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Here, we will give you brief notes on conditions for an angular harmonic oscillator to be angular shm and will be discussing all the terms used here. Before moving ahead, one should know the definitions of the basic terms given below. Simple harmonic motion is defined as a periodic motion along a straight line. Its acceleration is always toward a fixed point in the line and is proportional to its distance from the fixed point. Angular Oscillation is the oscillation when a body is allowed to rotate freely about a given axis.
Why is simple harmonic motion so important?
All around us are angular harmonic oscillators beating the human heart to the vibrating atoms that make up everything. Simple harmonic motion is a very important type of periodic oscillation where the acceleration (α) is proportional to the displacement (x) from equilibrium in the direction of the equilibrium position. In contrast, Angular Shm is defined as the oscillatory motion of a body in which the torque for angular acceleration is directly proportional to the angular displacement. Its direction is opposite to that of angular displacement.
Further, we will have a brief discussion on the topic below.
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Types of Simple Harmonic Motion
SHM or Simple Harmonic Motion can be classified into two types :
Linear SHM
Angular SHM
Linear Simple Harmonic Motion
Linear Simple Harmonic Motion occurs when a particle moves in a straight line around a fixed point (equilibrium position).
Take the spring-mass system, for instance.
Conditions for Linear SHM :
The restoring force or acceleration acting on the particle should always be proportional to the displacement of the particle and directed towards the equilibrium position.
Angular Simple Harmonic Motion
It’s a simple harmonic oscillator when a system’s oscillations are long-term and perpendicular to a predetermined axis.
Conditions to execute Angular Shm:
Force to return to its equilibrium position, the restoring torque (or angular acceleration) must always be proportional to the angular displacement of the particle.
Angular Harmonic Oscillator
An angular harmonic oscillator can also be a simple harmonic motion. SHM is produced by applying a restoring torque that is proportional to the angular displacement and is directed towards the mean position.
Consider a wire suspended vertically from a rigid support and let some weight be suspended from the lower end of the wire. When the wire is twisted through an angle θ from the mean position, a restoring torque acts on it, tending to return it to the mean position. Here restoring torque is proportional to angular displacement θ.
Hence r = -C θ ..(1)
Where C is called torque constant.
It is equal to the couple’s moment required to produce unit angular displacement. Its unit is N m rad−1.
The negative sign shows that torque is acting opposite to the angular displacement. This is the case of angular simple harmonic oscillators.
Examples: Torsional pendulum, the balance wheel of a watch.
But τ = I α …(2)
where τ is torque, I am the moment of inertia, and α is the angular acceleration
∴ Angular acceleration,
Α = τ /I = – C θ / I
This is similar to a = −ω2 y
Replacing y by θ and a by α, we get
α = −ω2θ = – (C/I) θ
ω = rt(C/I)
Period of SHM T = 2π rt(I/C)
Frequency n = 1/T = 1/2 π rt(C/I)
Angular Oscillation Formula
When a body can rotate freely about a given axis, then the oscillation is known as angular harmonic oscillator formula.
What we refer to as “mean position” refers to a location where the body’s torque is assumed to be zero. To put it another way, the resultant torque acts as if it is proportional to the angular displacement of the body, and this torque has a strong pull on the body toward its mean position.
Let θ be the angular displacement of the body, and the resultant torque τ acting on the body is κ is the restoring torsion constant, torque per unit angular displacement. If I is the moment of inertia of the body and a is the angular acceleration, then this is the differential equation of an angular harmonic oscillator and simple harmonic motion.
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Angular Simple Harmonic Oscillator
In an angular simple harmonic oscillator, the displacement of the particle is measured in terms of angular displacement theta. Here, the spring factor stands for torque constant, i.e., the couple’s moment to produce unit angular displacement or the restoring torque per unit displacement. In this case, the inertia factor stands for the body’s moment of inertia executing an angular harmonic oscillator.
Comparison of SHM with Angular SHM
In linear shm, the displacement of the particle is measured in terms of linear displacement r. The restoring force is F = -kr, where k is a spring constant or force constant, force per unit displacement. In this case, the inertia factor is the mass of the body executing shm.
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Conclusion
From the above, readers may be able to know that the study of the simple harmonic oscillator is important in classical mechanics and quantum mechanics. The reason is that any particle in a stable equilibrium position will execute simple harmonic motion if a small amount displaces it. A simple Harmonic Oscillator is applied in clocks, guitars, and violins. It is also seen in the car -shock absorber, where springs are attached to the car wheel to ensure a smoother ride. In this article, we have briefly discussed and covered every point regarding the angular harmonic oscillator.
