Day 34

Context

Having looked at several topics we now turn our attention to conservation of energy

Explanation

Energy can be divided into two very general categories: Kinetic energy and potential energy. The kinetic energy is associated with motion and is sometimes called the energy of motion.

There are several kinds of potential energy, but they are associated with some kind of force and are often called stored energy. Electrical force can result in electrical potential energy (like energy stored in a battery). A stretched spring provides an inward force that can result in stored energy inside of the spring. We know that the spring has stored energy because if we let go of it, it will contract toward the center of the spring. The force of gravity can result in gravitational potential energy. When an object is held up it has stored energy. Let go and it falls toward the ground due to the gravitational force.

We will focus on the gravitational potential energy.

After this class you should be able to:

• Define kinetic and potential energy.
• Express the gravitational potential energy mathematically.
• Show the relationship between kinetic and potential energy.
• Solve problems using conservation of energy.

Conservation Laws

Most people remember some version of the following statements as a statement of a conservation law. The example will be for energy, but it could be mass, momentum, or anything else that is conserved.

1. The total amount of energy in the universe is constant.
2. Energy can't be created or destroyed, it can only change form.
3. The total amount of energy you start with must equal the total amount of energy you end up with.

In this class we will focus on the last statement

Mathematical Model

Work and energy are intimately related; in fact, energy is often defined as the ability to do work. The law of conservation of energy, in classical mechanics, simply states that the initial kinetic energy plus the initial potential energy is equal to the final kinetic energy plus the final potential energy plus W. If other forms of energy are produced (heat, sound, light, etc.), they are incorporated into W. The mathematical model for conservation of energy is:

KE_i + PE_i = KE_f + PE_f + W

As stated earlier, the gravitational potential energy, which will be used here for potential energy, depends on the mass, gravity, and the height:

PE = mgh

where m is the mass, g is the acceleration due to gravity (9.8 m/sec²) and h is the height. The initial potential energy is obtained by using the initial height, h_i, and the final potential energy is obtained by using the final height, h_f. The position where h equals zero is an arbitrary choice. Since there is no absolute zero for height the change in position and the change in potential energy are the only measurable quantities. The assignment for h = 0 is usually at the lowest position.

Example Problem #1

If a 2 kg ball is dropped from rest at a height of 4 m above the floor, how much kinetic energy will it have at the instant that it is at a height of 3 m? Assume that no heat or other energy is generated during the fall.

The initial potential energy, PE_i = (2 kg)(10 m/sec²)(4 m) = 80 J. The initial kinetic energy, KE_i, will be zero because it starts from rest and isn't moving.

The final potential energy, PE_f = (2 kg)(10 m/sec²)(3 m) = 60 J.

Putting it into the conservation of energy equation:

KE_i + PE_i = KE_f + PE_f + W

0 + 80 J = KE_f + 60 J + 0

Solving for KE_f: KE_f = 80 J - 60 J = 20 J. The ball would have a kinetic energy of 20 J at the instant it is 3 m above the floor.

Example Problem #2

If a 2 kg ball is dropped from rest at a height of 4 m above the floor, how much kinetic energy will it have at the instant that it is at a height of 2 m? Assume that no heat or other energy is generated during the fall.

The initial potential energy, PE_i = (2 kg)(10 m/sec²)(4 m) = 80 J. The initial kinetic energy, KE_i, will be zero because it starts from rest and isn't moving.

The final potential energy, PE_f = (2 kg)(10 m/sec²)(2 m) = 40 J.

Putting it into the conservation of energy equation:

KE_i + PE_i = KE_f + PE_f + W

0 + 80 J = KE_f + 40 J + 0

Solving for KE_f: KE_f = 80 J - 40 J = 40 J. The ball would have a kinetic energy of 40 J at the instant it is 2 m above the floor.

Example Problem #3

If a 2 kg ball is dropped from rest at a height of 4 m above the floor, how much kinetic energy will it have at the instant that it is at a height of 1 m? Assume that no heat or other energy is generated during the fall.

The initial potential energy, PE_i = (2 kg)(10 m/sec²)(4 m) = 80 J. The initial kinetic energy, KE_i, will be zero because it starts from rest and isn't moving.

The final potential energy, PE_f = (2 kg)(10 m/sec²)(1 m) = 20 J.

Putting it into the conservation of energy equation:

KE_i + PE_i = KE_f + PE_f + W

0 + 80 J = KE_f + 20 J + 0

Solving for KE_f: KE_f = 80 J - 20 J = 20 J. The ball would have a kinetic energy of 60 J at the instant it is 1 m above the floor.

Example Problem #4

If a 2 kg ball is dropped from rest at a height of 4 m above the floor, how much kinetic energy will it have at the instant that it is at a height of 3 m? Assume that 1 J of heat or other energy is generated for every 1 m that the ball falls. (e.g. If the ball falls 2 m, 2 J of heat would be produced.)

The initial potential energy, PE_i = (2 kg)(10 m/sec²)(4 m) = 80 J. The initial kinetic energy, KE_i, will be zero because it starts from rest and isn't moving.

The final potential energy, PE_f = (2 kg)(10 m/sec²)(3 m) = 60 J.

In this case the ball will have fallen from 4 m to 3 m or a total of one meter. The problem states that 1 J of heat is lost for every 1 m that the ball falls, so W = 1 J.

Putting it into the conservation of energy equation:

KE_i + PE_i = KE_f + PE_f + W

0 + 80 J = KE_f + 60 J + 1 J

Solving for KE_f: KE_f = 80 J - 60 J - 1 J = 19 J. In this case the ball would have a kinetic energy of 19 J at the instant it is 3 m above the floor.

Example Problem #5

If a 2 kg ball is dropped from rest at a height of 4 m above the floor and bounces back up to a height of 3.5 m, how much energy was lost during the collision with the floor? Assume that no heat or other energy is generated while it is falling or while it is coming back up. In other words, W only comes from the collision with the floor. When doing this problem the initial conditions will be when the ball is dropped and the final conditions will be when the ball has risen to its maximum height after colliding with the floor.

The initial potential energy, PE_i = (2 kg)(10 m/sec²)(4 m) = 80 J. The initial kinetic energy, KE_i, will be zero because it starts from rest and isn't moving.

The final potential energy, PE_f = (2 kg)(10 m/sec²)(3.5 m) = 70 J. The final kinetic energy will be zero at the maximum height. Note: It stops for an instant at the maximum height.

Putting it into the conservation of energy equation:

KE_i + PE_i = KE_f + PE_f + W

0 + 80 J = 0 + 70 J + W

Solving for W: W = 80 J - 70 J = 10 J. The ball would have generated 10 J of heat or other energy during its collision with the floor.