Download 3/2 Lecture

Conservation of energy
•  CAPA homework due Tuesday night
•  LA applications due Monday, March 9.
1
Clicker question
Set frequency to BA
A block with an initial velocity v travels up a 40° slope to a height h
before sliding back. An identical block with the same initial speed
travels up a 20° slope. There is no friction on either slope. The
second block will reach a height of…
A.  2h
B.  h
C.  h/2
D.  Greater than h but not necessarily 2h
E.  Less than h but not necessarily h/2
2
1
K
+U
=
mv
Both start with initial energy which is all kinetic: i
g,i
i
2
At the top, velocity=0 so all kinetic energy was converted to
gravitational potential energy: K f +U g, f = mgy f
Since initial kinetic energy was the same, the final gravitational
potential energy must be the same so the height is the same.
2
Using conservation of energy
A spring loaded gun with spring constant of 5000 N/m is used
to fire a 0.01 kg BB off a 100 m cliff. The spring is compressed
0.1 m before launch. Assuming no air resistance, what is the
speed of the BB just before it hits the ground?
Use conservation of energy to find the kinetic energy at impact
We choose to set y=0 at the base of the cliff
Initial energy: E = K + U + U = 1 mv 2 + mgy + 1 k (Δx)2 =
2
2
i
i
g, i
s, i
i
i
i
2
0 + (.01 kg )(10 m/s 2 )(100 m) + 12 (5000 N/m )(.1 m) = 10 J + 25 J = 35 J
Final energy: E f = K f + U g, f + U s, f = 12 mv2f + mgy f + 12 k (Δx)2f =
1
2
(.01 kg )v2
By conservation of energy: 12 (0.01 kg )v 2 = 35 J so v = 2 ⋅ 35 J = 84 m/s
.01 kg
Don’t have to worry about x and y components of velocity
(and it doesn’t matter what angle we fire the gun at!)
3
Clicker question 1
Set frequency to BA
A spring-loaded dart gun shoots a dart straight up reaching a
height of 24 m. The same dart is shot straight up a second time
but the spring is compressed only half as far before firing. How far
up does the dart go, neglecting friction and assuming an ideal
spring?
K f +Ug, f +Us,f = K i +Ug,i +Us,i
A.  3 m
2
2
2
2
1
1
1
1
mv
+
mgy
+
k(Δx
)
=
mv
+
mgy
+
k(Δx
)
f
f
f
i
i
i
2
2
2
2
B.  6 m
C.  12 m
Before the dart is launched and when it reaches
D.  24 m
the maximum height the kinetic energy is 0
E.  48 m
Initial energy:Ei = 12 k (Δx)2 Final energy: E f = mgy f
By conservation of energy, Ei = Ef
If Δx is reduced by a factor of 2, yf will be reduced a factor of 4
4
Notice we did not have to worry about kinetic energy at all.
Pendulum
Generally set gravitational potential
energy zero-point at bottom of swing
L
θ
When pendulum is pulled back by an
angle θ, gravitational potential energy is
U grav = mgy = mg (L − L cosθ ) = mgL(1 − cosθ )
L cos θ
y=0
When released, the potential energy is converted to kinetic
energy and back to potential energy and so on.
This is a good example of conservation of energy
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What about non-conservative forces?
When a pendulum swings back and forth, the energy is being
transformed from potential to kinetic, back to potential, etc.
This is an example of conservation of energy:
E f = Ei where E = K + U and U = U g + U s
When is this energy not conserved?
The pendulum will eventually come to rest due to things
like friction (air resistance, rubbing at the pivot, etc.)
If I stop the pendulum, or when I start it by giving it a
push or pulling it back, I change the energy.
These are both examples of work
done by non-conservative forces
6
A more general conservation of energy
Conservation laws are probably the most important concept
in physics and conservation of energy is among the most
fundamental. Energy is always conserved when you
consider a large enough system.
When friction-like forces act, they transform kinetic energy
into thermal energy (heat). If we consider the pendulum
system to include the pivot and the air, no energy is actually
lost; it is transformed into thermal energy, Eth.
When an external force acts on the system (like my hand), it
is capable of energy transfer from the outside environment
to the system. This is labeled external work, Wext.
In reality, no energy is created or destroyed, it is simply
transferred from outside the system to inside the system.7
New conservation of energy
So far, we have been using the conservation of energy
equation when there are no non-conservative forces acting.
Remember, the full conservation
of energy equation is:
ΔK + ΔU = Wnc
or
K f + U f = Ki + U i + Wnc
Can divide Wnc into two more parts: Wnc = Wdiss + Wext
Wdiss is work due to dissipative forces (friction). It is always
negative and increases the thermal energy: ΔEth = −Wdiss
Wext are non-conservative external forces such as a hand
ΔK + ΔU = Wdiss + Wext
ΔK + ΔU + ΔEth = Wext
K f + U f = Ki + U i + Wdiss + Wext
K f + U f + ΔEth = Ki + U i + Wext
8 2
1
Remember U = U g + U s = mgy + 2 k (Δs )
The final conservation of
energy equation can be
written several ways:
Clicker question 2
Set frequency to BA
A mass slides down a frictionless ramp of height h and hits a carpet
with kinetic friction coefficient µk = 1.0. Its initial speed is zero. How
far does the mass slide along the carpet?
A.  h
B.  more than h
h
C.  less than h
D.  impossible to tell without the mass
No Wext but there is Wdiss from friction so K f + U f = Ki + U i + Wdiss
Kf = 0
Uf =0
Ki = 0
Wdiss is work by friction
Ui = mgh
Wdiss = − fk d = −µk nd = −µk mgd = −mgd
K f + U f = Ki + U i + Wdiss becomes 0 + 0 = 0 + mgh − mgd
So
d =h
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