Homework 2.

Chapter 1.

For discussion questions there are many points of view. One 
possible viewpoint follows.

1. If our world is to be self-sustaining, then we have to pay for 
what we use so that there is no net decrease in resources. This cost must 
be high enough so that we can replace any renewable resource (e.g. replant 
trees) or replace a nonrenewable resource (oil) with some equivalent 
resource (e.g. solar power in various forms). 

The cost of oil at present merely reflects the cost of production
(extraction, refining, transportation) plus whatever profits are realized.
This is a result of the fact that no one is presently planning to replace
oil with anything.

2. Most people agree that as the standard of living rises around the
world, the energy consumption per capita rises. This depends on the
definition of standard of living.  At present, a high standard of living
is equated with high rate of material consumption, so that one is
considered to be better off if one has more possessions and uses more
energy. Since most developing nations aspire to be like the developed
nations, their energy consumption per capita is rising.

3. Open question, requires some soul-searching. Use of public versus 
private transportation, change in home life style, etc.


Chapter 2. 

Questions

1.  (a) When you ride a bicycle your body converts chemical energy (food) to
mechanical energy (muscle) plus heat. For bike plus human, mechanical energy
is converted to kinetic energy (you accelerate to some velocity) and
potential energy, if you climb a hill. You also do work against friction (air
resistance, bearing and tire friction) which ulitmately ends up as heat. 
Most of the energy becomes work done (force x distance) moving you from A 
to B.

(b) A windmill converts the kinetic energy of the air (mass times velocity
squared) into mechanical energy. This is used to lift water, thereby changing
its potential energy and also giving it kinetic energy. Heat is produced
throughout.

2. When an automobile rolls up an incline and stops, some of the kinetic
energy (hopefully, most of it!) is converted to potential energy. The
potential energy can be converted back into kinetic energy when it rolls
downhill. Some energy is converted to heat through friction.

3. Any time that a moving object reverses direction, its velocity must be
zero for an extremely short time. However, it is almost always
accelerating! 

Think about a ball bobbing on a spring. The velocity is of the ball is
instantaneously zero at the endpoints of the motion, but the spring is
still exerting a force on it, so it is accelerating.

A ball thrown up is another example. It has a positive velocity going up,
which is becoming smaller and then negative as it begins to fall. At the
very top of the trajectory, the velocity is zero but the acceleration is
always g = 9.8 m/s2.

Problems:  

(It is OK to round "g", the acceleration due to gravity, off to  10 m/s2 
but I use 9.8 m/s2  here.)

1. A 2 kg object moving at 3 m/s has kinetic energy  KE = 1/2m (v squared) 
   KE = 1/2 (2 kg) (3 m/s squared) = 9 Joules
  
   If it were raised to some height it would have potential energy PE = 
   mgh

   Setting these equal mgh = 9 Joules = (2 kg) (9.8 m/s2) (h) = 9 
                                  or h = 9 / (19.8) = 0.45 m

2. Work = force x distance = change in energy

   In this case KE = (1/2) m (v squared) = Fd = 40 N x 5 m = 200 Joules
   To get the velocity we have 
      (1/2) (20 kg) (v squared) = 200 J
                      v squared = 200 J/ 10 kg = 20 (m/s)squared
                              v = 4.5 m/s

3. A ball with mass 0.5 kg is dropped 5 m. The final KE is equal to the
   change in PE which is PE = mgh = (0.5 kg)(9.8 m/s2)(5 m) = 24.5 J

   Final KE = 24.5 J = (1/2) m v squared = (1/2) (0.5 kg) v squared
                             v squared = 24.5/ 0.25 (m/s) squared
                             v = square root(98) = 9.8 m/s


   hmmm.... Since ALL OBJECTS accelerate in free fall at 9.8 m/s2, it 
   would take 1 second to fall 5 m, at the end of which, the velocity is 
   9.8 m/s! It doesn't matter what the mass is.


4. The kinetic energy of any object depends on its velocity squared. If 
   the velocity doubles, the kinetic energy increases by a factor of four.

   The brakes have to do work (exert a force times a distance) to stop 
   the car, that is to convert its velocity and kinetic energy to zero.

   In this problem, you are given that the brakes exert a constant force, 
   independent of speed. If the kinetic energy increases by a factor of 4, 
   the distance has to increase by a factor of 4.

   Mathematically:  
                  KE =  1/2 m (50 km/h) squared = F x 10 m    (given)

   at twice the speed:
                       1/2 m (100 km/h) squared = F x 40 m    (answer).


   It doesn't matter what the force or the mass is.