U represents the total potential energy for all
objects involved. If a system is isolated, it must have ΔEtotal = 0.
- Often, we can assume that the internal energy does not change, ΔEinternal = 0. This
requires, for instance, no kinetic friction or drag forces, which raise the temperature.
•
Learning objectives: After this lecture, you will be able to…
1. Define some of the the different forms of energy: kinetic, potential, and internal.
2. Use energy conservation to solve problems in which energy is converted from one
form into another.
3. Use mechanical energy to solve mechanics problems involving speed, height,
displacement, etc., in situations where the change in internal energy is negligible.
4. Use energy conservation to derive the equation of motion for an object falling
vertically, a mass on a spring, or other one-dimensional problems.
5. Calculate the dot product of two vectors.
6. Use the dot product to show that the elastic collision of one ball with an identical,
stationary ball will yield final velocities that are perpendicular to one another.
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Physical Sciences 2: Lecture 4a
September 22, 2016
Activity 1: Conservation of Energy
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A pole vaulter is running at a speed v. She plants her pole and vaults over the bar. As
she crosses the bar, her speed is essentially zero.
Assume that all of her potential energy as she crosses the bar comes from her initial
kinetic energy while running. What speed would she have to run in order to vault over a
bar 4 meters high?
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You climb to the 8th floor of the Science Center. If your body is 20% efficient in
converting food energy into mechanical energy, how many food Calories do you burn?
(1 food Calorie = 4184 Joules)
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Bonus! If all of the “lost” energy goes into heating up your body, estimate how much
your temperature would increase.
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Physical Sciences 2: Lecture 4a
September 22, 2016
Am I getting it?
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Consider a single isolated object with:
Kinetic energy K
1.
Potential energy U
Internal energy Einternal
Which of the following equations would express the conservation of energy for this
object? Choose all that apply. (Note the subscripts i = initial; f = final.)
a) ΔK + ΔU = ΔEinternal
b) ΔK + ΔU + ΔEinternal = 0
c) K i + U i = K f + U f + ΔEinternal
d) K i + U i + ΔEinternal = K f + U f
e) K i + U i = K f + U f
f) ΔK + ΔU = 0
2.
The mechanical energy of an object is its kinetic plus potential energy: Emech = K + U.
Which of the following must be true for mechanical energy to be conserved? Choose all
that apply.
a) ΔEmech = 0
b) ΔEinternal = 0
c) ΔEmech < 0
d) K i + U i = K f + U f
e) ΔK + ΔU = 0
f) There must be no kinetic friction.
g) There must be no static friction.
h) There must be no fluid drag.
i) There must be no chemical reactions taking place.
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Physical Sciences 2: Lecture 4a
September 22, 2016
Activity 2: Energy conservation for free-fall
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If we ignore air resistance, we can derive the equation for free fall in one dimension
(vertical motion only) by using conservation of energy. Let’s assume that the position of
an object as a function of time is given by a polynomial:
y(t) = A + Bt + Ct 2
where we don’t know anything about the values of A, B, or C.
1.
The y-velocity, vy, will be given by the time derivative of y. What is it?
vy =
2.
Write an equation for the total energy of the object (kinetic plus potential) in terms of its
position, y, and its velocity, vy. If energy is conserved, this equation must be constant.
3.
Now substitute in your expressions for y and vy into the energy equation. You’ll get an
equation that contains t that must be constant! Collect the various powers of t together.
4.
In order for this equation to be constant, it cannot depend on t! Use this fact to find some
constraints on the parameters A, B, and C in the original expression for y(t).
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Bonus! Show that if the original polynomial y(t) contained a term Dt3 that the coefficient
D would have to be zero. (Indeed, any additional terms with t4, t5, etc. must all be zero.)
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Physical Sciences 2: Lecture 4a
September 22, 2016
The Dot Product (a brief digression)
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You’ve seen how to multiply a vector
 times a scalar… now let’s see how to multiply two
vectors. Given two vectors a and b , we define the dot product:
Notice that the dot product combines two vectors and gives a scalar (a single number).
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There are some special cases of the dot product that are worth knowing:
If two vectors are parallel:
If two vectors are antiparallel:
If two vectors are perpendicular:
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There is another formula for the dot product that is very useful. Given
the components of two vectors:

