11.1.7.1: Illustrations

Illustration 1: Choice of System

The animation represents a ball sliding on a curved wire (position is given in meters, time is given in seconds, and energy on the bar graph is given in joules) subject to the forces of gravity, the normal force, and friction. Note that the wire does not depict the potential energy function of the ball (see Illustration 7.3 for the Illustration on Potential Energy Diagrams). There are also three bar graphs that accompany the animation. They represent the kinetic energy (orange), gravitational potential energy (blue), and the energy dissipated due to friction (red). The two animations represent two different systems in which to analyze the motion via energy. Restart.

First play Animation 1. Note that in this animation there is no potential energy due to gravity and no energy dissipated due to friction. How can this be? Well, in this case we have chosen the system to be just the ball. Animation 1: show system. As a consequence, the system is not isolated because the ball experiences an external force due to gravity in addition to the external dissipative force of friction. Gravity does positive and then negative work on the ball, changing the ball's kinetic energy. In addition, the force of friction dissipates energy by doing a negative amount of work on the ball.

Now play Animation 2. What is going on here? What is the system now?  Here there is potential energy due to gravity as well as energy dissipated due to friction. The system includes Earth and the room, and therefore the total energy must include gravitational potential energy and the frictional energy. Animation 2: show system. Given that we have defined a system that includes Earth and the room, the total energy (found by adding up all three bar graphs) should stay constant.

Illustration 2: Representations of Energy

There are many different ways to represent motion (as we have already seen). The same is true for energy. For example we can represent an object's kinetic energy using a graph of kinetic energy vs. time, a bar chart of kinetic energy that changes with time, or as a value in a table that changes with time (position is given in meters, time is given in seconds, and energy on the bar graph is given in joules). All three representations give us the same information, just in different forms. So why might we want to use a different representation? Well, it depends on what concept we are trying to illustrate and which representation gets to the heart of that concept. The collision occurs at a distance due to magnets on the front of each cart that allow the carts to collide without touching. Restart.

The graph shows us an instant-by-instant accounting of the kinetic energy of each cart. This is important if we wish to analyze every detail of the kinetic energy of the carts involved in the collision. Usually we are interested in whether or not energy is conserved in a given collision. In this case a graph gives this information, but it also gives us much more information. Notice that during the collision (kinetic) energy appears to be missing! This energy must be accounted for, so where is it? It is temporarily stored by the magnets attached to the carts. If there was a spring in between the carts, the energy would have been temporarily stored there instead. This energy is then transferred back to the carts by the end of the collision. This is why we compare the kinetic energies before the collision to those after the collision and often do not attempt to analyze the details of the collision itself.

Another way to answer the question of energy conservation, therefore, is with the bar chart (it is color coded) or with a table (the values are labeled). We simply compare the values—either the size of the bars or the values from the table—from before and after the collision. Are they the same? If yes, the energy, specifically the kinetic energy, of the two-cart system is conserved.

Illustration 3: Potential Energy Diagrams

A large \(2\text{-N/m}\) spring is shown attached to a \(1\text{-kg}\) red ball that is initially displaced \(5\text{ m}\) (position is given in meters, time is given in seconds, and energy on the bar graph is given in joules). The total energy and the potential energy are shown in the graph. Two bar graphs that depict the kinetic and potential energy are also shown. Finally, the values of the energy are shown in the table. Restart.

The potential energy diagram is an important diagram because it depicts the potential energy function, often just called the potential. This terminology is unfortunate since it can lead to confusion with the electric potential. The potential energy function is plotted vs. position, and therefore it tells you the potential energy of an object if you know its position. The potential energy function for a mass on a spring is just \(PE(x) = 0.5\ast k\ast x^{2}\). Here \(PE(x) = x^{2}\). Note that, depending on your text, you may have seen the potential energy function represented as either \(V(x)\) or \(U(x)\). We use the book-independent version \(PE(x)\). In addition to the potential energy function, a horizontal teal line represents the total energy of the system.

Because of the form of the above potential energy function, it is easy to get confused as to what it is actually showing and what it represents. If you have not done so already, run the animation. The red dot on the potential energy curve does NOT represent the actual motion of a particle on a bowl or roller coaster. In other words, it does NOT represent the two-dimensional motion of an object. It represents the one-dimensional motion of an object, here the one-dimensional motion of a mass attached to a spring. The motion of the red mass is limited to between the turning points represented by where the total energy is equal to the potential energy.

