1. General Physics I

2. Conservation of Energy
Integrated Newton’s 2nd Law wrt space  new expressions
Velocity
Scalar
Useful beyond mechanics
Demonstrations:
Basketball + tennis ball
Racing Pool Balls

3. Conservation of Energy
Σ𝑊 𝑒𝑥𝑡 + 𝑊 noncons. …=
Δ𝐾+Δ 𝑈 𝑔 +Δ 𝑈 𝑠 =Δ 𝐸 mech
Isolated, =0
Transfers
Transformations

4. Isolated Systems Only non-conservative forces do work:
𝑊 noncons. =Δ𝐾+Δ 𝑈 𝑔 +Δ 𝑈 𝑠 =Δ 𝐸 mech
No external work → 𝐸 𝑚𝑒𝑐ℎ is conserved:
0=Δ𝐾+Δ 𝑈 𝑔 +Δ 𝑈 𝑠 =Δ 𝐸 mech

5. Conservation of Energy
Work done by friction:
𝑟 𝑖 𝑟 𝑓 𝐹 𝑛𝑜𝑛𝑐𝑜𝑛𝑠 ⋅𝑑 𝑟
=− 𝑟 𝑖 𝑟 𝑓 𝐹 𝑛𝑜𝑛𝑐𝑜𝑛𝑠 𝑑𝑟
=− 𝐹 𝑛𝑜𝑛𝑐𝑜𝑛𝑠 Δ𝑟
“Lost” energy  internal energy.
Total distance traveled
Technically not valid for an extended object – only a particle.

6. Conservation of Energy
For isolated systems:
− 𝐹 𝑛𝑜𝑛𝑐𝑜𝑛𝑠 Δ𝑟=Δ𝐾+Δ 𝑈 𝑔 +Δ 𝑈 𝑠 =Δ 𝐸 𝑚𝑒𝑐ℎ
No friction: Δ 𝐸 𝑚𝑒𝑐ℎ =0

7. 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?

8. 5. impossible to determine
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
Answer: 4.The cart comes to a stop when all of the cart’s kinetic energy
is lost to friction.The frictional force times the stopping distance is equal
to the cart’s initial kinetic energy.

9. 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.
yi = 500m
yf = 400m
vi = 0m/s
Final Point

10. Power
How much energy is transferred (work done) or transformed per time:
𝑃= 𝑑𝐸 𝑑𝑡
𝑃 𝑎𝑣𝑔 = Δ𝐸 Δ𝑡
𝑃= 𝐹 ⋅ 𝑣
Units = 𝐽 𝑠 =𝑊 (Watts)
kWh – measure of?
General Expressions
Power from work on system
Unit of energy: kWh

11. A sports car accelerates from zero to 30 mph in 1. 5 s
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 s
4. 6 s
5. 9 s
6. 12 s
Answer: 4. In the absence of friction, the power of the engine is equal to
the kinetic energy of the car divided by the time it took to attain that kinetic
energy.

12. Reading Quiz: A 10 kg sled slides down a 30 degree slope
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?

13. 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.

14. Lab
Testing predictions of sliding objects using conservation of energy.
New Equipment:
None

15. 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 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?

16. Additional Problems

17. 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 Since the ski-jumpers bend over and wear very aerodynamic suits, you decide to neglect the air resistance to make your design.

18. 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?

19. the time it takes to return from the top to its original position.
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.
Answer: 3. If there were no friction, the sum of potential and kinetic energy
would have to be constant. So as the ball rose, its potential energy
would increase and its kinetic energy decrease. On the way down, the potential
energy would decrease again such that, at each point on the path,
the ball would have the same kinetic energy as on the way up. Because
there is friction, however, part of the total energy is dissipated. As a result,
the ball has less kinetic energy at each point on the way down. This means
it takes longer to go down than to go up.

20. 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

21. 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

22. A ball moves on a spring at a 30° angle below the horizontal
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.

23. 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

24. 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.

25. 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°

26. For the following situation, if it were twice the mass, the block would travel
higher.
lower.
The same height.
yf = 0.5m
vi = 5m/s
35°