Transcript 10/12, Nonconservative Forces and Power

General Physics I
Conservation of Energy
• Integrated Newton’s 2nd Law wrt space 
new expressions
– Velocity
– Scalar
– Useful beyond mechanics
• Demonstrations:
– Basketball + tennis ball
– Racing Pool Balls
Conservation of Energy
Isolated, =0
Σ𝑊𝑒𝑥𝑡 + 𝑊noncons. … =
Δ𝐾 + Δ𝑈𝑔 + Δ𝑈𝑠 = Δ𝐸mech
Transformations
Isolated Systems
• Only non-conservative forces do work:
𝑊noncons. = Δ𝐾 + Δ𝑈𝑔 + Δ𝑈𝑠 = Δ𝐸mech
• No external work → 𝐸𝑚𝑒𝑐ℎ is conserved:
0 = Δ𝐾 + Δ𝑈𝑔 + Δ𝑈𝑠 = Δ𝐸mech
Conservation of Energy
• Work done by friction:
𝑟𝑓
𝐹𝑛𝑜𝑛𝑐𝑜𝑛𝑠 ⋅ 𝑑 𝑟
𝑟𝑖
𝑟𝑓
=−
𝐹𝑛𝑜𝑛𝑐𝑜𝑛𝑠 𝑑𝑟
𝑟𝑖
= −𝐹𝑛𝑜𝑛𝑐𝑜𝑛𝑠 Δ𝑟
• “Lost” energy  internal energy.
Total
distance
traveled
Conservation of Energy
• For isolated systems:
−𝐹𝑛𝑜𝑛𝑐𝑜𝑛𝑠 Δ𝑟 = Δ𝐾 + Δ𝑈𝑔 + Δ𝑈𝑠 = Δ𝐸𝑚𝑒𝑐ℎ
• No friction: Δ𝐸𝑚𝑒𝑐ℎ = 0
Old Reading Quiz Question:
How do you know when to apply the
equations for nonconservative forces?
Should I always assume there is friction
unless problems tell me to ignore it, or is it
the other way around?
A cart on an air track is moving at 0.5 m/s when the
air is suddenly turned off. The cart comes to rest
after traveling 1 m. The experiment is repeated, but
now the cart is moving at 1 m/s when the air is
turned off. How far does the cart travel before
coming to rest?
1. 1 m
2. 2 m
3. 3 m
4. 4 m
5. impossible to determine
What would happen to the final speed of the cart if the bridge
shown were built between the two hills (the bridge produces the
same friction force as the track)?
1) The final speed would be greater than without the bridge.
2) The final speed would be less than without the bridge.
3) The final speed would be the same as without the bridge.
vi = 0m/s
yi = 500m
yf = 400m
Final Point
Power
• How much energy is transferred (work
done) or transformed per time:
𝑑𝐸
𝑃=
General
𝑑𝑡
Δ𝐸
Expressions
𝑃𝑎𝑣𝑔 =
Δ𝑡
𝑃 =𝐹⋅𝑣
𝐽
𝑠
• Units = = 𝑊 (Watts)
• kWh – measure of?
Power from
work on system
A sports car accelerates from zero to 30 mph in 1.5
s. How long does it take for it to accelerate from zero
to 60 mph, assuming the power of the engine to be
independent of velocity and neglecting friction?
1. 2 s
2. 3 s
3. 4.5 s
4. 6 s
5. 9 s
6. 12 s
Reading Quiz:
A 10 kg sled slides down a 30 degree slope.
Because of friction it maintains a constant
speed. For every meter of travel, how much
energy is lost to friction?
Reading Quiz:
A real world roller-coaster released at point A and coasting
without external power would traverse a track somewhat
like that shown in Figure 1. Friction is not negligible in the
real world. If we call the potential energy at point A relative
to the ground Uo what can you say about the potential and
kinetic energies UB, KB at point B and the potential and
kinetic energies UC, KC at point C.
Lab
Testing predictions of sliding objects using
conservation of energy.
