Oscillation Meaning: Class 11 Physics Notes, Displacement, Acceleration, and SHM Formula

An oscillation is a repeated back-and-forth or up-and-down motion of an object about a central or equilibrium position. This motion occurs after a displacement from the central position and a restoring force that pushes the object back to its original and stable position over and over again. This motion in its purest form is continuous and periodic, and one of its most common examples is a simple pendulum.  

Periodic and oscillatory motions are two essential concepts in this 13th chapter of Physics class 11. 

Periodic Motion 

It is the type of motion of an object when it takes the same path within a defined and regular time interval. 

Examples of Periodic Motion

•  Earth rotates around the sun every year. 
• A carousel horse moves up and down as the carousel rotates. 

Periodic motion follows Newton's Second Law of Motion. You can explore all of Newton's Laws of Motion quickly!

So, now when an object moves up or down with a constant acceleration, like the carousel horse with mechanical force, the displacement has the equation derived from Newton's Second Law. 
$$h=ut+\frac{1}{2}g{t}^2$$  

Oscillatory Motion

It’s a periodic motion where an object moves back and forth around an equilibrium position. Note that the equilibrium position is where the object rests without any external force.

You may even brush up now on two important concepts in early Class 11. 

Examples of Oscillatory Motions 

• A spring that stretches and compresses when we pull and release it.
• An analogue metronome arm that swings side by side.

Oscillations vs Vibrations - They’re Not the Same

When talking about motion in physics, oscillations and vibrations differ in definition. Vibrations are faster than oscillations.

An example may help here. A playground swing moves up and down, or a suspension bridge moves in oscillatory motion, while a string instrument’s tuning fork, when struck, vibrates. 

Period and Frequency

Having the above concepts clear, let’s get into two concepts. 

• Period - It means it’s the amount of time an object requires to complete one full cycle motion. To recap, the period for Earth’s motion is one year.
• Frequency - It’s the cycles per second. We calculate frequency in Hertz. Formula is v = 1/T

Displacement

Displacement in oscillations is learning how far the object has moved from equilibrium. Also, there are two different types of displacements. Linear and angular displacement.  

The displacement is linear for a spring, as it moves up and down. 

For a pendulum, the motion is right and left, and the displacement is angular. 

Aligning with your NCERT textbook's figures 13.2 (a) and 13.2 (b) we have recreated the diagrams for oscillations and displacement below to help you visualise the concepts highlighted in this section. 

Simple Harmonic Motion (SHM) is a pure form of oscillation. This is because it has a precise mathematical relationship between displacement and acceleration. 

Let's simplify this SHM definition. 

An object moves in a manner where displacement follows a sine or cosine wave. We use a sine wave to show the vertical height of the object at a point that's in motion around a circle over a specific period of time. The use of cosine wave is different here. It's the horizontal position of that same point over that same period of time.   

Then comes the force. It's that which pulls the object back. It's the restoring force and is directly proportional to how far it’s from the centre.

The farther it is from the centre, the stronger the force becomes to pull it back. Now, think in a way that the further you stretch a rubber band, the stronger its pull is.  

One essential thing to know when you are beginning to learn Simple Harmonic Motion is that this type of pure oscillation depends only on the position of the object, here, which is displacement. It does not depend on how fast the object is moving.  

To describe Simple Harmonic Motion in physics, we use these terms, especially in Class 11. 

Amplitude (A)

Amplitude or A is going to tell you the object’s distance when it swings away from its resting point. Just like how you’d pull a pendulum to one side, you can say that amplitude becomes the maximum distance from the middle point. We can also call it the maximum displacement from the mean or equilibrium position.

Period (T)

Using the same logic, the period tells us how much time it will take to complete one cycle for the object from its mean position to reach a specific amplitude and back to the same point. This is what we denote as a period. 

Frequency (ν)

The next thing to know after amplitude and period is frequency. This related concept takes into account the number of cycles or periods occurring within one single second. It has an inverse relationship with period. In Class 11 Physics, we should remember that if the period is long, the frequency is small. The frequency can only be higher when the period is short. 

Angular Frequency (ω)

To speed things up, we should know the speed or velocity of the object that moves within a cycle or period in terms of angles. 

The angular frequency formula is: ω = 2π /T. Here, 2π radians means one full circle, and T is the period that we calculate as per second.

Phase (φ)

Phase is about where in the cycle the motion is right now. Think of it like looking at a clock.
The angle of the hand tells you exactly what time it is, right? Similarly, the phase tells you the position in the oscillation. You can easily find out if the object is at the top, bottom, or somewhere in between. It even gives you the direction: is it moving up or down from that point?

Another minor thing to distinguish here is the (φ) symbol is the Phase Constant. The SHM phase in general is denoted by (ωt + φ). 

To understand all these terms used in SHM, we need to be clear about one more thing. So, in SHM, the object is moving constantly. It's not enough to know where the object is. Instead we should know when the object is where. So, if we know the specific moment in time (t), we can know the exact position (x) of the object at that particular instant. That's the displacement basically, and also what the SHM equation is all about.   

Equation of SHM

The equation of SHM tells us about the object's position at a specific moment (or the displacement) using five coordinates.

