The Secrets of SHM: The Importance of Understanding and Physics Formulae.

Physics is a fascinating subject that reveals the secrets of nature and the principles of motion. To describe the physical phenomena and the relationships between different quantities, physics uses a lot of formulae. However, learning physics is not just about memorizing these formulae and plugging in numbers. It is also about internalizing these formulae, which means understanding their origin, derivation, and interpretation. Internalizing formulae can help you to apply them more effectively, solve problems more creatively, and appreciate the beauty of physics more deeply. In this blog post, I will show you how to internalize physics formulae better, using the example of simple harmonic motion.

What is simple harmonic motion?

Simple harmonic motion (SHM) is a type of motion that happens when something moves back and forth around a fixed point, and the force that makes it move is always pointing towards that point, so the displacement from that point is always in the opposite direction to the force. For example, a ball bouncing on a spring, a swing, or a guitar string are examples of things that do SHM. SHM has a constant frequency and amplitude, which means that the thing moves with the same speed and distance every time it goes back and forth.

What are the formulae for SHM?

Three main things describe how something moves when it does SHM: displacement, velocity, and acceleration. Displacement is how far the thing is from its normal position (either in the y or x direction), velocity is how fast the displacement changes and acceleration is how fast the velocity changes. These things change with time and can be written using math functions, such as sine and cosine. 

The formulae for SHM in the y direction, like for a swinging pendulum are:

$$y(t)=A\sin (\omega t+\varphi )$$

$$v(t)=A\omega \cos (\omega t+\varphi )$$

a(t)=−Aω^2sin(ωt+ϕ)

$y(t)$ is the displacement in the y direction, $v(t)$ is the velocity, $a(t)$ is the acceleration, A is the amplitude (the maximum displacement), $\omega$ is the angular frequency (the time it takes to bounce up and down), $t$ is the time, and $\varphi$ is the phase angle (more on this later). These formulae are made from the definition of SHM which is that acceleration is proportional to the inverse of the displacement.

How are the formulae for SHM made?

To make the formulae for SHM, we need to use some basic mathematics. Mathematics is the study of the relationships between numbers and patterns. We can use mathematics to model the motion of something that does SHM as analogous to circular motion.

Imagine a dot moving around a circle of a radius of A with a constant angular speed $\omega$. The dot has a horizontal coordinate of x and a vertical coordinate of $y$, which change with time as the dot moves around the circle. We can draw a triangle with the dot, the centre of the circle, and the point where the vertical line from the dot meets the horizontal axis. The angle $\theta$ between the line from the centre to the dot and the horizontal axis is called the angular displacement, and it is related to the time by the formula:

$$\theta =\omega t+\varphi$$ 

$\mathrm{Where\; \varphi \; is\; the\; starting\; angle\; when }$$t=0$. At equilibrium, ϕ = 0

So now we need to find the x and y coordinates of the dot. To find these values, we need to go back to the basics of mathematics and see how these functions are derived from the properties of circles and triangles. In chapter 6 of her book Is Maths Real? Eugenia Cheng explains how to understand formulae better by using the idea of abstraction. Abstraction is the process of simplifying something by focusing on the essential features and ignoring the irrelevant details. For example, when we draw a circle, we don’t care about the thickness of the line, the colour of the paper, or the size of the circle. We only care about the shape of the circle, and the fact that all the points on the circle are the same distance from the center. This distance is called the radius of the circle, and it is an essential feature of the circle.

Similarly, when we draw a triangle, we don’t care about the angles, the sides, or the area of the triangle. We only care about the shape of the triangle, and the fact that it has three straight sides and three corners. These sides and corners are called the vertices of the triangle, and they are essential features of the triangle.

Now, from our SHM example, we have a circle of radius A, and we draw a triangle inside the circle, such that one of the vertices is at the centre of the circle, and another of the vertices is on the circle. We can label the vertices as D$and$ E, and the angle at the centre as θ (ωt). We can also draw a line from E to the horizontal axis, and label the point where they meet as F, this is the final of the vertices. We can see that the triangle DEF is right-angled because the angle is at 90∘. We can also see that the length of the side DE is equal to the radius of the circle, which is the amplitude (A). We can call this side the hypotenuse of the triangle, and it is the longest side of the triangle.

