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Guide 8-1. Conservation of Energy Fundamentals

Conservation is the third and last primary category of problem you'll encounter in this course. The other two categories were dvats and net force problems. Of these three types, conservation is the most important.

A. Fundamental Energy Concepts and Relationships

Here are some fundamental concepts needed for solving conservation of energy problems.

  • Forces come in two general types as far as energy is concerned:  conservative and non-conservative. The textbook gives two definitions of conservative force. Review these as needed in section 8-1. For our purposes, the conservative forces that we'll deal with the most this term are gravity and spring forces. In regard to spring forces, it's important to realize that the Hooke's Law force (Fspring = -kΔx) is the conservative one. (It's possible for a spring to obey a different force law that isn't conservative.)  The most common non-conservative force that we'll deal with is friction.

  • Potential energy is just a different way to deal with work. Potential energy is defined in terms of i) a change and ii) the work done by a conservative force. (Non-conservative forces don't have potential energy associated with them.)  If a conservative force does work, Wc , on an object, the object's potential energy changes. The work is the negative of the change in the potential energy:  Wc = -ΔU. Note that potential energy doesn't have meaning except in terms of how it changes.

  • Energy is always associated with a system. Consider, for example, an apple falling to the Earth. The gravitational potential energy is a property of the apple-Earth system. Without both the apple and the Earth, there would be no associated gravitational potential energy.

  • Mechanical energy is a term that means the sum of the kinetic and potential energies of a system. All forms of potential energy are included. For example, if an object is dropped onto a spring, there are 3 forms of energy that change:  kinetic energy of the object, gravitational potential energy, and elastic (or spring) potential energy.

Mechanical energy is conserved when all the forces in the system are conservative. Energy can change forms, but the total mechanical energy doesn't change with time when conservative forces only are present. (We'll see later how to modify this statement when non-conservative forces do work on the system.)

In the last example of an object dropped onto a spring, the system initially has only gravitational potential energy, Ug. As the object falls, some of the Ug is converted to kinetic, K.  As the object strikes and begins compressing the spring, both Ug and K are being converted to elastic potential energy, Ue. When the spring is compressed as far as possible and the object is momentarily at rest, all of the original Ug has been converted to Ue and there is no K.

B. The Conservation of Energy Equation

The goal in what follows is to present and apply a general conservation of energy equation to the solution of problems. We're going to use a more general statement than is used in the text. You need only one equation for all of your conservation of energy problems. Here's the general statement.

Wext = ΔEsys

The first thing to recognize in interpreting this relationship is that it applies to a system. ΔEsys represents all the energy changes within the system, while Wext represents the work done on the system by all forces external to the system. One must specify the system before applying the relationship in order to differentiate between what goes on the left and right sides of the equation.

For convenience, one typically selects for the system for a conservation of energy problem the objects that interact through conservative forces and experience potential energy changes in the system. While this is not required, it's the practice that the textbook author uses; therefore, we will usually follow this procedure.

Before looking at applications, let's see how the general conservation of energy equation above is consistent with that given in the text. One finds in many conservation of energy problems that no work is done by external forces on the system. In that case 0 = ΔEsys. The term on the left is what the textbook author calls ΔE without the sys subscript. We choose to use the subscript to emphasize that the right-hand side of the equation applies to the system. When all the objects in the system interact through conservative forces only, then ΔEsys is simply the mechanical energy of the system; that is, ΔEsys is the sum of the kinetic and potential energies associated with the system. Thus, the following statements in the textbook are all the same as saying "Mechanical energy is conserved when external forces do no work on the system."

Derived equations (assume Wext = 0): E = constant ΔE = 0 Ei = Ef

Throughout the course, always start with the general statement of conservation of energy, namely, Wext = ΔEsys. Do not start with the less general derived equations.

C. Systems and Conservation of Energy

Here's the plan of setting up a conservation of energy problem.

  • Identify the system.
  • Identify the forces acting external to the system.
  • For internal conservative forces, include a term for the potential energy change corresponding to that force on the right-hand side of the equation. Also include a ΔK term.
  • For external forces, include a term for the work done by that force on the left-hand side of the equation.