New Equipment:
- None
Problems
P8.19: A boy in a wheelchair (total mass 47.0 kg) has
speed 1.40 m/s at the crest of a slope 2.60 m high and 12.4
m long. At the bottom of the slope his speed is 6.20 m/s.
Assume air resistance and rolling resistance can be
modeled as a constant friction force of 41.0 N. Find the
work he did in pushing forward on his wheels during the
downhill ride. Additional question: where did the energy
come from for him to do this work?
P8.30: The electric motor of a model train accelerates the
train from rest to 0.620 m/s in 21.0 ms. The total mass of
the train is 875 g. (a) Find the minimum power delivered to
the train by electrical transmission from the metal rails
during the acceleration. (b) Why is it the minimum power?
Additional Problems
You have landed a summer job with a company that has been given
the contract to design the ski jump for the next Winter Olympics.
The track is coated with snow and has an angle of 25o from the
horizontal. A skier zips down the ski jump ramp so that he leaves it
at high speed. The winner is the person who jumps the farthest after
leaving the end of the ramp. Your task is to determine the height of
the starting gate above the end of the ramp, which will determine
the mechanical structure of the ski jump facility. You have been told
that the typical ski-jumper pushes off from the starting gate at a
speed of 2.0 m/s. For safety reasons, your design should be such
that for a perfect run down the ramp, the skier's speed before
leaving the end of the ramp and sailing through the air should be no
more than 80 km/hr. You run some experiments on various skies
used by the jumpers and determine that the coefficient of static
friction between the snow and the skis is 0.10 and its coefficient of
kinetic friction is 0.02. Since the ski-jumpers bend over and wear
very aerodynamic suits, you decide to neglect the air resistance to
make your design.
A 0.5 kg toy sinks into the lake with a starting
velocity of 2 m/s down. If the buoyant and
viscous forces of the water on the toy (drag,
effectively) do a total of -10J of work, what is the
speed of the toy 3m below the surface?
A stone is launched upward into the air. In addition
to the force of gravity, the stone is subject to a
frictional force due to air resistance. The time the
stone takes to reach the top of its flight path is
1. larger than
2. equal to
3. smaller than
the time it takes to return from the top to its original
position.
In order to find the speed of the cart at the top of the second bump, give an
expression for conservation of energy:
a) in terms of Ui, Uf, Ki, Kf, fk, and d.
b) in terms of m, d, fk and the given numbers.
vi = 0m/s
yi = 500m
yf = 400m
The cart on the roller coaster below barely makes it around the loop without falling off
the track. Give an expression for conservation of energy:
a) in terms of Ui, Uf, Ki, Kf, fk, and d.
b) in terms of m, d, fk and the given numbers.
vi = 0m/s
10m
r = 3m
A ball moves on a spring at a 30° angle below the horizontal. The equilibrium length
of the spring is 1.5m. The ball is released from rest at a total spring length of 1m.
Give an expression for conservation of energy:
a) in terms of Ui, Uf, Ki, Kf, fk, and d.
b) in terms of m, fk and the given numbers.
To find the final compression of the spring, give an expression for conservation of
energy:
a) in terms of Ui, Uf, Ki, Kf, fk, d, and k.
b) in terms of m, d, k, fk and the given numbers.
vi = 0m/s
yi = 500m
vf = 0m/s
yinitial contact = 100m
A ball is fired from a 2m long spring cannon at a 30° angle above the horizontal. The
equilibrium length of the spring is 1m and it is originally compressed to one quarter its
length. The cannon produces a constant friction force. To determine the maximum
height of the ball, give an expression for conservation of energy:
a) in terms of Ui, Uf, Ki, Kf, fk, and d.
b) in terms of m, fk and the given numbers.
For the following situation, give an expression for conservation of energy in terms of
Ui, Uf, Ki, Kf, fk, and d.
yf = 0.5m
vi = 5m/s
35°
For the following situation, if it were twice the mass, the block would travel
a) higher.
b) lower.
c) The same height.
yf = 0.5m
vi = 5m/s
35°