Derive the equation of motion for a simple harmonic oscillator

A simple harmonic oscillator is a system that exhibits periodic motion, where the restoring force is proportional to the displacement from the equilibrium position. 

Examples of simple harmonic oscillators include a mass attached to a spring, a pendulum, and a vibrating string. In this article, we will derive the equation of motion for a simple harmonic oscillator.

Derive the equation of motion for a simple harmonic oscillator-Consider a mass m attached to a spring with spring constant k. The mass is free to move along a horizontal axis. Let x be the displacement of the mass from the equilibrium position, and let F be the net force acting on the mass. According to Newton's second law of motion, the net force acting on an object is equal to the product of its mass and acceleration. Therefore, we have:

• F = ma

The restoring force of the spring is given by Hooke's law, which states that the force exerted by a spring is proportional to its extension or compression. Therefore, the restoring force acting on the mass is given by:

• F = -kx 

Derive the equation of motion for a simple harmonic oscillator-where the negative sign indicates that the force is directed towards the equilibrium position. Substituting this expression for F in the equation of motion, we get:

• -kx = ma

Dividing both sides by m, we get:

• a = -kx/m

Derive the equation of motion for a simple harmonic oscillator-This is the equation of motion for a simple harmonic oscillator. It is a second-order differential equation, since it relates the acceleration of the mass to its displacement from the equilibrium position. The solution to this equation is a sinusoidal function, which describes the periodic motion of the mass.

Also Read-

• Explain The Difference Between Kinematics And Kinetics In Mechanics

To solve this equation, we first make the assumption that the displacement of the mass from the equilibrium position is given by:

• x = A cos(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. The cosine function represents the oscillatory behavior of the mass, since it varies between -A and A as the mass moves back and forth along the axis. The angular frequency ω is related to the frequency f of the motion by the equation:

• ω = 2πf

where π is the mathematical constant pi. The phase angle φ represents the initial position of the mass at t=0.

Taking the first derivative of x with respect to time, we get:

• v = dx/dt = -Aω sin(ωt + φ)

where v is the velocity of the mass. Taking the second derivative of x with respect to time, we get:

• a = d^2x/dt^2 = -Aω^2 cos(ωt + φ)

Derive the equation of motion for a simple harmonic oscillator-Substituting these expressions for x, v, and a in the equation of motion, we get:

• -mAω^2 cos(ωt + φ) = -kA cos(ωt + φ)

Dividing both sides by -mA, we get:

• ω^2 = k/m

This equation relates the angular frequency of the motion to the spring constant and the mass of the oscillator. It is known as the angular frequency equation, and it describes the natural frequency of the oscillator, which is the frequency at which the oscillator oscillates without any external force acting on it.

Substituting the value of ω from the angular frequency equation into the expression for x, we get:

• x = A cos(ωt + φ) = A cos(√(k/m)t + φ)

This is the solution to the equation of motion for a simple harmonic oscillator. It describes the periodic motion of the mass, where the amplitude A and the phase angle φ depend on the initial conditions of the motion.

Conclusion

The equation of motion for a simple harmonic oscillator is a = -kx/m, where the acceleration of the mass is proportional to its displacement from the equilibrium position. 

Derive the equation of motion for a simple harmonic oscillator-This equation is a second-order differential equation, and its solution is a sinusoidal function that describes the periodic motion of the mass. The natural frequency of the oscillator is given by the angular frequency equation ω^2 = k/m, which depends on the spring constant and the mass of the oscillator. 

Derive the equation of motion for a simple harmonic oscillator-The amplitude and phase angle of the motion depend on the initial conditions of the motion. Understanding the equation of motion for a simple harmonic oscillator is important in various fields such as physics, engineering, and mathematics, as it helps in the design of machines and structures, and in the understanding of how objects move in the real world.

FAQ.

Q. What is a simple harmonic oscillator?

Ans. A simple harmonic oscillator is a system that exhibits periodic motion, where the restoring force is proportional to the displacement from the equilibrium position.

Q. What is the equation of motion for a simple harmonic oscillator?

Ans. The equation of motion for a simple harmonic oscillator is a = -kx/m, where the acceleration of the mass is proportional to its displacement from the equilibrium position.

Q. What is the solution to the equation of motion for a simple harmonic oscillator?

Ans. The solution to the equation of motion for a simple harmonic oscillator is a sinusoidal function that describes the periodic motion of the mass.

Q. What is the natural frequency of a simple harmonic oscillator?

Ans. The natural frequency of a simple harmonic oscillator is given by the angular frequency equation ω^2 = k/m, which depends on the spring constant and the mass of the oscillator.

Q. What factors affect the motion of a simple harmonic oscillator?

Ans. The motion of a simple harmonic oscillator is affected by the mass of the oscillator, the spring constant, and the initial conditions of the motion such as the amplitude and phase angle.

Q. What is the importance of understanding the equation of motion for a simple harmonic oscillator?

Ans. Understanding the equation of motion for a simple harmonic oscillator is important in various fields such as physics, engineering, and mathematics, as it helps in the design of machines and structures, and in the understanding of how objects move in the real world.

• Tweet
• Share
• Share
• Share