Lesson 27--Potential and Kinetic Energy

Potential and Kinetic Energy

Last week we discussed the significance and the origins of kinetic energy. Hopefully, you came to understand the reason we define kinetic energy as the quantity 1/2mv2. After all, it is just as easy to calculate 5mv6 or 3m2v and you could do that for any object. Remember, the quantity 1/2mv2 comes as a direct result of Newtonís 2nd law. The reason we use the 1/2mv2 relation as opposed to the others is because any force applied over some distance causes a change in that quantity which is also equal to the product of the force and the distance. Only 1/2mv2 can make such a claim.

The fact that a force applied over a distance creates predictable changes in kinetic energy means that some forces have the potential to create kinetic energy.

For example, if you lift Mr. Ochs 5 meters above the ground, the force due to gravity can only cause Mr. Ochs to go so fast by the time he hits the ground if you released him from that height. Essentially, if you were to let him go, then the force due to gravity would do a certain amount of work.

If you charge up the Van de Graaf generator with a positive charge and hold a positively charged toy poodle near it, the repulsive force between the two positively charged objects will cause the poodle to gain a certain amount of kinetic energy over a certain distance (the Van de Graaf generator would also gain KE, since every action has an equal and opposite reaction). Since the force between the generator and the poodle can be determined, then you could predict how much positive work the electrostatic force could potentially do to the poodle.

Letís say you have a spring-like bungie cord that you tie to a brick wall. You put your little brother, sister, or physics teacher in the middle of the cord and winch them back, stretching the cord as shown in the picture below. Although you would never do this, if you were to release the winch they would be propelled forward by a given force over a given distance. That is, the bungie cord would have the potential to do a certain amount of positive work on your physics teacher until it smashed me into the wall.

Regardless of whether you release any of these objects in the previous examples, once placed in that position the force on the object under consideration has the potential to do a certain amount of positive work. When objects are in positions where a force is present that could cause positive work that object is said to have POTENTIAL ENERGY. Simply stated, the potential energy of an object is the amount of kinetic energy that an object can possibly gain.

Not any old force is capable of giving an object potential energy. The force must be able to cause the speed of an object to increase without the presence of any other force.

So, letís think about which forces can do this by listing forces that we know and citing an example of how it could, or explaining why it couldnít create potential energy.

Gravity--Yup, if I pick something up and then let go of it, the object will fall and have its speed increase.

Electrostatic--Yes, lightning. Electrons move from an area of excess negative charge to a region of excess positive charge due to the repulsion of the net negative charge and the attraction of the net positive one.

Magnetic--Yes, hold a piece of iron near a strong magnet. When you let go of the iron it will move toward the magnet, increasing its speed and kinetic energy in the process.

Springs--Yes, push down on the front of your car and let go. Once you let go the springs cause the front of the car to start moving.

Friction--No, friction acting by itself on a sliding object causes that object to slow down.

Tension--No*, in order to create tension there must be something else on the other end on the rope.

*one might argue that on the microscopic level the bonds behave like springs. However, this argument is best saved for grad school.

Normal--No, it only exists because of other forces and it typically acts perpendicular to the motion of the object.

Drag--No, it always opposes the motion of an object. Consequently, it is only capable of providing resistance.

Lift--No, it acts perpendicular to motion of an object.

Forces, then, sort themselves into two categories, those that can create positive changes in kinetic energy and those that canít. The ones that can are deemed conservative forces and the ones that canít are non conservative forces. These two categories of forces have farther reaching implications is association with energy.

When only conservative forces act on an object, all of the energy of the system will remain in some combination of the two forms: kinetic or potential. This ought to make sense. Since potential energy represents the amount of work that a conservative force (or kinetic energy that an object can gain) would do where no other forces are present, then, in this case, the potential energy can only be transferred to kinetic and no other form.

For non conservative forces, the energy of the system does not remain as either potential or kinetic. Rather as energy is being transferred from potential to kinetic or back, a non conservative force transfers some of that energy out of the system. Usually, the transfer process is heat and it goes into changing the internal energy of the objects and surrounding.