• Amplitude
• Period
• Frequency
• Angular Frequency
• Phase

$$x(t)=A\cos (\omega t+\phi )$$

This basic equation of SHM is necessary to understand. Whether you are appearing for NEET Exams or other important competitive exams, score marks confidently that ask you any question on SHM. 

For a quick glance before your exams. 

Your CBSE Physics textbook introduces the concept of uniform circular motion (where speed is constant but velocity changes with direction) to help you better understand simple harmonic motion from a geometrical (visual) perspective. 

There could be a visual cue to learn SHM and UCM together.

Let’s say you are watching a particle moving in a circle from the side. It will be a straight line, isn’t it?

Now, we can project SHM equations of circular motions onto a straight line. 

Let's focus on the physics behind this relation between SHM and uniform circular motion. 

The NCERT text of the Oscillations Class 11 chapter in section 13.4 highlights how the “projection of uniform circular motion on a diameter of the circle follows simple harmonic motion.”

To put it simply, we just need to know or visualise that Simple Harmonic Motion (SHM) is just the side-view of an object in Uniform Circular Motion (UCM). It’s similar to the one-dimensional projection of an object that moves in a circle. 

To visualise, we can consider the Ferris Wheel. 

If we are watching it from the side, it spins steadily. We see that someone in a cart goes all the way up, then all the way down. This is a periodic motion, i.e., SHM. 

For the time being, the entire focus is on how high the person inside the cart is.

From this perspective, we are ignoring the forward and backward movement. Because, even though the rider is travelling in a circle, from your perspective, they are just oscillating up and down.

Now, we move from one-dimensional movement to two-dimensional movement. One easy way to look at it is a point that moves in a perfect circle around the origin (0,0) of a graph. This is our Uniform Circular Motion. 

From a 2D perspective, that is, the circle, at any instant, the point has both an x-position and a y-position, as you can see from the above image. 

As this point travels around the circle, these x and y values are constantly changing.

Y-axis movement 

To understand the Y-axis movement, we can ignore the side-to-side movement. Only focus on how high or low the point is. It goes up to the top of the circle, back down through the middle, to the bottom, and then back up again. This up-and-down oscillation is Simple Harmonic Motion (SHM).

The equation y(t) = A sin(ωt + φ) perfectly describes this vertical-only movement:

y(t): The vertical position at any given time t.

A: The amplitude, which is simply the radius of the circle (its maximum height).

sin(ωt + φ): The sine function naturally oscillates between -1 and +1. That perfectly mimics that up-and-down motion along the y-axis.

X-axis movement

Similarly, if you ignore the up-and-down movement and only watch its side-to-side motion, you'd see it go from the far right, through the middle, to the far left, and back again. This is also SHM.

This horizontal movement is described by the cosine function: x(t) = A cos(ωt + φ).

So, the smooth, two-dimensional Uniform Circular Motion is simply the combination of two separate Simple Harmonic Motions happening at the same time. One projected onto the y-axis and one projected onto the x-axis.

 

Let’s see how exactly velocity, force, and acceleration work in Simple Harmonic Motion. 

Velocity in Simple Harmonic Motion

We calculate the Velocity in SHM through this
When an object/particle moves in a circle, we show it as 

$v=\omega A$

Here, 
ω  is the angular frequency

A is the radius

For SHM, side-to-side, the velocity changes with time. And we represent that as

$v\left(t\right)=-\omega A\sin (\omega t+\phi )$

Note that the negative sign in this equation highlights that the direction is opposite to the displacement when it moves back to its centre.

Acceleration in SHM

When it is a uniform circular motion (UCM), there is a centripetal acceleration, and it remains constant. It will always point towards the centre of the circle.

$a={\omega }^{2}A$

When we project this acceleration onto the x-axis (our diameter of motion), we get the acceleration for SHM. 

a(t) = -ω²Acos(ωt + φ)

And since we already know that the displacement is x(t) = Acos(ωt + φ), the equation simplifies into this. 

a(t) = -ω²x(t)

Therefore,

$a\left(t\right)=-{\omega }^{2}A\cos (\omega t+\phi )=-{\omega }^{2}x\left(t\right)$

This equation tells us two important things. 

1. Magnitude: The acceleration is directly proportional to the displacement (a ∝ x). The farther the object is from the centre, the greater the magnitude of its acceleration.

    

2. Direction: The negative sign is the most important part to remember about SHM in Class 11. It means the acceleration is always directed opposite to the displacement. If the object is to the right (positive displacement), the acceleration is to the left (negative). Because it always pulls it back towards the centre. This is why the force is called a restoring force.

Force plays a vital role in Simple Harmonic Motion. One of the givens here is that there should always be a restoring force to bring the oscillating object to its equilibrium. And in equilibrium there is no force. 

Also, the magnitude of the restoring force is directly proportional to the displacement from the equilibrium position. 

Now, this relationship can be understood better with Hooke’s Law. This pretty much applies to any solid material's elastic behaviour that can return to its original shape after being deformed up to a certain threshold. 

$$F=-kx$$

F is the restoring force

K is the constant

X is the displacement