Simple Harmonic motion

We will cover the importance of Simple Harmonic Motion, give a brief introduction to the topic, and also provide refernces to useful content

Importance

Simple Harmonic Motion is in the syllabus for both JEE mains and JEE Advanced. In the last 5 years nearly 13 questions came in JEE mains. Counting the online exams as separate exams, we see that alteast 1 question comes from this topic comes every year in JEE mains. Similarly, in JEE Advanced, 12 questions were asked in the last 5 years.

Periodic motion

There are two basic ways to measure time: by duration or periodic motion. In case of measurement by duration of time we can use clocks and calenders. One of the most important examples of periodic motion is simple harmonic motion (SHM), in which some physical quantity varies sinusoidally.

SHM Quantitative study

Simple Harmonic Motion or "SHM" is a standard topic in introductory mechanics because it is a powerful illustration of how Models are applied in real-world problem solving. Simple harmonic motion (also called simple harmonic oscillation) is defined to be one-dimensional motion such that the position is a sinusoidal function of time (a sine function and/or cosine function, both at the same frequency). This motion is by definition periodic (it repeats itself at regular intervals with period T).
Here is a simple video for better understanding of SHM

Suppose a function of time has the form of a sine wave function,
!!y(t)=Asin((2pit)/T)!! where !! A > 0!! is the amplitude (maximum value). The function y(t) varies between !!A!! and !!− A!!, because a sine function varies between +1 and −1. A plot of y(t) vs. time is shown above.
The sine function is periodic in time. Simple harmonic motion is general sinusoidal motion and can be represented by the sum of a sine function and a cosine function, provided that we choose our coordinates such that the origin is located at the equilibrium position. (!!sumF=0!!)
!!x(t)=Acos(w_0t)+Bsin(w_0t)!!

where !!A!!, !!B!!, and !!ω_o!! are constants: !!ω_o!! is determined by the force and dynamics of the particle, and A, B depend on details of the particular motion (e.g. the amplitude of oscillation).

Angular Frequency

Simple harmonic motion is characterized by a time period (T) and a frequency (f). The period is the amount of time it takes to complete one full cycle of the motion; that is, to return to the same position with the same velocity. !!f=1/T!!. Its Unit is hertz.
The angular frequency is given as !!w_o=2pif=(2pi)/T!!.
A full cycle of the motion corresponds to !!2pi!! radians of motion, hence the angular frequency is the number of radians per second covered by the object's motion.

The Position at t = 0

The constant !!A!! in our equation for the position is equal to the position of the oscillating object when !!t = 0!!. You can see this by substituting !!t = 0!! into our equation for the position:

!!x(0)=x_o=Acos(0)+Bsin(0)=A!!

The Velocity at t = 0

!!v_x(t)=(dx)/(dt)=-w_0Asin(w_0t)+w_0Bcos(w_0t)!!

so that at t = 0:

!!v_x(0)=(dx)/(dt)=-w_0Asin(0)+w_0Bcos(0)=w_0B!!

Thus !!B=v_0/w_0!!

Now we can perform one more derivative to find the accelaration experienced by an object executing SHM :

!!a_x(t)=(d^2x)/(dt^2)=-w_0^2Acos(w_0t)-w_0^2Bsin(w_0t)!!

So !!a_x(t)=-w_0^2x(t)!!

So we see that the acceleration is proportional to the position and points toward the origin. This type of acceleration is called a linear restoring acceleration, because the acceleration is always pointing back towards the midpoint of the motion (trying to restore the object to its "natural" position).

SHM with springs

A mass connected to a spring experiences a linear restoring interaction: the force of the spring on the mass is !!F_s=−kx!! . The minus sign (very important!) means that the force is always opposite in direction to the displacement. If the mass has moved to the right, the spring will pull it left and vice versa. The force always tries to restore the mass to its equilibrium position, which is taken to be at x = 0 above.

If the mass is pulled to the right and released, the force is leftward until the mass reaches the origin. Hence it will be moving towards the left when it arrives at the origin and will continue moving left until its velocity goes to zero and changes direction (due to the now rightward restoring force).

Now we apply Newton's Second Law: !!F_(t)=ma!!

The only force is the force of the spring !!F_s=−kx!!

!!-kx=ma!!

!!a_x(t)=(d^2x)/(dt^2)=-k/mx(t)!!

We also know that !! a_x(t)=-w_0^2x(t)!!

!!w_0=sqrt(k/m)!!

You can click here for better understanding

Spring and Mass concept

Some points to remember :
1.The Mass is treated as a point particle.
2. Interactions is between the mass and the spring.

The object is attached to one end of a spring. The other end of the spring is attached to the wall at the right. Assume that the object undergoes one-dimensional motion.
The spring is initially stretched a distance !!l_0!! and given some initial speed !!v_0!! to the left away from the equilibrium position. The body again tries to comes back to its original position due to the restoring force.

Applying Newton's Second Law: !!F_t=-kx=ma!! . So we get !!a=-k/mx!!

This is a linear restoring acceleration, giving rise to oscillations at !!ω_0!! with !!sqrt(k/m)!!

The time period of ossilation is given as !!T=(2pi)/w_0=(2pi)sqrt(m/k)!!

For better understanding of the topic click here

Mechanical energy in SHM

Simple harmonic oscillators have experience only the force !!F=−kx!!. This force is conservative, hence mechanical energies is constant and the effects of the force can be accounted for using the potential energy, !!U=1/2kx^2!!.
Just one important thing to keep in mind is there always conservation of energy during the motion.

Introduction to SHM

SHM analysis

Calculating the period, frequency, and amplitude of the mass spring system.

[References : Khan Academy , edX]

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