Now, we can define the functions sine, and cosine using the lengths of the sides of the triangle. Sine is the ratio of the side opposite the angle we are using, and cosine is the ratio of the side adjacent to the angle we are using. In other words, we can write: 

sin(θ)=y/A

cos(θ)=x/A 

which we can rearrange to give the coordinates of the dot as: 

x=Acos(θ)

y=Asin(θ)​

We can see that these functions are the same as the coordinates of the dot on the circle, except that we have divided them by the radius of the circle, which is A. This means that the functions sine and cosine are the same as the ratios of the coordinates of the dot on the circle. This is why we can use these functions to model the motion of something that does SHM.

By deriving the functions sine and cosine from the properties of circles and triangles, we can see how they are related to each other and to the motion of a system undergoing SHM. We can also see how they change with the angle $\theta$, and how they can be used to calculate the lengths of the sides of any right-angled triangle. For example, we can see that sine and cosine are opposite functions, which means that they are shifted by π​/2 radians or 90∘from each other. This means that when $\theta$ is 0, sine is 0 cosine is 1, and when $\theta$ is π/2 radians, sine is 1 and cosine is 0. Using this triangle model, we can think of the motion of the dot around the circle as a combination of two motions: one along the horizontal axis, and one along the vertical axis. The vertical component of the circular motion follows the same pattern as a simple harmonic motion with the same frequency and amplitude as the circle. The horizontal component of the circular motion is also a simple harmonic motion, but it is shifted by a quarter of a cycle (π/2 radians) or 90 degrees from the vertical component. This means when θ is π/2​ radians or 90∘ the displacement in the y direction is a maximum and the displacement in the x direction is a minimum and vice versa.

Hence, for our pendulum example, the vertical coordinate y undergoes SHM in the y direction, and the horizontal coordinate x also observes SHM, but at right angles to the y direction. We can then use calculus to find the velocity and acceleration of an object undergoing SHM, through differentiation of the displacement to find the velocity, and acceleration functions. The analogous equations for motion in the horizontal axis can be derived from the displacement equation in the x direction.

As mentioned at the beginning of this blog post, in SHM, the displacement of the object from its equilibrium position in the y direction is given by:

$$\mathrm{<span\; class="base"><span\; style="font-family:\; times;"><span\; class="mopen">y(</span><span\; class="mord\; mathnormal">t</span><span\; class="mclose">)</span><span\; class="mspace"></span><span\; class="mrel">=</span><span\; class="mord\; mathnormal">Asin(\omega </span><span\; class="mord\; mathnormal">t</span><span\; class="mspace"></span><span\; class="mbin">+</span><span\; class="mord\; mathnormal">\varphi </span><span\; class="mclose">)</span></span></span>}$$

$$\mathrm{<span\; class="base"><span\; style="font-family:\; times;"><span\; class="mclose"><br></span></span></span>}$$

where A is the amplitude, ω is the angular frequency, and φ is the phase angle. The velocity of the object is the derivative of displacement concerning time, so we can differentiate to get:

$$\mathrm{<span\; class="base"><span\; style="font-family:\; times;"><span\; class="mord\; mathnormal">v</span><span\; class="mopen">(</span><span\; class="mord\; mathnormal">t</span><span\; class="mclose">)</span><span\; class="mspace"></span><span\; class="mrel">=</span><span\; class="mord"><span\; class="mfrac"><span\; class="vlist-t\; vlist-t2"><span\; class="vlist-r"><span\; class="vlist"><span\; class="mord"><span\; class="mord\; mathnormal">dy/dt</span></span></span><span\; class="vlist-s"></span></span><span\; class="vlist-r"><span\; class="vlist"></span></span></span></span><span\; class="mclose\; nulldelimiter"></span></span><span\; class="mspace"></span><span\; class="mrel">= </span><span\; class="mord\; mathnormal">A</span><span\; class="mord\; mathnormal">\omega cos</span><span\; class="mopen">(</span><span\; class="mord\; mathnormal">\omega </span><span\; class="mord\; mathnormal">t</span><span\; class="mspace"></span><span\; class="mbin">+</span><span\; class="mord\; mathnormal">\varphi </span><span\; class="mclose">)</span></span></span>}$$

$$\mathrm{<span\; class="base"><span\; style="font-family:\; times;"><span\; class="mclose"><br></span></span></span>}$$

Similarly, the acceleration of the object is the derivative of velocity with respect to time, so we differentiate again to get: 

a(t)=dv/dt​=−Aω^2sin(ωt+ϕ)

Notice that the acceleration function is proportional to the displacement function but with a negative sign. This means that the acceleration is always opposite to the displacement, which is consistent with the restoring force that causes SHM.

