Summary for Chapter 7: Energy
Kinetic Energy
Kinetic energy is energy a object has because it is moving. A mass M moving with a speed v has a kinetic energy
A mass of 10kg moving with a speed of 10m/s has a kinetic energy of 500kgm2/s2 = 500J. Note that a Joule, J, is defined as 1 kgm2/s2. If the speed is increased to 20m/s the KE becomes 2,000J, so doubling the speed quadruples the KE.
Work
When a force, F, acts on an object as the object moves a distance d, the force will do
work, W=F´d if
the force and the displacement, d, are in the same direction. If the force and
displacement are in
W=F´d (parallel)
W = – F´d (opposite directions)
W = 0 (perpendicular)
When several forces act on a moving object the total work done is the sum of the individual works, which is the same as the work done by the total or net force. If one force does +10J of work and the other –3J, the total work is +7J.
Work and Kinetic Energy
The net or total work done on an object will change its kinetic energy y the relation
Wnet = Change in KE
If I do + 7J of net work on an object I increase its KE by 7J. If I do –7J of net work on it I will decrease its KE by 7 J.
If I throw a ball upward, as it rises gravity does negative work on it and its KE decreases, i.e. it slows down. On the way down gravity is doing positive work and its KE increases, i.e. it speeds up.
We transfer energy to an object when we do positive work on it and take energy away from it when we do negative work on it. We transfer energy by doing work.
Potential Energy
Sometimes there is a potential for a force to do work on an object because of its position. A ball resting on a table y meters above the floor has the potential for gravity to do work on it if it falls to the floor. Because gravity has the potential to do work on it, i.e. transfer energy to it, we say is has a gravitational potential energy. In this case the potential energy is the work gravity will do on it.
The gravitational potential energy of a mass on the earth is its mass times g, times the height, or Mgh, relative to the earth's surface, because the ammount of work gravity will do on a mass M as it falls a distance h parallel to the force is F×h = Mg×h. If I have a 3kg mass that is 6m above the earth's surface, it's potential energy is 3kg × 10m/s2 × 6m = 180kgm2/s2 = 180J. This means that gravity will do 180J of work on it if it falls to the ground.
What if it only falls 2m? If it falls 2m, its height is 6m - 2m = 4m and its P.E. (potential energy) = 3kg × 10m/s2 × 4m = 120kgm2/s2 = 120J. It is 60J less, because gravity did 60J of work (of the 180J of possible work) on it. It is sort of like a bank account. If you have $180 in your savings account it represents a "potential" money. If you withdraw $60 and put it in your wallet, you have "realized" some of that potential and there is only $120 left in the account. So if gravity does 20J of work on an object its gravitational potential energy decreases by 20J.
If a force has a potential energy to describe the work it does, then the work done by that force = minus the change in Potential Energy, or
W = - (Change in PE)
If the only force doing work on an object has a potential energy to describe it, then the quantity
KE + PE = Etotal
stays constant. The Quantity (KE + PE) is usually called the total or mechanical energy of the object, and it's given the symbol E. (Sometimes it is just called the Energy.) It has this energy by virtue of its motion (i.e. KE) and its position relative to some other object that exerts a force on it and can do work on it (i.e. PE). A quantity that does not change is said to be conserved, so we would say that the objects energy is conserved if the only force or forces doing work on it are included in the potential energy.
Example 1:
If I hold a 2kg mass at the top of a cliff, 5m above the ground, it has a PE = 2kg × 10m/s2 × 5m = 100J. If it's not moving, its KE = 0J. What happens if I drop it? If I can neglect air resistance, gravity is the only force doing work on it as it freely falls, at least until it hits the ground. Since the work done by gravity is accounted for in the PE, the quantity
E = KE + PE
will stay constant. In this case it starts out at E = 0J + 100J (since the KE = 0J at the instant I release it and its PE is 100J at that instant.) As it falls the PE will decrease, since its height is decreasing. To keep PE + KE = 100J, the KE must increase. Half way down, at h = 2.5m, its PE = 50J, so the KE = 50J to keep KE + PE = 100J. Just before it hits at the bottom, the height is almost zero, so the PE is almost 0. Therefore the KE has to be almost 100J at that instant.
Example 2:
Consider that same 2kg ball at the top of the 5m cliff. What happens if I throw it upward with a speed of 5m/s? If it's 5m above the ground when I release it, its PE = 100J again, but its KE = (1/2) × 2kg × (5m/s)2 = 25J. Therefore its total energy is
E = PE + KE = 100J + 25J = 125J.
If I can neglect air resistance, its total energy with stay 125J until it hits the ground and the ground does work on it. Since I threw it upward, it will rise until it stops, but only for an "instant", and then fall back down. When it reaches the top, its speed is 0, so its KE = 0. This means that its PE = 125J, all the energy is PE. { Since PE = Mgh, you should be able to show that the height at that instant is 6.25m.} When the ball reaches the ground, i.e. just before it hits the ground, h = 0m again and all the energy is KE, so the KE = 125J.