 

Study but don't memorize the examples below. Memorization doesn't help you in applying conservation of energy to new situations. What you should learn is the method of applying conservation of energy to any situation. You'll learn the method by doing problems that use it.
Example 1. Ball falling vertically in a vacuum

Consider a ball that falls from rest in the absence of air friction. Suppose the goal is to find the velocity of the ball after it has fallen a certain distance. What are the interacting objects? The answer is the ball and the Earth. They interact through gravity, which is a conservative force. The Earth pulls on the ball, and the ball pulls back on the Earth. Therefore, the system here includes the ball and the Earth. Mechanical energy, which includes Ug and K, remains constant for this system, since the only force is a conservative one. Since gravity is internal to this system, and there are no external forces, Wext = 0. ΔEsys is composed of two terms, the change in kinetic energy, ΔK, and the change in gravitational potential energy, ΔUg. Thus, the conservation of energy equation becomes

0 = ΔK + ΔUg.

Click here to see how the energies change with time. The energy of the system is conserved (total system energy remains constant).
Example 2. Ball falling vertically in air

Suppose now that the ball is falling in air. This means there will be a force of air friction. In conformance with the note in the yellow box above, we choose not to include the air in the system, because air friction is not a conservative force. Therefore, the system is the same as that chosen for Example 1. You may wonder how we know that air friction is a non-conservative force. The test to apply is whether the total work done over a closed path is 0. This means that there must be as much negative work done as positive work so that the total work is 0. This can never be the case for air friction. That's because the force of air friction always opposes the motion of the object; therefore, the work done by air friction is always negative. Of course, the presence of air friction means that the ball achieves a lower velocity than it would in falling the same distance in a vacuum. This means that mechanical energy isn't conserved. Wext will be the work done by friction, Wf, and will not be 0. The right-hand side of the equation will remain the same as in Example 1. Therefore, the result is

Wf = ΔK + ΔUg.

Click here to see how the energies change with time. The energy of the system decreases with time as a result of the work done by friction on the system.
Example 3. Block sliding down a frictionless inclined plane

A box slides down a frictionless, inclined plane. Again, the goal is to find the speed of the box after it has slid a particular distance. Interestingly, we need only include the box and the Earth in the system. They interact through the conservative force of gravity. Mechanical energy is conserved for this system. Let's see why the plane isn't included in the system. The plane exerts a normal force on the box, but that force doesn't change the energy of the box. Work must be done on a system to change its energy. The normal force can't do work, because it's always perpendicular to the displacement of the object. We select as the system the block and the Earth. Gravity is a force internal to the system as in the previous two examples. However, the normal force on the block is external. Thus, the work done by the external force is Wn, and we have

Wn = ΔK + ΔUg.

Since the normal force is always perpendicular to the displacement of the block, Wn = 0 and

0 = ΔK + ΔUg.

The result is the same as for Example 1. Note that you would also get the same result if the block were sliding on a curved surface such as the interior of a bowl. The surface still does no work on the block. Perhaps you can begin to see how seemingly different problems are actually the same problem in terms of energy conservation. When you see such similarities, you're really understanding physics. This makes your problem solving more efficient.

 
Click here to see how the energies change with time. The energy of the system is conserved.
Example 4. Block oscillating on a frictionless surface

A block oscillates horizontally across a frictionless surface under the action of a spring. Knowing the block's maximum displacement from the equilibrium position, the goal is to find the speed of the block as it passes through the equilibrium position. What would you pick for the system? The spring force is conservative, so it's convenient to include the block and the spring in the system. We don't need to pick the Earth this time, because gravity doesn't change the energy of the block. We don't include the plane, because the normal force can't change the energy of the block either. For the system of block and spring, mechanical energy is conserved. Consider what is included in ΔEsys. There will of course be a ΔK, since the block is accelerating and experiences a change in kinetic energy. There will also be a change in the elastic potential energy, ΔUe, since the compression of the spring is changing. There will be no ΔUg term, since the block isn't changing height. Therefore, the conservation of energy equation applied to this situation is

0 = ΔK + ΔUe.

 
Click here to see how the energies change with time. The energy of the system is conserved.
An example for you to do. The simple pendulum

Consider a bob swinging in a pendulum motion at the end of a string and in the absence of friction. The goal is to find the speed of the bob after it has swung through a particular arc. What would you pick for the system and why? Would you include the string which exerts tension force on the bob? Would mechanical energy be conserved? Apply the general conservation of energy equation Wext = ΔEsys to the solution. Use the above examples as a guide. Try drawing the energy bars at different times.



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