Now also show the kinetic energy on the graph. Watch the kinetic energy and potential change as the mass moves and the spring ceases to be stretched and then gets compressed. Notice that the potential energy added to the kinetic energy always adds up to the total energy. Therefore, if you know the total energy and the potential energy function, you know the kinetic energy of the object at any position in its motion.

Clearly, the force depicted is a spring force. How can we be sure? Well, there is a relationship between the force and the potential energy function. This relationship is expressed as \(F_{x} = - d (PE)/dx\). Therefore, since \(PE(x) = x^{2}\), \(F_{x} = - 2 x\), which tells us that \(k = 2\text{ N/m}\) (as stated in the first line of the Illustration).

Illustration 4: External Forces and Energy

When we talk about energy we tend to focus on the change in kinetic energy and change in potential energy, where the change in potential energy is the negative of the work done by conservative forces. But what happens with nonconservative and external forces? (Note that some books lump external forces with nonconservative forces.) Well, these are the forces that cause the total energy of the system to change. In other words, without a nonconservative force or an external force, the total energy would never change. This is what we mean by the statement of energy conservation, \(\Delta KE + \Delta PE = 0\).

If there are nonconservative or external forces, the total energy will change. When we say nonconservative forces we usually are thinking about kinetic friction. Kinetic friction is a special force that always decreases the total energy of the system (the amount of work that it does on an object is always negative). If friction exists in a system, and you wait long enough, all of the energy will dissipate. What about external forces? Do they add or take energy from the system? Well, it depends.

Consider the cart in the animation. The cart interacts with the two-handed image if the image is near the left-hand or right-hand end of the cart (position is given in meters and time is given in seconds). The arrow below the cart shows the direction and strength of the external force applied. Reset the animation if the cart goes off the end of the track. Restart.

Move the cart around and look at each of the graphs. Now focus on the |F| cos(theta) vs.position graph, which tells you about the work done by the external force (the hand). Is it always positive or is it always negative? It can be positive or negative depending on the circumstances. If the work done by the force is positive, the energy of the system (the cart and Earth) increases. Since the potential energy of the cart remains fixed (since it does not change height), all of this energy is seen as kinetic energy. If the work done by the force is negative, the energy of the system decreases. Again, the change in energy is seen as kinetic energy.

Illustration 5: A Block on an Incline

A block is on an incline and slides without friction. Partway down the incline, it hits a spring as shown (position is given in meters, time is given in seconds, and energy on the bar graph is given in joules). You can add the protractor by checking the box. Also shown are the force vectors, one for each force (the red ones) and one for the total force acting (the blue one). The energy of the system is shown in the three bar graphs on the right: kinetic energy (orange), gravitational potential energy (blue), and elastic potential energy (green). Restart.

Let's begin by analyzing this situation as we would have in Chapters 3 and 4. First, we need to define a convenient set of axes. A convenient set of axes has one axis along the incline and the other axis perpendicular to the incline. This choice allows us to have one direction where there is no acceleration (the direction perpendicular to the incline) and one direction where there is an acceleration (parallel to the incline). There is also another reason for this choice of axes. It allows us to decompose only one force instead of two. We have to decompose the gravitational force into a component along the incline and one perpendicular to the incline. How do we deal with the spring force? Well, the honest answer is that while we can analyze the forces to determine the acceleration, it is not tremendously useful since the spring force is not constant.

Run the animation and look at the normal force and the gravitational force vs. the spring force. The spring is not compressed initially, then it compresses, and then it uncompresses. During this time the net force on the block changes dramatically. Look at the blue net force vector. As a consequence, the acceleration of the block changes dramatically as well. (Note that the net force still points parallel to the incline; its size is what changes dramatically.)

Since the forces change over the course of the motion of the block, the acceleration of the block is not constant throughout the motion of the block. Newton's laws and kinematics clearly fall short in analyzing the motion here. What to do? Use energy! At the starting point of the motion of the block, it has no kinetic energy, and no elastic (spring) potential energy, but it does have gravitational potential energy. As the block moves down the incline some of the gravitational potential energy is converted to kinetic energy. When the block hits the spring, the kinetic energy and the gravitational potential energy get converted to elastic (spring) potential energy.

Watch the animation and describe how all of the potential energy due to the compressed spring gets converted to other types of energy.

Illustration authored by Mario Belloni.
Script authored by Steve Mellema and Chuck Niederriter and modified by Mario Belloni.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.