Physics Tutorial: Simple Harmonic Motion

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In this Physics tutorial, you will learn:

  • Definition of simple harmonic motion
  • Differences between periodic motion and simple harmonic motion
  • Types of simple harmonic motion
  • Conditions for the existence of simple harmonic motion
  • The equation of simple harmonic motion
  • How to calculate the velocity and acceleration in a simple harmonic motion
  • How does simple harmonic motion occur in oscillating springs?

Introduction

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.

What is Simple Harmonic Motion?

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.

  1. In periodic motion, the object's displacement may or may not be in the direction of the restoring force (remember the concept of restoring force explained in Physics tutorial "Types of Forces III. Elastic Force and Tension", in which elastic force was a kind of restoring force because it tries to send the object back to the original position). As an example of periodic motion but not SHM we can mention the circular motion, in which the centripetal force - which is a kind of restoring force - is perpendicular to the object's motion as centripetal force is directed towards the centre of curvature while displacement is according the tangent to the circle. On the other hand, in simple harmonic motion, the object's displacement is always in the opposite direction of the restoring force.
  2. In periodic motion, the movement may or may not be oscillatory. For example, we can move along the perimeter of a square several times, but this motion is not oscillatory as it is not back and forth but only forth. On the other hand, in simple harmonic motion the movement is only oscillatory.

Types of Simple Harmonic Motion

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.

Physics Tutorials: This image provides visual information for the physics tutorial Simple Harmonic Motion

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.

Physics Tutorials: This image provides visual information for the physics tutorial Simple Harmonic Motion

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.

Conditions for the Existence of Simple Harmonic Motion

There are two conditions for a periodic motion to be considered as SHM. They are:

  1. In system there should be an elastic restoring force F. If the system is disturbed away from the equilibrium, the restoring force will tend to bring the system back towards equilibrium position which is at the centre of the oscillation. This restoring force can be either of the known forces, such as gravitational, elastic, electrostatic, magnetic force etc. Restoring force is proportional in magnitude and opposite in sign with the displacement.
  2. The acceleration of the system should be directly proportional to the displacement and it is always directed towards the equilibrium position.

(Displacement here must be understood as a shift from the equilibrium position).

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.

Physics Tutorials: This image provides visual information for the physics tutorial Simple Harmonic Motion

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.

Physics Tutorials: This image provides visual information for the physics tutorial Simple Harmonic Motion

Here are the reasons why sine or cosine functions are better in this regard.

  1. 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.
  2. 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

x(t) = x0 × sin⁡ω × t

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

x(t) = x0 × sin⁡(ω × t + φ)

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.

Physics Tutorials: This image provides visual information for the physics tutorial Simple Harmonic Motion

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

T = t/N = 6 s/3/4 = 8 s

Thus,

ω = /8 = π/4 rad/s

Therefore, the equation of SHM shown in the graph becomes

x(t) = (2 cm) × sin (π/4 × t + π)

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.

Velocity and Acceleration in a Simple Harmonic Motion

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):

v(t) = dx/dt
=d[x0 × sin⁡(ω × t + φ)]/dt
= x0 × ω × cos⁡(ω × t + φ)

and for the acceleration a in respect to the time, a(t):

a(t) = d2x/dt2
= dv/dt
= d[x0 × ω × cos⁡(ω × t + φ)]/dt
= -x0 × ω2 × sin⁡(ω × t + φ)
= -ω2 × x(t)

Therefore, we obtained a very important formula of SHM that relates acceleration a, angular frequency ω and displacement x:

a = -ω2 × x

SHM in Elastic Springs

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:

F = m × a
= -k × x

where m is the mass of hanged object and k is the spring constant. Thus,

-m × ω2 × x = -k × x
m × ω2 = k

Hence,

ω2 = k/m

Or

ω = √k/m

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.

Example 1

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.

Physics Tutorials: This image provides visual information for the physics tutorial Simple Harmonic Motion

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.

Physics Tutorials: This image provides visual information for the physics tutorial Simple Harmonic Motion

Calculate the following quantities.

  1. The spring constant
  2. The maximum and minimum velocities of the object
  3. The maximum and minimum acceleration
  4. The magnitude of velocity and acceleration when the object has moved halfway to the centre from its original position

Solution 1

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

F = k × x1
k = F/x1 = 20 N/0.05 m= 400 N/m

b The maximum velocity vmax is obtained by the equation

vmax = ω × x0

where

x0 = Amplitude = 20 cm = 0.20 m

and

ω = √k/m = √400/0.2 = √2000 = 44.7 rad/s

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

a = -ω2 × x

we obtain for the maximum acceleration amax:

amax = -ω2 × x0
= -(√k/m)2 × x0
= -k/m × x0

Considering only the magnitude of acceleration (i.e. not considering the negative sign), we obtain

amax = k/m × x0
= 400 N/m/0.2 kg× 0.20 m
= 400 m/s2

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

x(t) = x0 × sin⁡(ω × t + φ)

