simple harmonic

SIMPLE HARMONIC MOTION

Definition: A motion which repeats itself after equal intervals of time is called a periodic motion.

Definition: A periodic motion in which a body moves back and forth repeatedly about a mean position is called oscillatory motion.
To execute oscillatory motion, a body must be displaced from its equilibrium position and acted upon by a restoring force directed towards the equilibrium position.
Restoring Force: This force allows the body to return to its mean position, overshoot due to kinetic energy, and then lose speed until kinetic energy is zero, resulting in oscillation.

Definition: If a body moves such that its restoring force is directly proportional to its displacement from a fixed point and always directed towards that point, it is said to execute simple harmonic motion (SHM).
Mean Position: The fixed point around which the body oscillates is called the mean or equilibrium position—where the body would rest if it loses all energy.

The restoring force (F) for displacement (y) is described by: F = -ky
- Where k is the force constant defined as the restoring force per unit displacement, with units of N/m (Newton per meter).
Newton's Second Law: Considering mass (m) of the body and acceleration (a), we can come to the equation: ma = -ky
- Rearranged leads to:
  rac{d^2y}{dt^2} + rac{k}{m}y = 0
- Letting rac{k}{m} = w^2, thus:
  rac{d^2y}{dt^2} + w^2y = 0
- Where:
- rac{d^2y}{dt^2} is the acceleration of the body.
- y is the displacement from equilibrium.
- w is the angular frequency.
Negative Sign Implication: The negative sign ensures that the acceleration is always directed towards the equilibrium position.
- Thus:
  a = -w^2y

Consider a circular motion of radius r, center O, and uniform angular velocity w. The relationship is: y = r ext{sin}( heta)
- For one full cycle:
  heta = wt
- Therefore, the y-coordinate can be represented as:
  y = r ext{sin}(wt)

If heta_0 is the initial phase or phase constant, then:
y = r ext{sin}(wt + heta_0)

The instantaneous velocity v when the displacement is y is given by:
v = rac{dy}{dt} = rw ext{cos}(wt)
From trigonometric identities:
- ext{sin}^2(wt) + ext{cos}^2(wt) = 1
- Squaring both yields other relationships used to derive further motion equations.

The derived relationship for velocity as a function of displacement is: v = rac{ heta}{2} w ext{sqrt}(a^2 - y^2)
- At maximum displacement y = a (extreme positions), velocity is zero since:
  v = w ext{sqrt}(a^2 - a^2) = 0
- At equilibrium position y = 0 (center), velocity is maximum:
  v_{ ext{max}} = aw

Acceleration: Given by: a = rac{d^2y}{dt^2} = -w^2y
- At equilibrium (y = 0), acceleration is zero.
- At extreme positions (y = ext{±}a), maximum acceleration is:
  a = ext{±}aw^2

Amplitude (a): The maximum value of displacement from the mean position.
Time Period (T): The smallest interval of time after which the motion repeats.
Frequency (f): The number of complete cycles (oscillations) per second.
Angular Frequency (w): The rate of angular displacement, defined as: w = 2 ext{π}f
- Unit: radians per second.

A harmonic oscillator in SHM experiences potential and kinetic energies:
Potential Energy (P.E.):
- Derivable from the restoring force:
  P.E. = U = rac{1}{2}ky^2 = rac{1}{2}mw^2y^2
Energy conservation ensures total:
E = ext{K.E.} + ext{P.E.}

Total Energy (E):
- At any displacement: E = rac{1}{2}mw^2a^2
Kinetic Energy (K.E.):
- Given by:
  K.E = rac{1}{2}mv^2 = rac{1}{2}mw^2(a^2 - y^2)
At extreme positions (max displacement): P.E. is max, and K.E. is zero.
At equilibrium position: K.E. is max, and P.E. is zero.

Graphs illustrate how K.E., P.E., and Total Energy vary with displacement.
- Total energy remains constant and unaffected by displacement.
- As displacement increases towards extremes, P.E. increases and K.E. decreases correspondingly.
At maximum displacement, total energy is purely potential; at equilibrium, purely kinetic.