00:01
In our question we have to write down the differential equation for simple harmonic motion, damp harmonic oscillation and driven damp harmonic oscillation.
00:09
Consider simple harmonic motion.
00:11
Now the force equation is given as f being equal to minus k x where k is the spring constant.
00:18
Let this be equation 1.
00:19
Here the negative sign is due to opposite direction of force and displacement.
00:23
Now there will be a dragging force to oppose the motion and it will depend on velocity.
00:28
This force can be given as f, d being a equal to minus b v here the negative sign is due to the opposite direction of dragging force and velocity b is a constant therefore the net force in dam motion can be given as f being a equal to minus k x minus b v let this be equation 3 but we know that force is equivalent to a product of mass into acceleration which can be given as mass multiplied by d square x x by d t square.
01:11
Substituting the same, in our equation we get m b square x by d t squared equal to minus k x minus b velocity being d x by d t further simplifying we have d square x by d t square plus k x plus b d x x x equal to this is the equation for damped oscillation.
01:45
Now the boundary solution between an under -damped oscillator and an over -damped oscillator occurs at a particular value of friction coefficient and is called critically damped.
02:12
Now if an external time -dependent force is present, the harmonic oscillator that is described as driven oscillator and the concept of the phase is a way of comparing the two oscillations which are occurring at the same time...
