Periodic motion characterises all simple harmonic motions. The item oscillates in SHM, moving back and forth between its extreme and mean positions. The restoring force, which is directly proportional to the size of an object's displacement from its mean position but acts in the opposite direction as the displacement, is felt by the oscillating object during the whole oscillation process. It is possible to write it as F α -x. Cradle, swing, pendulum, guitar, bungee leaping, and other real-world instances of SHM include motions that have their restoring force opposite the displacement.
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In this article, we will cover the concept of the composition of two SHM. This concept is part of Oscillations and Waves, which is a crucial chapter in Class 11 physics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, VITEEE and more. Over the last ten years of the JEE Main and NEET (from 2013 to 2023), eight and three questions have been asked on this concept respectively.
If a particle is acted upon by two forces such that each force can produce SHM, then the resultant motion of the particle is a combination of SHM.
Composition of two SHM in the same direction
Let a force
Let another force
Now if force
Note: Here the frequency of each SHM's are the same
And the resulting phase is given by
Let a force F1 on a particle produces an SHM given by
and a force F2 alone produces an SHM given by
Both the force F1 and F2 acting perpendicular on the particle will produce an SHM whose resultant is given by:
The above equation is the general equation of an ellipse. That is two forces acting perpendicular on a particle execute SHM along an elliptical path.
It is a straight line with a slope
When
which is represented by below straight line with slope
When
It represents a normal ellipse
Example 1: The SHM of a particle is given by the equation
1) 5
2) 1
3) 7
4) 12
Solution:
Resultant Amplitude of Two SHM -
Hence, the answer is the option (1).
Example 2: The motion of a particle varies with time according to the relation
1) The motion is oscillatory t but not SHM
2) The motion is SHM with amplitude
3) The motion is SHM with amplitude
4) The motion is SHM with amplitude
Solution:
Both SHMs are along the same direction and of the same frequency.
Hence, the answer is the option (3).
Example 3: A particle executing simple harmonic motion along
1)
2)
3)
4)
Solution:
The amplitude is the maximum displacement from the mean position.
This question is based on the concept of shifting of mean position.
At mean position
and the equation is
So the Amplitude is A.
Hence, the answer is the option (1),
Example 4: A simple harmonic oscillator of angular frequency 2 rad s-1 is acted upon by an external force F=sint N. If the oscillator is at rest in its equilibrium position at t=0, its position at later times is proportional to:
1)
2)
3)
4)
Solution:
From the equation of motion, we have
The general solution of equation (2) consists of a sum of two parts,
The first part is the solution let's say x=P(t) which satisfies equation (2), is called a particular solution.
The second part is the solution let's say x=S(t) which satisfies equation (2) with F(t)=0, is called a specific solution.
for x=P(t)
We try a solution of type
So
and the specific solution is given by
For which the solution is given as of SHM
Now The general solution is given as
Given
So using x(t)=0 at t=0
Now using
we get
taking k=0
Hence, the answer is the option (4).
Example 5: A particle executes simple harmonic motion and is located at
Solution:
Hence, the answer is the option 1.
Adding two oscillatory motions of two Simple Harmonic Motions (SHM) with the possibility of different amplitudes, frequencies and phases leads to a compound motion. The resultant would be found by adding their individual displacements using vector addition provided they are along one line. In addition, if they have similar frequencies, the resultant motion will be SHM but with a different amplitude and phase.
Q 1. What is Simple harmonic motion?
Ans: Simple harmonic motion is the simplest form of oscillatory motion in which the particle oscillates on a straight line and the restoring force is always directed towards the mean position and its magnitude at any instant is directly proportional to the displacement of the particle from the mean position at that instant i.e. Restoring force α Displacement of the particle from the mean position.
Q 2: Give the example of periodic motion.
Ans: Circular motion with uniform speed.
Q 3: What is Osillation?
Ans: An Oscillation is a special type of periodic motion in which a particle moves to and fro about a fixed point called the mean position of the particle.
Q 4: Which of the following is a necessary and sufficient condition for SHM?
Ans: Mean Position: A position during oscillation where the particle is at the equilibrium position, i.e. net force on the particle at this position is zero.
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