The Physics of Swinging
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My son begs to watch me swing on one of the swing sets at the park. I tell him that there is so much work involved and I don’t know if I have the energy to do all of the
many things it takes to make a swing move. It’s such an innocent plea, but complicated in the terms of the actual process of it. The physics of swinging has so many components. From resonance to force, and from the period of the swing to the conversion of energy, the process of swinging is actually a complicated matter.
While you watch a person swing, place your hand at the maximum point of the swing’s achieved height and then count how many seconds it takes to return back to that same height. You have just measured the period of the swing. The period of the swing is the time it takes the swing to make one full move back and forth. The equation used to solve for the period mathematically is T = 2p (square root of L/g), where L is the length of the pendulum, and g is gravity. There are a few things that can change the period of a pendulum. As length increases and as the force of gravity increases so will the period. Likewise, when both gravity and length of the chains decrease, the period does also. My reference Mark Nethercott says that if there are no outside influences, the period stays constant at about 15 degrees of arch, but the amplitude must be low. This statement corresponds with Newton’s first law of motion (law of inertia) that says, “Every object remains at rest or in motion in a straight line at constant speed unless acted on by an unbalanced force.” (Physics, A World View p.31).
A force other than gravity and the length of the swing can alter the outcome of a period. While standing with your hand out, measuring the period, give the person on the swing a push.
“Periodic motion is motion that repeats itself at regular intervals of time and resonance is periodic increases of the amplitude of periodic motion due to a force
at a constant interval. So while you push the person on the swing, you are creating a form of resonance for the swing.” –Mark Nethercott.
There is one last force that changes the period of a swing, and that is squatting and standing, or leaning back and forth.
In his article “It Don’t Mean A Thing If You Ain’t Got That Swing” A.G. LeBlanc asks the question “Squatting and standing at appropriate times will result in great motion and terrific heights as any kid will tell you, but why?” This question can be answered, but first the descriptions of certain energies within the operation of a swing need to be defined.
Starting out on the swing, you are at no movement, your feet are just dangling and your hands grip the chains. You are now at the equilibrium of the swing. According to Mark Nethercott, there is only the energy between you, gravity, and the ground. Essentially there is no kinetic energy or gravitational potential energy and you use this point as the starting and stopping position. Nethercott states that “ . . . if you use a force, such as work, and pull the (swing and person) to a point, it then has a potential energy that has mass of object (m) times the acceleration of gravity times the height (h) of the object, from the ground.” This means that the gravitational potential energy equals mgh, where the mass is the weight of the person and leather strap combined, gravity is the pull of the earth and the height is the distance from the ground to the person on the swing. Essentially, gravitational potential energy is defined in Physics, A World View as “the work that would be done by the force of gravity if an object fell from a particular point in space to the location assigned the value of zero…” (p.151). As Mark Nethercott stated, you must perform work on the swing in order to create the gravitational potential energy. Work equals the force multiplied by the distance, or in this case the height (W=fh.)
Once you achieve the gravitational potential energy through the applied work, let go of the swing. Gravity, the ever-acting force, pushes the swing down through the equilibrium and to a point equal in degrees away from the equilibrium, but opposite the point from which it was dropped. The reason the swing moves to this position is that it has the beginning gravitational potential energy, which then converts to kinetic energy. Kinetic energy is the product of one-half the mass (m) times the velocity squared (v^2). KE = 1/2mv^2, where mass is once again the weight of the person and the leather strap combined, and velocity is the distance per time (m/s). The book Physics, A World View states “kinetic energy: the energy of motion. . .” (p151). According to the law of conservation of energy the gravitational potential energy pushing the person toward the earth is converted to kinetic energy as the person reaches the end of the chain’s length and moves upward.
Now A.G. LeBlanc’s answer to his question of why squatting and standing changes the period of the swing. LeBlanc explains that swinging “ . . .can be related to conservation of angular momentum, because this is a radial force and therefore does not create torque, angular momentum stays constant.” Angular momentum is defined as “a vector quantity giving the rotational momentum. For an object orbiting a point, it is the product of the linear momentum and the radius of the path, L = mvr.” (Physics, A World View p.122). In the case of angular momentum (L) and a swing, the mass is still the weight of the person and the leather strap, the velocity (v, usually measured in meters per second) and the radius (r) is the length of the chains to the bottom of the swing. A.G. Leblanc notes that the loss of momentum of inertia translates to a greater velocity, which increases the kinetic energy. LeBlanc goes on to explain that changing your center of mass and the distribution of it by standing at the point of equilibrium you have decreased the radius of the angular rotation and increased your kinetic energy at the point of maximum kinetic energy. Remember that kinetic energy is what moves you and the swing upward, so when kinetic energy is increased you achieve a point higher than the point where gravitational potential energy was highest previously. LeBlanc’s article goes on to clarify that squatting, again, redistributes your center of mass altering the necessary potential energy.
The processes of standing-squatting, pumping, and pushing are actually products of resonance and timing is a huge factor when applying it to swinging. Physics, A World View tells us “The inputs must be given at the natural frequency of the swing. If you push, pump, stand, or squat at the wrong times, your attempts will seem effortless.
When a swing can and can’t operate. As long as there is gravity, the swing will work. If you take a swing to the moon get into it, attempting to swing, you will do so. Although the force of gravity isn’t as strong, in fact only 1/6 what it is on earth, it is still present. Looking at the equation of gravitational potential energy, GPE = mgh, note that gravity is a constant in that equation and even though it is micro-gravity on the moon there is still motion. Because the gravity is smaller, the gravitational potential energy will be smaller, hence the kinetic energy, and the period of the swing will decrease as well. Where won’t a swing work? Where there is no gravity. If GPE = mgh and g = 0, then the entire equation totals zero. Gravity is the ultimate mover of the swing without it; there is no swing!
At the thought of explaining this to my two year old son, I looked down into his big brown eyes and walked over to sit in the leather strap of the swing. I guess it could
be said that it takes a lot to make one little boy happy. From changing my center of mass and determining the timing for resonance, to establishing the gravitational potential energy and the change to kinetic energy, there’s a lot of work in such a small occurrence.
LeBlanc, A.G. “It Don’t Mean A Thing If You Ain’t Got That Swing” -http://aci.mta.ca/TheUmbrella/Physics/P3401/Investigations/RealSwin
Kirkpatrick, Wheeler. Physics, A World View. Orlando; Harcourt, 2001.