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:

v(t) = x0 × ω × cos⁡(ω × t + φ)

We have

ω = √k/m = √400/0.2 = √2000 = 44.7 rad/s

Thus, substituting the known values in the equation of velocity for this SHM, we obtain

v(t) = 0.20 × 44.7 × cos (44.7 × t + π/2)
= 8.94 × cos⁡(44.7 × t + 3.14/2)
= 8.94 × cos⁡(44.7 × t + 1.57)

and for acceleration a at any instant t:

a(t) = -x0 × ω2 × sin⁡(ω × t + φ)
= -0.20 × (44.7)2 × sin⁡(44.7 × t + π/2)
= -400 × sin⁡(44.7 × t + 1.57)

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,

ω = /T ⇒ T = /ω
= 2 × 3.14/44.7
= 0.14 s

From the initial position to halfway to the centre, the object has moved by 1/8 of a complete cycle. Hence,

t = T × 1/8 = 0.14 s/8 = 0.0175 s

Thus, for t = 0.0175 s, we obtain

v(0.0175) = 8.94 × cos⁡(44.7 × 0.0175 + 1.57)
= 8.94 × cos⁡(44.7 × 0.0175 + 1.57)
= 8.94 × cos⁡2.35225 = 8.94 × (-0.70)
= - 6.258 m/s

We need only the magnitude of velocity, so v = 6.258 m/s.

Also,

a(0.0175) = -400 × sin⁡(44.7 × 0.0175 + 1.57)
= -400 × (-0.71)
= 284 m/s2

Summary

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.

  1. In periodic motion, the object's displacement may or may not be in the direction of the restoring force. On the other hand, in simple harmonic motion, the object's displacement is always in the opposite direction of the restoring force.
  2. In periodic motion, the movement may or may not be oscillatory, while in simple harmonic motion the movement is only oscillatory.
    1. Based on the trajectory (path) of the oscillating object, there are two types of simple harmonic motion.

  3. 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.
  4. 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

The function that describes better mathematically a SHM, is the sine (or cosine) function. This is because of two main reasons:

  1. 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.
  2. 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

x(t) = x0 × sin⁡ω × t

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

x(t) = x0 × sin⁡(ω × t + φ)

The initial angular displacement φ here is known as the "phase shift".

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.

Applying the above derivation rules for the SHM motion, we obtain for the velocity v in respect to the time, v(t):

v(t) = dx/dt
= x0 × ω × cos⁡(ω × t + φ)

and for the acceleration a in respect to the time, a(t):

a(t) = d2 x/dt2
= dv/dt
= -x0 × ω2 × sin⁡(ω × t + φ)
= -ω2 × x(t)

Therefore, we obtained a very important formula of SHM that relates acceleration a, angular frequency ω and displacement x:

a = -ω2 × 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:

F = m × a = -k × x

where m is the mas of hanged object and k is the spring constant. Thus,

-m × ω2 × x = -k × x
m × ω2 = k

Hence,

ω2 = k/m

Or

ω = √k/m

Then we can substitute this value of angular frequency in the sine equation of SHM if an object hanged on an elastic spring causes oscillations.

Simple Harmonic Motion Revision Questions

1. A spring attached to the left end of a horizontal platform is pulled by 10 cm when a 80 N pulling force acts on it.

Physics Tutorials: This image provides visual information for the physics tutorial Simple Harmonic Motion

Then, the spring is released and a 2 kg object is attached to the hook. The spring is pulled again by 30 cm and released.

Physics Tutorials: This image provides visual information for the physics tutorial Simple Harmonic Motion

What is the angular frequency of oscillations?

  1. 400 rad/s
  2. 20 rad/s
  3. 800 rad /s
  4. 28 rad/s

Correct Answer: B

2. The equation of a simple harmonic oscillation is

x(t) = 12 × sin (π/6 × t + π/3)

where x is in cm.

Which option below shows the correct values based on the above equation?

  1. Amplitude = 12 cm, Period = 12 s, Phase shift = π/3 rad, Initial position = 6√3 cm
  2. Amplitude = 12 cm, Period = π/6 s, Phase shift = π/3 rad, Initial position = 12 cm
  3. Amplitude = 6 cm, Period = 3 s, Phase shift = 0 rad, Initial position = 6√3 cm
  4. Amplitude = π/6 cm, Period = 12 s, Phase shift = π/3 rad, Initial position = 12 cm

Correct Answer: A

3. A spring oscillates according the equation of SHM

x(t) = 4 × sin⁡(π/2 × t + π/6)

where x is in cm.

What is the velocity and acceleration at t = 2 s?

  1. v = - 2π × 3 m/s, a = - 2 × π2 m/s2
  2. v = π × 3 m/s, a = 1/2 × π2 m/s2
  3. v = - π × 3 m/s, a = 2 × π2 m/s2
  4. v = - π × 3 m/s, a = 1/2 × π2 m/s2

Correct Answer: D

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