Simple Harmonic Motion Formula

We see different kinds of motion in every day of life. The motion of the hands of a clock, motion of the wheels of a car, etc is such type of motion. We easily find easily these types of motion keep repeating themselves. Such motions are periodic in their nature. One such type of periodic motion is simple harmonic motion i.e. SHM. In this topic, we will discuss the simple harmonic motion formula with examples. Let us learn the concept!

                                                                                                                                      Source: en.wikipedia.org

Simple Harmonic Motion Formula

What is Simple Harmonic Motion?

A motion that repeats itself in equal intervals of time is periodic in nature. We will see what periodic motion is to understand simple harmonic motion.

Periodic motion is the motion in which an object repeats its path in some equal intervals of time. We see many examples of periodic motion in our everyday life. The motion of the hands of a clock is a type of periodic motion. The rocking of a cradle, swinging on a swing, leaves of a tree moving to and fro due to wind breeze, etc are examples of periodic motion.

The particle performs the same set of movements repeatedly in a periodic motion. One such set of movements is an Oscillation. An example of such an oscillatory motion is Simple Harmonic Motion. When an object moves to and fro along some line, then the motion is simple harmonic motion. Oscillations of a pendulum is a type of simple harmonic motion.

The formula for SHM

Suppose that there is a spring fixed at one end. When there is no force applied to it, it is at its equilibrium position. Now, if we pull it outwards, then there is a force exerted by the string which is directed towards the equilibrium position.

If we push the spring inwards, then there is a force exerted by the string towards the equilibrium position. Therefore, we can see that the force imposed by the spring is towards the equilibrium position. This force is known as the restoring force.

Let the force be F and the displacement of the string from the equilibrium position be x.

Therefore, the restoring force is given by,

F= – kx

Here, the negative sign indicates that the force is in the opposite direction.

Here, k is the constant known as the force constant. Its unit is Newton per meter.

Now, for a string, let its mass be m. Then the acceleration of the body is:

\(a = \frac{F}{m}\)

\(a = – k \times \frac{x}{m}\)

 \(= – {\omega}^2 x\)

Here, \(\frac{k}{m} = {\omega}^2\)

The time taken by an object for completing its one oscillation is called time period. The frequency of SHM is the number of oscillations that a particle performs per unit amount of time. Thus, the frequency of the oscillatory motion is:

\(f= \frac{1}{T}\)

Where,

a	Acceleration
F	Force
T	Time Period
m	Mass
f	Frequency
k	force constant
\(\omega\)	Angular frequency

Solved Examples

Q.1: What is the value of acceleration at the mean position of simple harmonic motion?

Solution: At mean position, x =0

Acceleration =\( -{\omega}^2 x\)

= \(-{\omega}^2 \times 0\)

Acceleration = 0.

Therefore, the value of acceleration at the mean position is minimum and it will be zero.