Hopefully, at this point, you understand the basis for each form of energy. The next step is to develop representative equations for the potential energy expressions of different forces.

Now that we've discussed potential energy conceptually. Let's look at how it should be treated mathematically. We already understand that potential energy is related to the work that a conservative force could do. Mathematically we would write this as:



In this case U is the symbol for potential energy. The rest of the equation is exactly what you have seen before only with a negative sign. So, how does this help us determine the potential energy of an object.

Let's start with something relatively simple--the potential energy near the surface of the earth. You may remember from last year that the equation was mgh, let's see if we can show this to be true using the equation shown above.



However, potential energy exists in more places than near the Earth and has more causes than gravity.

For example, springs can have potential energy and they don't even have to be on Earth. I remember once, when I was visiting Slartibartfast while drinking a pan galactic gargle blaster . . .. Well, let's just say that I believe there could be springs on other planets. Using a similar approach to the one we just used to find the potential energy near the surface of the Earth, we can determine the potential energy an object has when it is attached and stretched (or compressed) some distance from the equilibrium position of the spring. Once again we start with the calculus definition of potential energy.



Another term for potential energy is "elastic energy." It means the same thing, but no one seems to bother explaining themselves when they use it. Since conservative forces create potential energy and conservative forces only transfer energy from kinetic to potential or vice versa, then no mechanical energy is lost. Which is elastic behavior, hence potential energy is referred to as elastic energy. Or, at least that's what I have reasoned.

As we have discussed earlier, gravity exists throughout the universe. Therefore, the equation for gravitational potential energy that we found earlier has limited applications. Instead, we will use Newton's universal gravitational force to determine the potential energy some distance away from a massive object. Here, we will employ a trick. Since most objects have different dimensions, defining the potential energy as zero at the surface would mean that zero would vary significantly from object to object. The one point that is a common distance away from all objects is infinity. So rather than lift a mass off the surface to some point, we will bring the mass in from infinity to the same point. The result will startled and amaze you.

Once again, we begin the definition of potential energy.



How can the potential energy be less than zero? Remember that potential energy is a function of position and where you declare the position to be zero is up to you. However, once you choose that position, you must remain consistent. Therefore, points on one side of that position will be negative and on the other side, positive. The consequence is that you can have negative potential energy and negative total energy due to the arbitrary choice of what position you define as zero. Universal gravitational potential energy is a perfect example. Since we defined infinity to have zero potential energy, then everything is going to have negative potential energy, because it is very difficult to get farther away than infinity from an object.

A good follow up question is: Can kinetic energy ever be less than zero? I leave this for you to ponder for a while.

Potential energy also appears in the electromagnetism realm. In both electric fields and in circuits, charges may be positioned such that they have potential energy. Rather than derive their expressions now, I'll just give you the equations.



Regardless of the origin of the potential energy, all expressions represent the potential ability of a conservative force to perform work. Knowing how to calculate potential energy is one thing, but applying it is quite another. When using energy considerations to solve problems, you typically consider two points and compare the change in kinetic and potential energies from one point to the other. The neat thing is, where conservative forces are only considered, the path taken between the two points is irrelevant. Meaning you only need to look at the starting and ending positions.

This is frequently referring to as path independence. So, even if you know nothing about the nature of the force acting on an object, you can tell whether or not the force is conservative by the amount of work done by a force going between two points along different paths. If the amount of work differs, then the force acting on the object is not conservative. However, should the amount of work be the same, you do not necessarily know that you have conservative forces acting. It is possible that a nonconservative force could do equal amount of work along two different paths. However, if you make a round trip (return to the initial point), the work done by a conservative force will be zero. That cannot happen for a nonconservative force.

