Question

How to prove that 'coswt+sin2wt+cos4wt' represents a periodic motion ?

Answer 1

Hi

yes they represent period and periodic is

LCM of (2pi/w,2pi/2w,2pi/4w)=2pi/2w=pi/w

hence period is pi/w

Answer 2

We need to prove that cos wt + sin 2 wt + cos 4 wt represent the periodic motion.

We know that :

omega = 2 π / t

= > t = 2 π / omega

I am using omega as w .

t = time period .

In the first case :

t = w

Second one :

t = 2 w

Third one ,

t = 3 w

Clearly the time period is periodically varying .

To prove they are periodic , we can find the H.C.F of the data :

H.C.F of 2 π / w , 2 π / 2 w and 2 π / 4 w

            = 2 π / w

Hence the time period is changing by :

2 π /  w - 2 π /2 w  =  π / w

Next time :

2 π / 2 w - 2 π / 4 w = 2 π / w

Thus the change is 2 π / w - π / w = π / w

π / w is always constant and hence the time period is constant !

and hence it is periodical.

Hence proved.