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Background The pendulum is a classic of physics. It involves the concepts of periodic motion, falling, circular motion, potential and kinetic energy, and centripetal force. It's concepts can be used to explain the g-force felt by a fighter pilot when coming out of a dive, or why you feel the air-time (lift out of your seat) on a roller coaster ride when you go over a hill. Its applications are far reaching, and yet, it is so simple. Anyone can make one, anywhere they are. If you have a strap on your car keys, hold the strap and pull your keys back and voila, a pendulum. The picture depicts a simple pendulum:  The simple pendulum consists of a ball attached to a string, and when let go, it moves back and forth in a motion called periodic motion. Periodic motion is simple motion that is repetitive; in other words, it has a (a) starting point (in this case the ball pulled to the left), (b) an equilibrium position (the point where the ball passes the dotted line), and (c) an opposite point (when the ball is all the way to the right). The picture below has each position marked:  Basic Concepts Before we get to the period of a pendulum, let's take a moment to discuss the motion of a pendulum with regards to energy, displacement, and acceleration. The first thing that is obvious about a pendulum's motion is the fact it stops for an instant at points A and C. This is akin to a ball being thrown straight into the air. When a ball thrown straight up reaches its peak, it stops for an instant, and then begins to fall back. There is a very interesting depiction of this concept in the animated movie Horton Hears a Who. In the movie, a Who catapults himself straight into the air, and is traveling very fast at first, but like all objects, he slows down on the way up. The cool part is where the steps to his house are located: they are precisely at the height where the who hits the peak of his flight. Because he stops for that instant at the top, he can easily just step out onto his steps.
At points A and C, the energy is all potential, because for those two instants, the ball is not moving, so there is no kinetic energy. The Potential Energy of an object is just what it sounds like: it is the energy that is stored, and has potential to do Work. The cool thing about potential energy is it only depends on the height from the equilibrium position: it doesn't matter how it gets to its height (such as in a circular sweep for a pendulum).
 All of that potential energy is converted to kinetic energy on the way down, until it reaches the equilibrium position. It is at that point the potential energy is zero, and the Kinetic energy is at a maximum; furthermore, according to the conservation of energy, the kinetic energy equals the potential energy the ball had at point A.  The graphs of position and velocity The velocity and position of a pendulum share a interesting relationship. If we establish left of the equilibrium position as negative displacement, and right positive as well as motion towards the left as negative and vice versa, we can graph both the position and velocity. Let's start with the position. The picture below shows the graph of displacement versus time.  It looks like a cosine wave, because that's what it is. In fact, this graph could be produced if the pendulum was a cone full of sand, and it was held over a conveyor belt like contraption made with paper. When you release the pendulum, the sand slowly leaks from the cone, and leaves behind the curve you see above as the paper moves at a constant rate. There are a couple of things to note with this graph. (1) notice how the graph moves from the negative position, passes the x-axis, and moves to the positive position, and does so with a smooth curve. The graph below shows the corresponding positions shown in figure 2. It should not be surprising that the graph passes the x-axis at the bottom of the path, position B. Remember that is where the displacement is zero, so it has to be zero on the graph as well. Now, figure 6 only shows the three positions: A, B, and C. However, notice how the graph just repeats itself: after position C, it passes the x-axis again (zero displacement), and returns to the negative displacement (point A).  This is why it's called periodic motion: it just keeps repeating the process, the energy is constantly changing or transforming from potential to kinetic, and as long as there is no air resistance, the pendulum will continue forever. Of course, in the real world, air resistance slows the pendulum down, which means it doesn't make it all the way to it's original height, and in fact with each swing the height it reaches keeps reducing until finally it stops. You might ask where does the energy go? Isn't there something about energy can't be created or destroyed? Is energy destroyed when the pendulum slows to a stop? The answer is a definite no! In fact, energy conservation (another way of expressing the more recognizable "energy cannot be created or destroyed") is a staple of physics: it has never been found to be violated...ever! So, where does the energy go when a pendulum slows to a stop? Simple: it is lost due to heat caused by the friction between the pendulum and the air. Heat is another form of energy, and if we were able to measure the temperature of the surrounding air as the pendulum slowed, we would detect a very slight increase in the