# Simple Harmonic Motion

Simple Harmonic Motion

In the preceding sections, the motion of a body when acted upon by a constant force was considered. The motion is one of constant acceleration. In this section, the motion of a body subjected to a variable force is considered. There is one common and important type of non-uniformly accelerated motion that can be analysed easily. This motion is called periodic motion. A motion is said to be periodic if it repeats at regular intervals of time.

The motion of the hands of a clock, movement of a planet around the sun and the motion of the blades of a fan are examples of repetitive motion. Here, any point in the path of the body is crossed by the body in the same direction, at regular intervals of time. On the other hand, the motion of a body attached to a suspended spring, the motion of the prongs of an excited tuning fork, the motion of the plucked string of a musical instrument, the motion of the pendulum of a clock, the motion of the balance wheel of a watch are also examples of periodic motion but oscillatory. i.e., a to and fro motion. A given point in the path is crossed by the body in opposite directions at regular intervals. This is called oscillatory motion.

What is Simple Harmonic Motion?

We already know about oscillatory and periodic motions. Here we shall be talking about a special case of periodic motion that is simple harmonic motion. So what is simple harmonic motion? We can define simple harmonic motion as a motion in which the restoring force is directly proportional to displacement of the body from the mean position. And this restoring force always acts towards the mean position.

Mathematically it is represented as, F = – Kx

Where,

K = Force Constant

x = Displacement from the mean positon

Now we will try to find the displacement as a function of time. We have,

F = – Kx

ma = – Kx

$$m\frac{{{d}^{2}}x}{d{{t}^{2}}}+{{\omega }^{2}}x=0$$

Where,

$$\omega =\sqrt{\frac{K}{m}}$$

Solving this we get,

x (t) = A sin (ωt + φ)

Where,

A = Amplitude of Oscillation

Φ = Initial Phase

So graphically Simple Harmonic Motion (SHM) can be represented as a sine wave as shown below. At time t = 0, the displacement is zero. Hence,  for this case. Some examples of simple harmonic motion are a pendulum, a block connected to a spring etc.

Now we know that displacement is a function of time. We can find velocity and acceleration of Simple Harmonic Motion (SHM).

Velocity of SHM = $$\frac{dx}{dt}$$

v(t) = Aω cos (ωt + φ)

Hence, velocity and displacement are 900 out of phase. Also velocity is maximum at mean position and minimum at extreme positons.

Now for acceleration,

Acceleration of SHM = $$\frac{dx}{dt}$$

a (t) = – ω² x sin (ωt + φ)

We can also say that,

a (t) = – ω² x (t)

Acceleration is acting always opposite to the displacement. It is maximum at extreme positions and minimum at mean position.

For finding the time period of the Simple Harmonic Motion (SHM) we consider,

F = ma

F = – m ω² x (t)

Also, F = – K x (t)

Using above equations,

mω² x(t) = K x(t)

$$\omega =\sqrt{\frac{K}{m}}$$

Therefore,  $$T=\frac{2\pi }{\omega }$$

$$T=2\pi \sqrt{\frac{m}{K}}$$.

So, now we can see how time period is related with mass and force constant.