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AP Physics 1: Exam Prep12 chapters | 136 lessons

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Lesson Transcript

Instructor:
*David Wood*

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this lesson, you will be able to explain what simple harmonic motion is, and use the kinematics equations for simple harmonic motion (both conceptually and numerically) to solve problems. A short quiz will follow.

**Kinematics** is a branch of physics that deals with the motion of objects, without reference to the forces that cause the motion. Or, in a more practical sense, it is the study of motion in terms of displacement, velocity, and acceleration. In regular kinematics, we use the equations of constant acceleration to study how these three quantities vary over time.

In today's lesson, we're going to talk about how we can analyze the kinematics of simple harmonic motion. **Simple harmonic motion** is any motion where a restoring force is applied that is proportional to the displacement, in the opposite direction of that displacement. Or in other words, the more you pull it, the more it wants to go back the opposite way. An example of this is a block on a spring, because more you stretch it, the more force you feel back towards the equilibrium position. Another example would be a pendulum, where gravity and tension pull the pendulum back towards the center.

The displacement, velocity and acceleration of an object undergoing simple harmonic motion are sinusoidal in nature. That is, if you drew a displacement-time, velocity-time and acceleration-time graph, their shapes would all be some kind of sine curve. But those sine curves are what's called out of phase, meaning that the peak of one curve doesn't happen at the same time as the peak of the others.

We can think about this conceptually by picturing a pendulum.

A pendulum is moving fastest as it swings through the very middle, and for an instant it reaches a velocity of zero at the very edges. So the velocity-time graph would look like this:

assuming that we start our stop-watch when the pendulum is released out at the far side, not in the middle. This forms a negative, or upside down, sine curve. It's negative because the velocity is pointed to the left on our diagram.

The displacement, on the other hand, starts out positive because it starts out on the right hand side. Here, we're assuming that the middle is the original. As it gets to the middle, the displacement reduces to zero, and then it becomes negative as it swings past the origin, and so on. So the equation for acceleration is a positive cosine graph.

What about acceleration? Well, acceleration is greatest at the left side and the right side - at the edges. This is because these are the points where the pendulum bob feels the most force pushing it towards the center. Once it reaches the center, the acceleration, at least for a moment, is zero. So the acceleration starts out negative, goes to zero, becomes positive, returns back to zero and so on. So acceleration will be a negative cosine graph.

If you're familiar with calculus, these equations will make a lot of sense:

If you differentiate a cosine equation, you get negative sine, and if you differentiate a negative sine, you'll get a negative cosine. If you don't know calculus, there's no need to worry. The important thing is that you understand what the shapes of these graphs mean in terms of the pendulum itself. Of course, these same equations apply to any example of simple harmonic motion. Instead of holding a pendulum to the right and releasing it, we could be pulling a mass on a spring to the right; it would still work.

These three equations assume you release the pendulum on the right hand side at *t* = 0. If you start to the left hand side, the signs will flip, and if you start the stopwatch as the pendulum is speeding through the very middle, the sines and cosines will swap.

For the sake of problem solving, you in particular need to be able to use the first of these equations: the displacement equation. In this equation, *A* is the amplitude of the oscillation (otherwise known as the maximum displacement - the displacement when it's at the far right or far left) measured in meters, *f* is the frequency of the oscillation, or the number of complete cycles per second, measured in Hertz, *t* is the time after you release the pendulum (on the far right, like in the diagram) measured in seconds, and *x *is of course the displacement in meters.

**Kinematics** is a branch of physics that deals with the motion of objects, without reference to the forces that cause the motion. Or in a more practical sense, it is the study of motion in terms of displacement, velocity, and acceleration. Kinematics can be used not only to study linear motion, but also other motion, including simple harmonic motion. **Simple harmonic motion** is any motion where a restoring force is applied that is proportional to the displacement, in the opposite direction of that displacement. Or in other words, the more you pull it, the more it wants to go the opposite way. An example of this is a block on a spring or a pendulum.

Three graphs and equations describe the displacement, velocity and acceleration of an object undergoing simple harmonic motion. They assume that when you start the stopwatch at *t* = 0, the pendulum is positioned off to the right hand side (with a positive displacement).

Strengthen your ability to complete these goals when you've reviewed the lesson:

- Recollect the focus of kinematics
- List the simple harmonic equations and their assumptions
- Interpret the graphs that describe the simple harmonic equations

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AP Physics 1: Exam Prep12 chapters | 136 lessons

- Hooke's Law & the Spring Constant: Definition & Equation 4:39
- Simple Harmonic Motion (SHM): Definition, Formulas & Examples 6:57
- Simple Harmonic Motion: Kinetic Energy & Potential Energy 5:44
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- Pendulums in Physics: Definition & Equations 5:51
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