Back To CourseAP Physics 1: Exam Prep
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David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.
Simple harmonic motion is any motion where a restoring force is applied that is proportional to the displacement and in the opposite direction of that displacement. Or in other words, the more you pull it one way, the more it wants to return to the middle. The classic example of this is a mass on a spring, because the more the mass stretches it, the more it feels a tug back towards the middle. A mass on a spring can be vertical, in which case gravity is involved, or horizontal on a smooth tabletop.
If you imagine pulling a mass on a spring and then letting go, it will bounce back and forth around an equilibrium position in the middle. Like with all simple harmonic motion, the velocity will be greatest in the middle, whereas the restoring force (and therefore acceleration) will be greatest at the outside edges (at the maximum displacement). Another example of simple harmonic motion is a pendulum, though only if it swings at small angles.
There are many equations to describe simple harmonic motion. The first we're going to look at, below, tells us that the time period of an oscillating spring, T, measured in seconds, is equal to 2pi times the square-root of m over k, where m is the mass of the object connected to the spring measured in kilograms, and k is the spring constant (a measure of elasticity) of the spring. The time period is the time it takes for an object to complete one full cycle of its periodic motion, such as the time it takes a pendulum to make one full back-and-forth swing.
All simple harmonic motion is sinusoidal. This can best be illustrated visually. As you can see from our animation (please see the video at 01:34), a mass on a spring undergoing simple harmonic motion slows down at the very top and bottom, before gradually increasing speed again as it approaches the center. It spends more time at the top and bottom than it does in the middle. Mathematically, any motion that has a restoring force proportional to the displacement from the equilibrium position will vary in this way.
Because of that, the main equation shown below is shaped like a sine curve. It says that the displacement is equal to the amplitude of the variation, A, otherwise known as the maximum displacement, multiplied by sine omega-t, where omega is the angular frequency of the variation, and t is the time. This displacement can be in the x-direction or the y-direction, depending on the situation. A vertical mass on a spring varies in the y-direction sinusoidally. A horizontal mass on a spring varies in the x-direction sinusoidally. A pendulum has such a variation in both directions.
This equation has a sine in it, and a sine graph starts at zero. Using this equation is like starting your mathematical stopwatch in the middle of a pendulum swing: t = 0 is in the center of the oscillation. If, on the other hand, you replace sine with a cosine, then the equation is still correct; you're just starting to measure time at the maximum displacement instead.
But we also need to define angular frequency. Angular frequency is the number of radians of the oscillation that are completed each second. A full 360 degrees is 2pi radians, and that represents one complete oscillation: from the middle, to a fully stretched spring, back to the middle, to a fully compressed spring and then back to the middle again. You can convert angular frequency to regular frequency by dividing it by 2pi. Regular frequency, f, just tells you the number of full cycles per second, measured in hertz.
It's also worth noting that time period, T, such as was used in the first equation, is equal to 1 divided by the frequency, f. So, because of that connection, you can get a question that crosses over between all three of these equations.
Okay, let's go through an example. We have a mass on a spring, with mass 4 kilograms and spring constant 6 N/m. The maximum displacement of the spring is 0.8 meters. What is the time period of the mass on the spring? And what is the displacement after 0.6 seconds? The stopwatch starts as the mass on the spring passes through the equilibrium position (through the middle).
Okay, so first of all, we should write down what we know. The mass on the spring, m = 4 kg. The spring constant, k is 6 N/s. The amplitude (or maximum displacement), A, is 0.8 m, and the time, t, is 0.6. We're trying to find T and x.
First of all, to find T, we just plug numbers into the first equation and solve. 2pi times the square-root of 4 divided by 6. That comes out as 5.13 seconds.
Now, we need to find the displacement, x. We'll use the sine version of the displacement equation, since the stopwatch starts as the mass on the spring passes through the middle position. To find the displacement, we first need the angular frequency, omega. We have everything else in the displacement equation: both the time and the amplitude. We don't have angular frequency, and we don't have regular frequency, f, but we do have the time period from the previous part. One divided by the time period is the frequency, so 1 / 5.13 = 0.195 Hz. And to get the angular frequency, we just multiply this regular frequency by 2pi, which comes out as 1.23 radians per second.
At last we can finally plug numbers into the displacement equation. The displacement equals, 0.8 meters, multiplied by sine of 1.23 times 0.6. Make sure your calculator is in radians mode, type that in and you should get 0.54 meters. And that's it; we're done!
Simple harmonic motion is any motion where a restoring force is applied that is proportional to the displacement and in the opposite direction to that displacement. In other words, the more you pull it one way, the more it wants to return to the middle. The classic example of this is a mass on a spring, because the more you stretch it, the more you feel a tug back towards the middle. With any simple harmonic motion, the velocity is greatest in the middle, but the restoring force (and therefore the acceleration) is greatest at the maximum displacement (that's when the spring is most stretched or most compressed).
The equations discussed in this lesson can be used to solve problems involving simple harmonic motion. In these equations, x is the displacement of the spring (or the pendulum, or whatever it is that's in simple harmonic motion), A is the amplitude, omega is the angular frequency, t is the time, g is the acceleration due to gravity (which is always 9.8), T is the time period of the oscillation, m is the mass of the object moving under simple harmonic motion and f is the regular (non-angular) frequency.
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Back To CourseAP Physics 1: Exam Prep
12 chapters | 136 lessons