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Physics Lesson 10.1.4 - Equation of Simple Harmonic Motion
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Welcome to our Physics lesson on Equation of Simple Harmonic Motion, this is the fourth lesson of our suite of physics lessons covering the topic of Simple Harmonic Motion, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
Equation of Simple Harmonic Motion
Let's consider an object moving back and forth from -x to + x and again to -x through the equilibrium position 0 as shown in the figure below.
You can see that the farther from the equilibrium position, the slower the object moves. From here, we can deduce that the acceleration becomes zero it a short instant when the object needs to turn back, i.e. when it reaches the maximum displacement from the equilibrium position. Such a situation is similar to that of an object thrown vertically upwards. Remember that in such cases, the object reaches a maximum position, stops for a while and then it turns back (falls down). The only difference is that in SHM this process occurs in both sides of the trajectory.
If we look for an appropriate function to describe mathematically the simple harmonic motion, we will understand that the function, which fits more to it, is the sine (or cosine) function. Below, a sine function (y = sin x) is shown.
Here are the reasons why sine or cosine functions are better in this regard.
- Both sine and cosine functions have an equilibrium position and two extremities: one maximum and one minimum, just like the SHM. When used to describe SHM, the maximum and the minimum of the sine function represent the turning points, i.e. the distance from the origin to these points gives the amplitude A, which is the maximum displacement from the origin. In many cases, the amplitude is also denoted as x0 or xmax.
- Like in rotational motion (which has many similarities to SHM as they both are periodic), we use the concept of period T to describe the time needed to do one complete oscillation. As a result, we can also make use of related concepts such as frequency f, where f = 1/T, angular velocity (which here is known as angular frequency) ω = 2π / T = 2π × f, and so on.
In mathematics, the simplest sine function in which the time t is the independent variable and the position x is the dependent one, has the form
If the graph does not start at the origin but it is shifted from it, we insert the angular displacement φ in the above equation, so it becomes
The initial angular displacement φ here is known as the "phase shift". Look at the figure, from which we will explain how to interpret a SHM sine graph.
The solid line represents the x(t) = x0 × sin ω × t. It is shown only for a better understanding of the dotted line graph that represents the x(t) = x0 × sin (ω × t + φ) graph, for which we are interested.
Thus, since the dotted graph starts at the origin but is first goes down and then it moves up, there is a half oscillation shift in respect to the normal sine graph. Therefore, the phase shift is π radians (π radians = 1800 = half a rotation or cycle).
Also, you can see that the maximum displacement from the origin is 2 cm. This means the amplitude x0 = 2 cm.
At last, there is an information regarding the time. Thus, three quarters of a complete oscillation is done in 6s (N = 3/4 and t = 6s). This means one compete oscillation (period T) is done in
Thus,
Therefore, the equation of SHM shown in the graph becomes
This equation helps us find the position x of the oscillating object at any instant t. Obviously, this position fluctuates between x = + 2 cm and x = -2 cm as the magnitude of x(t) cannot be greater than the amplitude x0.
You have reached the end of Physics lesson 10.1.4 Equation of Simple Harmonic Motion. There are 6 lessons in this physics tutorial covering Simple Harmonic Motion, you can access all the lessons from this tutorial below.
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