temperature. Furthermore, that heat dissipates or spreads out further from the pendulum until it becomes undetectable, but it is still present nonetheless. Every process involves friction and energy loss due to heat. It is interesting to think that lost energy is all around us. In fact, all of the extra energy in the form of heat distributes itself throughout the entire atmosphere. Have you ever heard that in approximately six years, the molecules from one exhale of breath distributes itself evenly around the earth? Energy in the form of heat loss does the same thing. Ok, so back to the pendulum, and we will neglect air resistance (pretend it's in the vacuum of space) and assume it will continue indefinitely. Figure 7 shows the graph of velocity vs. time of a pendulum, and you will notice it is not that different from displacement vs. time. In fact, it is identical in form to the displacement graph in figure 5, only it starts at the x-axis, or zero velocity, which it should since when you pull the pendulum back, it is not moving until you let it go.  Also, notice how the graph curves upwards, and is above the x-axis. This means the velocity values are positive, which again agrees with the pendulum's motion: it does start at negative displacement, but is moving towards the right when it is let go, which is positive velocity. Figure 8 is similar to figure 7 in that is shows the corresponding points: A, B, and C.  Now, here is the payoff for breaking down the motion of a pendulum with these two graphs. With careful analysis, you will notice that the displacement graph and velocity graph for a pendulum are opposites of each other: when one is maxed, the other is crossing the x-axis and vice versa. Figure 9 shows this cool relationship.  I have included a few vertical lines to show how they line up. It should be no surprise that this relationship exists based on our analysis of a pendulum's motion so far, but it is fun to see it, and further shows the beauty of physics and mathematics. There really is nothing more to say for our purposes (of course there is a lot more to say about the pendulum's motion, but that's for another time). Now on to the period! The Proof using Differential Equations The term Period is defined as the time it takes to complete one cycle. Of course, period only applies to periodic motion (hence the term periodic), because if motion doesn't repeat itself in cycles, how can one measure the time it takes to complete one cycle? A lot of proofs in physics can be done without calculus, or more precisely differential equations; however, the period of a pendulum is not one of them (sort of: more on that in a moment). So, how does one know to use differential equations instead of geometry and algebra? It has to do with the variables. If any variable is constantly changing over time (meaning it is not remaining a constant), then one has to incorporate calculus in order to include what is called a rate of change. In the case of a pendulum, the changing variable is the angle made by the string and the ceiling as shown below.  There is an interesting side story to this topic of geometric proofs versus differential equations. Dr. Richard Feynman was one of the most prolific physicists in the 20th century. In the book (which includes the actual lecture on CD) The Lost Lecture Feynman gave a special lecture proving that all orbiting bodies follow elliptical paths using exclusively geometry. The interesting thing is it takes an entire book (the transcribed lecture) to prove this fact, when it only takes three pages of notebook paper to prove the exact same thing using differentrial equations (which will be included in my Special Proofs section soon, the differential equations method...not the geometric method!). The point is, differential equations (referred to for now on as DE) greatly simplify proofs, which might sound absurd since most people have no idea what the subject of DE entails. I assure you though that you can have no idea what DE are, but can recognize certain patterns that allow you to derive things like the period of a pendulum using them, and all without having to take a DE class. Interestingly, there is a way to prove the period of pendulum without DE (which I will show after the DE approach), but it is much longer, and looks more complicated. [Note: The figures will be reset to one, and each equation will be numbered in order to compare the two methods: the one with DE, and the one without] To begin, a force diagram must be drawn as shown in figure 1.  There are a few things worth poiting out. The maroon vector pointing up the string is the actual full tension in the string. The black vector pointing straight down is the pendulum's weight, and it is made up of two parts as shown: one part is the one pulling the pendulum in the opposite direction the string's tension is pulling. Those two cancel, and we know they cancel because the pendulum isn't moving along those two directions. The other part is the one labeled mgsin(theta). That is the one that is not canceled, and is the component of gravity that is pulling the pendulum towards the right. It is that force that we need to remember. Now, we start with the very basic equation of Newton's Second Law:  Now here is where it will get a little tricky. Instead of using a, we are going to write a in it's calculus form. Recall that acceleration is change in velocity over change in time. The other way to look at it is this: velocity is found by taking the derivative of displacement (x), and acceleration is found by taking the derivative of velocity. That means, acceleration is actually the second derivative of displacement. Now all this talk of derivatives is confusing, but all you need to know is they are calculus tools for finding change in distance over change in time for speed, and so forth.  