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In this Physics tutorial, you will learn:
Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
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10.1 | Simple Harmonic Motion |
What happens when you pull an elastic spring and then you release it?
What about a pendulum wall clock? The needle of a sewing machine? A tree when wind breeze flows? What kind of motion do these examples involve?
All situations mentioned above are examples of simple harmonic motion, which we will explain in this tutorial, accompanying it with numerous examples for a better understanding.
By definition, "Simple harmonic motion (in short SHM) is a repetitive movement back and forth through an equilibrium (or central) position, so that the maximum displacement on one side of this position is equal to the maximum displacement on the other side."
In other words, in simple harmonic motion the object moves back and forth along a line. These back and forth movements are known as "oscillations". Simple harmonic motion is part of a wider category of motion known as "periodic motion", which includes other types of motions that repeat themselves such as circular, vibrational and other similar motions. However, simple harmonic motion and periodic motion are not the same thing. There are two main differences between them.
Based on the trajectory (path) of the oscillating object, there are two types of simple harmonic motion.
1) If oscillations occur according a straight line, such as when we pull a spring and then we release it, we obtain a linear simple harmonic motion, For example, an oscillating spring makes linear simple harmonic motion as shown in the figure below.
In this case, elastic force acts as a restoring force because it tries to oppose the springs elongation or compression.
2) If a hanged object swings around a central line (usually a vertical line dictated by the existence of gravity), we have an angular simple harmonic motion as shown in the figure below.
The angle θ is an important parameter of SHM for which we will discuss in the next paragraph. Here, the direction of restoring force F is determined by the sum of the two force vectors acting on the object: gravitational force Fg and tension T of the string, whose sum gives the restoring force F.
We will discuss angular SHM in the next Physics tutorial. In this topic we will focus exclusively on linear SHM.
There are two conditions for a periodic motion to be considered as SHM. They are:
(Displacement here must be understood as a shift from the equilibrium position).
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.
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.
From calculus it is known that the first derivative of position with respect to time (dx/dt) represents the velocity v and the second derivative of position with respect to time d2x/dt2 (or the first derivative of velocity in respect to time dv/dt) gives the acceleration a.
Also, the first derivative of sin x is equal to cos x and the first derivative of cos x = - sin x. Likewise, the first derivative of k × sin a × x = k × a × cos a × x and the first derivative of k × a × cos a × x = - k × a2 × sin a × x.
Applying the above derivation rules for the SHM motion, we obtain for the velocity v in respect to the time, v(t):
and for the acceleration a in respect to the time, a(t):
Therefore, we obtained a very important formula of SHM that relates acceleration a, angular frequency ω and displacement x:
Since the restoring force F is in the opposite direction to the acceleration a, we obtain from Newton's Second Law of Motion for the acting forces in springs:
where m is the mass of hanged object and k is the spring constant. Thus,
Hence,
Or
Then we can substitute this value of angular frequency in the sine equation of SHM if oscillations are caused by an object hanged on an elastic spring.
A spring is mounted in a horizontal plane with one end as stationary. A 20 N force exerted on this spring, causes an elongation of 5 cm as shown in the figure below.
Then, we stop pulling the spring, release it and attach a 0.2 kg object at the hook. Then we pull the spring by 20 cm and release it. As a result, the spring starts oscillate in SHM as shown below.
Calculate the following quantities.
a Spring constant is calculated by considering the initial pulling force F and applying the Hooke's law F = k × x. Thus, since F = 20 N and x1 = 5 cm = 0.05 m, we have
b The maximum velocity vmax is obtained by the equation
where
and
is the angular frequency.
This maximum velocity is at the equilibrium position, while the minimum velocity occurs at the maximum elongation or compression of the spring, where the object stops and makes ready to turn back. Hence, vmin = 0.
c From the equation
we obtain for the maximum acceleration amax:
Considering only the magnitude of acceleration (i.e. not considering the negative sign), we obtain
Again, the minimum acceleration occurs at the turning positions where the object stops, i.e. amin = 0.
d) Velocity and acceleration at a given instant are calculated through the sine or cosine functions adapted for SHM as discussed earlier in the theory.
Thus, since
where φ = π/2 because the object is initially at the maximum position which means it is shifted by one quarter of a complete cycle (the cycle starts from the equilibrium position, i.e. at middle of trajectory),
we have for the velocity v at any instant t:
We have
Thus, substituting the known values in the equation of velocity for this SHM, we obtain
and for acceleration a at any instant t:
Now, let's find the time t in which the object is at halfway to the centre from the original position. But before, we must calculate the period T. Thus,
From the initial position to halfway to the centre, the object has moved by 1/8 of a complete cycle. Hence,
Thus, for t = 0.0175 s, we obtain
We need only the magnitude of velocity, so v = 6.258 m/s.
Also,
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