General Physics I: Day 17
Conservation of Energy & Energy Diagrams
Using Energy Diagrams
Tool for visualizing how energy is transformed
Example: Object oscillating on a spring
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WarmUp: Adding Kinetic
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A truck initially at rest at the top of a hill is allowed to roll
down. At the bottom, its speed is 14 m/s. Next, the truck is
again rolled down the hill, but this time it does not start
from rest. It has an initial speed of 14 m/s at the top before
it starts rolling down the hill. How fast is it going when it
gets to the bottom?
~24% → 14 m/s
~9%
True if no energy were added (flat).
→ 17 m/s
~18% → 20 m/s
~9%
→ 24 m/s
~40% → 28 m/s
2x the energy means… 2x the speed?
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An object hangs motionless from a spring. Think
about the sum of the elastic potential energy of the
spring and the gravitational potential energy of the
object and Earth (𝑈Spr + 𝑈g ). When the object is
pulled down and held, this sum
A) increases.
B) stays the same.
C) decreases.
Worked-Example: Spring Safety System
Mcar = 600 kg
k = 36 kN/m
vmax=?
Mpassengers = 600 kg
h = 25 m
xspring = ?
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Suppose the system does fail
when the car is at an elevation of
25 meters. If the 600-kg car is
carrying six 100-kg passengers,
(a) how fast is the car moving
when it connects with the spring
and (b) how much does the
spring compress when it finally
stops the car?
Worked-Example: Spring Safety System
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Worked-Example: Spring Safety System
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Worked-Example: Spring Safety System
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Worked-Example: Spring Safety System
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Sample Problem (tricky, but cool)
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A small cap is placed on top of a smooth inverted
spherical mixing bowl (as shown on the board). If
the cap is nudged slightly, and we ignore friction, at
what height will the cap leave the surface of the
bowl?
Warm-Up: Coaster
A real-world roller coaster is shown. A coaster car is
released at point A and coasts without external power.
Friction is not negligible in the real world.
a) Does the roller coaster have the same mechanical
energy at points B and C?
~33% → Yes
~67% → No
b) Is the total energy conserved during the coaster
ride?
~77% → Yes
~23% → No
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Warm-Up: Coaster
c) Give a qualitative statement about what forms
the energy has when it is halfway down the hill
after B.
~17% → Described 𝐾, 𝑈𝑔 and 𝑈int
~42% → Left out 𝑈𝑔
~17% → Left out 𝑈int generated by friction
~17% → Left out 𝐾
~17% → Described energy “lost” to friction
~8% → Described energies that don’t exist
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Warm-Up: Coaster
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Conceptual miss-steps
“C) force energy, potential energy”
“c. The energy is still there. It turns into different
energies: kinetic and potential.”
“c. Some of the energy is lost due to friction and air
resistance. The total energy it had at the beginning
is not equal to the total after it reached point C.”
Watch out for “potential energy” is too vague.
Energy is never lost. It has to be there somewhere.
Warm-Up: Coaster
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“a) No, because Emech = U + K, we know that U is
dependent upon the height per the equation U =
ugh. So the coaster will have different mechanical
energies at points B and C because they are
different heights. b) The total energy is conserved
but the mechanical energy is not conserved because
the sum of KE and PE constantly decreases due to
friction.”
Warm-Up: Coaster
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“a)For mechanical energy solely, yes. At the roller
coaster hits points B and C, it has a relationship
between potential energy and kinetic energy. 100%
PE occurred at Point A, while at the other points the
car converts some KE to PE as it climbs each hill.
b)Yes, it is continuously converted between
potential, kinetic, and internal/non-conservative
energies (is friction and air resistance external or
internal?) c)U[potential],U[internal], K”
Conservation of Energy!
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If we only allow conservative forces (quite limited):
U i  Ki  U f  K f
What if we allow other forces?
They can appear in two ways:
K i  U i  Wother  K f  U f
K i  U i  K f  U f  U int
Applying Conservation Energy
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• Choose a system!
• Choose an initial and final situation.
• Are there non-conservative forces? Do they do
work on your system?
Alternatively, did the internal energy of the
system change?
• Write down knowns and unknowns for initial and
final potential and kinetic energies.
K

U

K

U


U
i
i
f
f
int
• Go:
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Empire State Penny: If you drop a penny from the
top of the Empire State Building (373 m), how fast
will it be going at the bottom?
First, lets ignore air resistance
(a horrible approximation):
A) 60 m/s
B) 86 m/s
C) 7300 m/s
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Empire State Penny: A real penny (2.5 g after 1982)
has a terminal velocity. One empirical measurement
gave 25 mph (11 m/s).
How much thermal energy is created as the penny
falls from the top of the Empire State Building (373
m) down to the pavement below?
A) 9.0 J
B) 9.1 J
C) -9.1 J
D) 9000 J
Friction & 𝐸mech.
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Someone says:
“friction causes a loss of mechanical energy”. True?
Lots of counter-examples. Often this is about how
we define our system.
Static friction can do positive or negative work:
• Sit in the back of a truck, change speed (+ or –).
If we choose our system poorly, even kinetic
friction can increase Emech.:
• Pulling a table cloth out from under dishes.
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A child gets on a shallow playground slide. After
pushing himself to get started down the slide he
slides at a constant speed all the way to the bottom.
Compare the change in gravitational potential
energy of the child to the work done on the child by
non-conservative forces.
A) 𝑊𝑓 is greater than Δ𝑈𝑔
B) 𝑊𝑓 is less than Δ𝑈𝑔
C) 𝑊𝑓 is equal to Δ𝑈𝑔
Coming up…
Tuesday (10/21) → 8.1 – 8.2
WarmUp due Monday by 10:00 PM (w/Image)
Ch. 7 Homework due Sunday by 11:59 PM
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