Oscillation and Simple harmonic oscillations – Definition

Definition of Oscillation


The term Oscillation refers to a Periodic Motion of any object moving at a distance about the equilibrium position and repeats itself over and over for a period of time. Repeating back and forth movement over the same path about an equilibrium position is called Oscillatory motion.
Example of oscillation: Up and down of a spring. The motion of the pendulum.

Simple harmonic oscillations

Oscillations, where the net force on the system is a restoring force, known as simple harmonic motion and the system is known as a harmonic oscillator. This can be understood with a simple explanation, take a spring resting on a horizontal surface. Fixed its one end with an unmovable object and the other end with a movable object. In equilibrium condition, nothing moves and the spring at its relaxed length.

Now, disturbing the equilibrium, by Pull or push the object of mass ‘m’ parallel to the axis of the spring and stand back. The system will oscillate back and forth under the restoring force of the spring. A force acts in the direction opposite to the displacement from the equilibrium position called restoring force. If the spring obeys Hooke’s law and restoring force is directly proportional to extension then the system is called simple harmonic oscillator and the way it moves is called simple harmonic motion.

Oscillation and Simple harmonic oscillations - Definition

Begin the analysis with Newton’s second law of motion.

∑F = ma eq.1

The restoring force of the spring is actually negative as it works in the opposite direction to the displacement (let’s’) of the mass ‘m’ from the equilibrium state. On Replacing net force with Hooke’s law, and put acceleration, a= d2x/dt2 into equation 1, we get

F=k.d eq.2 (hooks law)

Wherek=spring constant,

d =spring stretched or compressed

On Equating Eq. 1 and 2 we get.

− k d= m d2x


Rearrange the terms we get eq.3 written as

− k d = d2x

m dt2

The above equation is called a second-order, linear differential equation. In this equation, on the lefthand side, we have a function with a negative sign in front of it and some coefficients. On the right-hand side, we have the 2nd derivative of that function. So the solution for this type of equation is a function whose 2nd derivative is itself with a negative sign. We have two possible functions that satisfy the given requirement is— sine and cosine —these two functions are principally the same since one is a phase-shifted version of the other. A derivative of a trigonometric phase-shifted function is also a phase-shifted.

The solution of our2nd order differential equation is a sinewith its phase shift. Here a term angular frequency comes into consideration it counts the radians/second. A sequence of events that repeats itself is known as a cycle. So, the sine function repeats itself after moving 2π radian. The motion of a harmonic oscillator repeats itself after moving through one complete cycle of simple harmonic motion.so mathematically it can be written as
ω=ϕ/t (2 π radian/ 1 period) eq.4

                                             f = n/t (1 cycle/ 1 period)                eq. 5

Divide eq 4 by 5 we get

ω/f = ϕ / n (2 π radians per cycle)

As both radians and cycles are unit lessquantities that means…

            ω   /f =    2π radian   /1 cycle    
                ω = 2   πf      

Here’s the general form of solution to the simple harmonic oscillator can be written as

X1 = A1 sin(2πft + φ)

where the terms define as.

X1 = position

A1 = amplitude

f = frequency

t = time

φ = phase

X1 = A1 sin(2πft + φ)

Find its first derivative…dX1/dt = 2π f A1cos(2 π f t+ϕ)

also we can find its second derivative…d2X1/dt2= -4 π2 f2A sin(2 πft+ϕ)

Feed the equation and its second derivative back into the differential equation…3.After solving we get the result

k/m=4 π2 f2

in terms of frequency, it can be written as

f=1/2 π√k/m

or inverting it we get simply period of harmonic motion

T=2 π√m/k

• So, the simple harmonic motion evolves over a period of time with a frequency that depends on upon stiffness of the restoring force (f) and the mass (m) of the object in motion

• A heavier mass oscillates with a longer period and a stiffer spring oscillates with a smaller period.

• Frequency and time period are not affected by the amplitude.

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