Simple harmonic oscillator

Simple harmonic oscillator

1. Simple Harmonic Oscillator

2. Group Members : Group 6 Shishir Karmoker Md. Nahid Ahosan Uthpol Kisor Mithu Tanjina Zaman Shosy 2016-2-55-008 2016-2-55-011 2016-2-55-009 2015-1-60-196

3. What is simple harmonic oscillator? Simple harmonic oscillator (SHO) is the oscillator that is neither driven nor damped. • The motion is periodic and sinusoidal. • With constant amplitude; T he acceleration of a body executing Simple Harmonic Motion is directly proportional to the displacement of the body from the equilibrium position and is always directed towards the equilibrium position .

4. General Equation = A cos(   H ere, x = Displacement A = Amplitude of the oscillation f = Frequency t = Elapsed time Φ = Phase of oscillation Hooke’s Law   W here, F = Elastic force k = Spring constant x = Displacement

5. Equation Displacement x is given by: Differentiating once gives an expression for the velocity at any time And once again to get the acceleration at a given time:  

6. Simplifying acceleration in terms of displacement Acceleration can, Acceleration can also be expressed as:  

7. Simple Harmonic Oscillator – Quantum theory The Schrödinger equation with a simple harmonic potential energy is given by ……………..(1) Where is h-bar, m is the mass of oscillator, is the angular velocity and E is its energy. The equation can be made dimensionless by letting, ……….(2) ……..(3)  

8. Then, ……..(4 ) Becomes, = …………(5 ) Now define, ……………..(6 )  

9. = = ………..(7 ) Then (5) simplifies to, ………………(8 )  

10. Examples

11. Mass on a spring A mass M attached to a spring of spring constant k exhibits simple harmonic motion in space with , Alternately, if the other factors are known and the period is to be found, this equation can be used, The total energy, E is constant, and given by ,  

12. Mass on a simple pendulum In the small-angle approximation, the motion of a simple pendulum is approximated by simple harmonic motion. The period of a mass attached to a string of length with gravitational acceleration g is given by,  

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