## Energy in simple Harmonic Motion – Class 11 | Chapter – 14 | Physics Short Notes Series PDF for NEET & JEE

Energy in simple Harmonic Motion: In Simple Harmonic Motion (SHM), the total mechanical energy of the system is constant and is equal to the sum of the kinetic energy and potential energy of the object. The potential energy of the object is stored in the spring that is responsible for the SHM.

## Energy in Simple Harmonic Motion

Let us consider an object of mass m attached to a spring of spring constant k, free to move along a horizontal axis. The equilibrium position of the object is defined as the position where the spring is in its natural, unstretched state.

At any point in its motion, the object has a potential energy U and a kinetic energy K. The potential energy of the object is given by:

U = (1/2) kx2

where k is the spring constant of the spring, and x is the displacement of the object from its equilibrium position.

The kinetic energy of the object is given by:

K = (1/2) mv2

where m is the mass of the object and v is its velocity.

Using the equation of motion for SHM, we can express the velocity of the object as:

v = ± Aωsin(ωt + Φ)

where A is the amplitude of the motion, ω is the angular frequency, t is the time, and Φ is the phase angle.

Substituting this expression for the velocity into the equation for the kinetic energy, we get:

K = (1/2) m(Aω)2 sin2(ωt + Φ)

Using the trigonometric identity sin^2(θ) = (1/2)(1 – cos 2θ), we can simplify this expression as:

K = (1/2) m(Aω)2 [1 – cos 2(ωt + Φ)]

The total mechanical energy of the system is given by the sum of the kinetic and potential energy:

E = K + U

Substituting the expressions for K and U, we get:

E = (1/2) kx^2 + (1/2) m(Aω)2 [1 – cos 2(ωt + Φ)]

Simplifying this expression using the trigonometric identity cos 2θ = 2cos2(θ) – 1, we get:

E = (1/2) kA2 – (1/2) kx2 = constant

We can see that the total mechanical energy of the system is constant and is equal to the sum of the potential energy and kinetic energy of the object. The kinetic energy of the object is maximum at the equilibrium position, while the potential energy is maximum at the extreme positions. As the object moves back and forth, the kinetic and potential energy interchange, but the total energy of the system remains constant.

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