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The Kinematics of Simple Harmonic Motion

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  • 0:01 What is Kinematics?
  • 1:00 Equations
  • 4:00 Example Problem
  • 5:04 Lesson Summary
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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.

What is Kinematics?

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.

Equations

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:


velocity-timegraph


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.


pendulum 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:


simpleharmonicmotion


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.

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