Example:



Since no energy is removed from the object under consideration in conservative problems, then any change in kinetic energy is offset by an equal and opposite change in potential energy. Therefore, the following equation holds in situations where only conservative forces act:



Mechanical Energy

Kinetic and potential energies comprise what is called mechanical energy.The sum of kinetic and potential energy, then, is the total mechanical energy of a system. As stated previously, in conservative systems the sum of the changes in kinetic and potential energy is zero. Consequently,for a conservative system, then the total mechanical energy doesnít change--that is, if the total energy started at 5 J, it is 5 J half way through, and 5 J at the end. For that matter, it is 5 J anywhere in between. Thus regardless of the position, the total energy would always be 5 J for this example. This fact allows us to determine the kinetic or potential energy at another point easily, provided we know the total energy and either the kinetic or potential energy.

Examples:





Rotational Kinetic Energy I should have discussed this in the previous lesson, but I didn't. If it is any consolation, it will be in the previous lesson next year. In the previous lesson we only considered translational kinetic energy (for point to point motion). However, a torque applied to an object through some known angle will produce a predictable change in angular velocity. That is, work can be done to a rotating object changing the rotational kinetic energy of that object. Thus, when analyzing problems using conservation of energy methods, you'll need to consider whether the object is merely translating or rolling as well. Any time an object is rolling or rotating as well as translating, you will need to consider rotational kinetic energy.

The expression for rotational kinetic energy is similar to the expression for translational. In fact, if you remember that the rotational equivalent of mass is moment of inertia (I) and the equivalent of velocity is angular velocity (w), then the expression is basically the same. Or,





Factoring in rotational kinetic energy should not change your mindset when solving problems using conservation of energy. You should remember that the you still have path independence and that mechanical energy is composed of kinetic and potential energies. In conservative systems, whatever is lost by one is gained by the other, so the total must always remain the same. However, when you add in rotational kinetic energy you may also transfer between the two types of kinetic energies.

PROBLEMS:

1. What does it mean for something to be said to have potential energy?


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2.Does the path an object takes influence the amount of kinetic energy it gains if only conservative forces act on that object? What about the rate at which it gains KE?


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3.How does work relate to potential energy?


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4.How do conservative and non conservative forces differ?


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5.What does it mean to say that potential energy is the energy of position?


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6. How is it possible to have negative total energy?


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7.Is it possible to have negative potential energy? kinetic energy? Explain each case.




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8.While bungie jumping from a 40 meter high bridge Serena (m = 50 kg) notices that the bungie cord is 20 meters long when unstretched. Assume the bungie cord acts like a spring whose k = 200 N/m.

a) How fast is she going when she just barely starts stretching the bungie cord?

b) How far below the bridge is Serena when she has stretched the bungie cord such that the net force on her is zero?

c) Justify whether you believe she is moving up, down, or is stopped at this point.

d) Does Serena hit the ground?




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9.What is the kinetic energy of a 3 kg sphere of radius 0.4 meters that is spinning at 8 rad/s?


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10.What is the difference between translational and rotational kinetic energy?


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11.A ball rolls down an incline from a height of 8 m to a height of 3 m, what is the speed of the ball as it reaches the bottom of the incline?


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12.An electron (m = 9.1 x 10-31 kg, q = - 1.6 x 10-19 C) jumps from the negative (-12 Volts) to the positive plate (0 Volts) of a parallel plate capacitor.

a) If it does not experience any other forces, how fast will it be going as it reaches the positive terminal?

b) The positive terminal is 10 cm from the negative terminal how long does it take the electron to move between the terminals--assume that the acceleration is constant.

SS
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13.How fast must an object be shot to escape the earthís gravitational pull if launched from space 200 km above the earth's surface? The earth's mass is 5.98 x 1024 kg and its radius is 6.37 x 106 m.


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14.An object starting at a height of 100 m has 1000 Joules of Potential Energy and zero Joules of kinetic energy. State the relative amounts of potential and kinetic energy at the heights listed below:

a) h = 90 m
b) h = 50 m
c) h = 20 m
d) h = 12.7 m
e) h = 0.001 m



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15.From the diagram shown below, determine the relative amounts of kinetic and potential energies, given the total energy is 2 J and only conservative forces act on the object described by the graph. The line describes the various values of what the potential energy would be if you could be at that position. Also, state whether it is possible to reach the indicated location.



a. x = 0m

b. x = 2m

c. x = 4m

d. x = 8m

e. x = 10m
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