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;
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