So, this is still F = ma, only written a little more technically. There is a reason for writing it that way, which will become apparent soon. Now, remember we needed to know the second component of gravity.  Since that component of gravity is the net force (the only force acting on the pendulum that doesn't cancel), then we can combine equations 13 and 14:  Now we need to to a little rearranging of terms using substitution of terms. For example, since we are dealing with circular motion (only a partial circle in the case of a pendulum), we can use the relationship between the angle, the arc length distance, and radius shown in equation five.  Technically, the L in equation 5 would be r for radius, but since the radius in this example is the length of the string, we use L. Now, solving for x, we get x = the angle times the Length, and that can be substitued into equation 4.  Because we replaced one changing variable with another (the angle theta in place of x), it becomes part of the second derivative. Since the L is a fixed variable, it can be written with the mass to give equation 7.  NOw all that is left is to move the term on the right side of the equation to the left side of the equation. In other words, add the mgsin(theta) term to both sides of the equation.  There is one more step to this proof, and it may seem like a trick of mathematics, but sometimes such tricks are needed, used, and work when applied correctly. This trick happens to be the fact that the sine of theta for very small angles is approximately equal to theta itself. If you had the chance to read through the tidal forces proof (which is quite daunting compared to this), you will recall that the same kind of trick was used. That's one thing I love about physics: you can make approximations in a proof, and yet, the results still yield real-world applications. Now, at the same time we are going to substitute theta in for sin(theta), we are going to rewrite the second derivative term, which is for another good reason you will see quickly.  So, notice how the second derivitive of the angle was changed to just the angle (theta) with two dots above it. That's another way to express the same thing, and it's quicker. But, for our purposes, it will allow us to see the pattern necessary to extract the period of the pendulum. First, we will divide out by mL, which will leave the theta double dot without a coefficient.  Almost there! Notice how we have the second derivative (the theta double dot) by itself, with just the theta term (with m/L as a coefficient), and it equals zero. This is the differential equation we want, because it holds a special property. First, let me show you the general form of this particualr differential equation.  If you look at equation 11, you can see the x double dot (which just means acceleration), and the x with a k in front. The x is position, and the k can consist of constants, like the g/L in equation 10. The reason for the double dots is so that the whole equation is uniform. The same equation can be written as equation 12.  As you can see, it's not as obvious to spot the form of the differential equation that is necessary, but equation 12 works nonetheless. We'll stick with equation 10 though. Here's the cool thing: when the differential equation is in the form as that shown in equation 11, it is describing periodic motion, and the k (g/L) is all we need to find the period of this particular periodic motion. In fact, we'll see this again in another proof (coming soon) where one falls through the Earth, which elicits periodic motion as well. The k in that equation is different, because the periodic motion is different. OK, so what do we do with the g/L? Observe equation 13 below.  The funny t is called tau, and is the special Greek letter designated for period. Notice how 2pi/tau equals the square root of k, or the coefficient in front of the position variable in the differential equation. For the pendulum, k is g/L. Now, solving equation 13 for tau is all that is left to find the period of a pendulum!  And this is precisely what experiment confirms when period is found using sensitive equipment. (It should be noted that a simple pendulum is simple when dealing with small angles, and gets a little more complicated as the angle of the swing gets larger, but for all practical purposes, even with large angles, this equation for period holds). Ahhh, the beauty of mathematics and how it can describe physical reality! Oh, and it should be noted that the period does not have mass in the equation: only length and g. That means no matter the mass of the pendulum, it will have the same period! The only thing that effects the period is the length of the string (shorts strings yields small periods and vice versa), and the force of gravity. In other words, a pendulum will have a larger period on the moon as compared to the Earth. I'm going to leave that here for now and call it done. I will come back and show the other proof that does not need differential equations. For now though, we will call this one good!!! Timothy K Loper, PhD |