What if the motion starts at the maximum displacement?

The formulae for SHM that we have derived assume that the motion starts at the equilibrium position, which is the point where the displacement is zero and the velocity is maximum. However, this is not always the case. Sometimes, the motion may start at a different position, such as the maximum displacement or somewhere in between. In this case, we need to adjust the formulae for SHM by adding a phase angle ϕ which is the starting angular displacement of the motion when t=0. The phase angle ϕ tells us how much the motion is shifted from the equilibrium position, and it can be positive or negative depending on the direction of the shift. 

If the motion starts at the maximum displacement, then the phase angle $\varphi$ is π​/2 radians, which means that the motion is shifted by π/2 radians at t = 0 π/2 radians at t = 0 This means that the dot starts at the top of the circle wand moves downwards. The formulae for SHM when the motion starts at the maximum displacement are:

y(t)=Asin(ωt+π/2​)

v(t)=Aωcos(ωt+π/2​)

a(t)=−Aω^2sin(ωt+π/2​)

We can also simplify these formulae by using the identity cos(α+β)=cosαcosβ−sinαsinβ

In this case, we have α=θ and β=π/2. Therefore, we can substitute these values into the formula and simplify:

cos(θ+π/2​) = cosθcosπ/2​ − sinθsinπ/2​

cos(θ+π/2​) = cos(0) − sin(1)

cos(θ+π/2​) = −sinθ. 

Therefore, the simplified formulae for SHM when the motion starts at the maximum displacement are:

$$y(t)=A\cos (\omega t)$$

$$v(t)=-A\omega \sin (\omega t)$$

a(t)=−Aω^2cos(ωt)

We can see that these formulae are the same as the ones we have derived before, except that we have swapped the sine and cosine functions. This means that the displacement and the velocity of the motion are out of sync by π/2 radians or 90∘ and the displacement and the acceleration are in sync but with opposite signs.

How does understanding the formulae for SHM help us learn better?

By deriving the formulae for SHM from mathematics, we can see how they are related to each other and to the motion of something that does SHM. We can also see how the things that affect SHM, such as amplitude, frequency, and phase, change the way something that does SHM moves. For example, we can see that:

• The amplitude $A$ is the maximum displacement of something that does SHM from its normal position, and it determines the size of the motion.
• The angular frequency $\omega$ is the speed of change of the angular displacement, and it determines the speed of the motion. The frequency $f$ and the period $T$ of the motion are related to the angular frequency by the formulae: $\omega =2\pi f$ and T=1/f
• The phase angle, ϕ is the starting angular displacement of something that does SHM when t=0, and it determines the position of something that does SHM at any time. The phase difference between two things that do SHM is the difference between their phase angles, and it determines how they move together or apart.

By understanding these formulae, we can also see how sine and cosine are related to each other and to the motion of something that does SHM. For example, we can see that:

• Sine and cosine are opposite functions, which means that they are shifted by π​/2 radians or $90^{\circ }$from each other.

• This means that the displacement and the velocity of something that does SHM are out of sync by π/2​ radians or $90^{\circ }$, and the displacement and the acceleration are in sync by π radians or 180∘ so have opposite signs.

• The relationship between sine and cosine indicates that when displacement is at its highest point, velocity is at its lowest point. Conversely, when displacement is at its lowest point, velocity is at its highest point. In addition, when velocity is at its lowest point, acceleration is at its highest point, and when displacement is at its highest point, acceleration is also at its highpoint but negative. Finally, when displacement is at its lowest point, acceleration is at its highest point, but it is a positive maximum.

Physics is a wonderful subject that explores the mysteries of nature and the laws of motion. To express the physical phenomena and the connections between different variables, physics uses a lot of formulae. However, learning physics is not just about memorizing these formulae and substituting numbers. It is also about internalizing these formulae, which means grasping their source, derivation, and meaning. Internalizing formulae can help you to apply them more efficiently, solve problems more inventively, and enjoy the beauty of physics more profoundly. In this blog post, I have shown you how to internalize the formulae for SHM better, using mathematics. I hope this blog post has been useful and fun for you, and I urge you to try internalizing other physics formulae as well. Remember, physics is not just about numbers and symbols, it is also about ideas and concepts. Happy learning! 😊