Periodic Motion

Class 12 Periodic Motion Notes and Numerical Solution

Topics To be covered in Periodic motion

• Periodic or Simple Harmonic Motion
• Equation of Simple Harmonic
• Motion
• Energy of a Particle Executing
• S.H.M.
• Application of Simple Harmonic
• Motion
• Simple Pendulum
• Angular Simple Harmonic Motion
• Types of Oscillation and Resonance

Simple Harmonics Motion

If a particle moves to and fro about a mean position in a straight line such that the acceleration is directed towards the mean position and is directly proportional to the displacement from that position is called SHM.

Simply , The motion of a body is said to be in SHM if its acceralation is directed towards the mean position and is directly proportional to that position.

i.e $acceleration\:\infty \:displacement$

$acceleration\:=-k\cdot displacement$ :- where $k$ is constant and the negative sign shows that acceleration is directed in opposite to the the motion of the object.

Characteristics of SHM

Relation betwen the acceleration and displacement of the particle executing SHM

Let us consider a body having mass '$m$' is moving in a circular path of radius $r$ with constant speed $v$ .Suppose body is initially at point A and after time $t$ it reaches to point B describing angular displacement $ \theta $ 

Let us draw perpendicular BN on verticle diamenter $yy'$.Also let when the particle at point $B$ , the displacement by $y$ such that ON = $y$

From Fig 

$$\sin \left(\theta \right)=\frac{y}{r}$$

$$or,\:y=r\sin \left(\theta \right) \:---\:i\:$$
$$ \:If\:\omega \:be\:the\:angular\:velocity\:then\:\theta \:=\:\omega t\:$$

$$ \therefore y =r\sin\left(\omega t\right) \:---\:ii\: $$

This is the displacement equation for a particle at SHM ,Now velocity of SHM be -

$$ v=\frac{dy}{dx}$$

$$=\frac{d\left(r\sin \left(\omega t\right)\right)}{dt}$$

$$ \therefore  v= v=\:r\omega \cos \omega t\:---\:iii$$

$$ eqn iii can also be written as $$

$$ $$

I.e. a $ \propto $ y where, ‘a’ is acceleration and ‘y’ is displacement.

Or, a = -k y, -ve sign show that ‘a’ is opposite to y

Displacement(y):

It is defined as the distance from mean position (XX¹ of O) of a body executing SHM. It is denoted by ‘y’.

In right angle $\Delta $OPM, sin$\theta $ = OM/OP = y/r or, y = rsin$\theta $

We know, angular velocity, $\omega $ = $\frac{\theta }{{\rm{t}}}$∴$\theta $ =${\rm{\: }}\omega $t

Putting the value of $\theta $ in eqni) we get,

y = r sin${\rm{\: }}\omega $t