a = (ax , ay )

b = (bx , by )
the dot product is equal to:
 
a ⋅ b = ax bx + ayby
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

Calculate the dot product of the vectors a = (–2, 3) and b = (5, 4). Can
you confirm graphically that the sign of this dot product is correct?
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 
What is the dot product of a ⋅ a ?
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Physical Sciences 2: Lecture 4a
September 22, 2016
Activity 3: Pool ball collisions
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The collision of two billiard balls is (approximately) elastic. That means that both
momentum and kinetic energy are conserved in the collision.

Consider the pool scenario: Ball 1 (mass m) is moving at initial velocity v1i . It strikes
ball
has the same mass and is initially
 2, which

 at rest.
 The two balls have final velocities
v1f and v2f , respectively. Show that either v1f and v2f must be perpendicular (at an
angle of 90°), or one of the velocities must be zero.
1.
Write the equation for conservation of momentum in this system in terms
of the masses

and velocities of the balls. Solve this equation for the initial velocity v1i .
2.
Now take the dot product of this equation with itself; on the left-hand side you should
have v1i2 . (Dot products are like any other product: commutative, distributive, etc.)
3.
Write the equation for conservation of kinetic energy in this system, in terms of the
masses and velocities of the balls. Solve this equation for v1i2 .
4.
Set the expressions for v1i2 in parts (2) and (3) equal, and what do you get? What does
this mean?
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Bonus! Can you find a simple geometric argument that the outgoing velocities must be
perpendicular (at an angle of 90°)?
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Physical Sciences 2: Lecture 4a
September 22, 2016
Oscillations
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Lots of physical systems oscillate: they go “back and forth” with a well-defined period of
oscillation. Consider, for instance, a pendulum: a mass hanging from a string.
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One hallmark of oscillators is that the energy of an oscillator will trade off between
potential energy and kinetic energy as the system oscillates:
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Do I believe that energy is conserved? I’ll bet my face on it…
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Now let’s look at another oscillating system: a mass on a spring. Graph the kinetic
energy of the mass as a function of time. What must the potential energy look like?
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Qualitatively, what should be the elastic potential energy of a spring that is stretched (or
compressed) a distance x from its equilibrium length?
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Physical Sciences 2: Lecture 4a
September 22, 2016
Activity 4: Energy conservation for an oscillator
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Now let’s derive the equation of motion for a mass on a spring. We know it will have
elastic potential energy given by its displacement x from equilibrium:
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U elastic = kx 2
2
Let’s assume that the position x as a function of time is given by:
x(t) = Asin(Bt)
where we don’t know anything about the values of A or B.
1.
The x-velocity, vx, will be given by the time derivative of x. What is it?
vx =
2.
Write an equation for the total energy of the object (kinetic plus potential) in terms of its
position, x, and its velocity, vx. If energy is conserved, this equation must be constant.
3.
Now substitute in your expressions for x and vx into the energy equation. You’ll get an
equation that contains t that must be constant! Is there anything you can do to make
this equation so it does not depend on t? (Remember: you can choose the values of A or
B to be anything you want. And here’s a hint: for any angle θ, sin 2 θ + cos 2 θ = 1 .)
4.
The frequency of an oscillatory function like sin(Bt) is given by f = B 2π . What is the
frequency of the oscillator you found in your result from part (3)?
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Super-duper bonus! If you had assumed a polynomial for x(t) = A + Bt + Ct 2 + Dt 3 +… ,
show how you could end up with the solution x(t) = Asin(Bt) .
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Physical Sciences 2: Lecture 4a
September 22, 2016
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Physical Sciences 2: Lecture 4a
September 22, 2016
One-Minute Paper
Your name: _____________________________
Names of your group members:
TF: _____________________________
_________________________________
_________________________________
_________________________________
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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 during class.
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What single topic left you most confused after today’s class?
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Any other comments or reflections on today’s class?
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