Simple harmonic oscillator
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This document discusses the simple harmonic oscillator. It defines a simple harmonic oscillator as an oscillator that is neither driven nor damped, with motion that is periodic and sinusoidal with constant amplitude. The acceleration of a body in simple harmonic motion is directly proportional to and directed towards the displacement from the equilibrium position. General equations for displacement, velocity, and acceleration as a function of time and other variables are provided. Quantum mechanical treatment of the simple harmonic oscillator is also summarized. Examples of simple harmonic oscillators include a mass on a spring and a simple pendulum in small angle approximation.
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; The 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 𝒙(𝒕) = A cos( 𝟐𝝅𝒇𝒕 + 𝝋) Here, x = Displacement A = Amplitude of the oscillation f = Frequency t = Elapsed time Φ = Phase of oscillationHooke’s Law 𝑭 = − 𝒌𝒙 Where, F = Elastic force k = Spring constant x = Displacement
- 5. Equation Displacement x is given by: 𝒙 𝒕 = 𝑨 𝐜𝐨𝐬(𝝎𝒕 + 𝝋) Differentiating once gives an expression for the velocity at any time 𝒗 𝒕 = 𝒅𝒙 𝒕 𝒅𝒕 = −𝑨𝝎 𝐬𝐢𝐧(𝝎𝒕 + 𝝋) And once again to get the acceleration at a given time: 𝒂 𝒕 = 𝒅 𝟐 𝒙 𝒕 𝒅𝒕 𝟐 = −𝑨𝝎 𝟐 𝐜𝐨𝐬(𝝎𝒕 + 𝝋)
- 6. Simplifying acceleration in terms of displacement Acceleration can, 𝒂 = 𝒅 𝟐 𝒙 𝒅𝒕 𝟐 = − 𝝎 𝟐 𝒙 Acceleration can also be expressed as: 𝒂 𝒕 = − 𝟐𝝅𝒇 𝟐 𝒙(𝒕)
- 7. Simple Harmonic Oscillator – Quantum theory The Schrödinger equation with a simple harmonic potential energy is given by − ћ 𝟐 𝟐𝒎 𝒅 𝟐 𝒅𝒙 𝟐 + 𝟏 𝟐 𝒎ѡ 𝟐 𝒙 𝟐 𝝋 = 𝑬𝝋……………..(1) Where ћ is h-bar, m is the mass of oscillator, ѡ is the angular velocity and E is its energy. The equation can be made dimensionless by letting, 𝒙 ≡ 𝒂𝒚……….(2) 𝒅𝒙 ≡ 𝒂 𝒅𝒚……..(3)
- 8. Then, − ћ 𝟐 𝟐𝒎𝒂 𝟐 𝒅 𝟐 𝒅𝒚 𝟐 + 𝟏 𝟐 𝒎ѡ 𝟐 𝒂 𝟐 𝒚 𝟐 𝝋 = 𝑬𝝋……..(4) Becomes, ( 𝒅 𝟐 𝒅𝒚 𝟐 − 𝒎 𝟐 𝝎 𝟐 𝒂 𝟒 ћ 𝟐 𝒚 𝟐)𝝋 = − 𝟐𝒎𝒂 𝟐 𝑬 ћ 𝟐 𝝋…………(5) Now define, 𝒂 ≡ ћ 𝟐 𝒎ѡ ……………..(6)
- 9. 𝝐 ≡ 𝟐𝒎𝒂 𝟐 𝑬 ћ 𝟐 = 𝟐𝒎𝑬 ћ 𝟐 ћ 𝒎𝝎 = 𝟐𝑬 𝒎𝝎 ………..(7) Then (5) simplifies to, 𝒅 𝟐 𝝋 𝒅𝒚 𝟐 + 𝝐 − 𝒚 𝟐 𝝋 = 𝟎………………(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, 𝝎 = 𝟐𝝅𝒇 = 𝒌 𝑴 Alternately, if the other factors are known and the period is to be found, this equation can be used, 𝑻 = 𝟏 𝒇 = 𝟐𝝅 𝑴 𝒌 The total energy, E is constant, and given by, 𝑬 = 𝒌𝑨 𝟐 𝟐
- 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, 𝑻 = 𝟐𝝅 𝒍 𝒈