The following graph shows the variation of displacement 'y' of a particle undergoing SHM with a quantity 'R'. Then 'R' is likely to be 

The correct answer is option 3) i.e. Total energy of the particle

CONCEPT:

• Simple harmonic motion (SHM): It is a type of oscillatory motion in which the restoring force is directly proportional to the displacement of the body from its mean position.
  • ​This means that the net force which in turn is the acceleration is proportional to the displacement of the object and acts in the opposite direction of the displacement.
  • The displacement in an SHM for a system starting at equilibrium (x = 0) is given by the equation:

⇒ x = asinωt

​​​Where x is the displacement, a is the amplitude, ω is the angular frequency and t is the time taken. 

• The energy in simple harmonic motion: 
• The kinetic energy is given by the equation: 

\(⇒ KE = \frac{1}{2}mω ^2 (a^2 - y^2)\)

• The potential energy is given by the equation: 

\(⇒ PE = \frac{1}{2} mω ^2y^2\)

Where m is the mass, ω is the angular frequency, y is the displacement of the particle from the mean position and a is the amplitude.

EXPLANATION:

• The given graph tells us that the quantity R is not dependent on displacement 'y'. 
• The total energy of a particle executing SHM is the sum of its kinetic and potential energy.

The total energy, E = KE + PE

\(⇒ E = \frac{1}{2}mω ^2 (a^2 - y^2) + \frac{1}{2} mω ^2y^2\)

\(⇒ E = \frac{1}{2} mω ^2a^2\)

• It is inferred from the above equation that the total energy 'E' does not depend on the displacement of the particle 'y' in SHM.
• Therefore, the quantity 'R' in the given graph represents the total energy of the particle.