# Find Frequency of Combined Motion Help!

Find the frequency of the combined motion of:

a)sin(2(pi)t)+cos(13(pi)t-pi/4)
b)sin(3t)-cos[(pi)t]

I know to find the period of the combined motion you do T=n1T1=n2T2 where the n's are integers, so I believe the frequency of the combined motion is just the inverse of the combined period.

For a), I found the period of the sine function to be 1/6, and the period of the cosine function to be 2/13, so T=12(1/6)=13(2/13)=2s, so the frequency is 1/2 Hz...but the answer is actually 6.25 hz.

For b) I found the period of the sin function to be (2/3)pi, and that of the cosine function to be 2, so n1(2/3)pi=n22 has no integer solutions, so there is no determined combined period and thus no determined combined frequency either...but the answer is 0.49 Hz...

Can someone help me please? I already successfully did one of them, but I can't get these..

Also, does it make a difference whether the functions are being added or subtracted?

ehild
Homework Helper
It does not matter if the functions are added or subtracted.

If T1, T2 are the periods, the following condition has to be hold for the new period: T=n1*T1=n2*T2, where n1 and n2 are relative primes.

In case a, T1 = 1, T2= 2/13, so n2=13, n1=2 and T=2, f=0.5

In case b, such integers can not be found as one of the periods is irrational. If it were sin(3(pi)t) -cos((pi)*t), then

T1=2/3 and T2=2 ---> T = 3T1=1T2 = 2, f =0.5

I attach a picture that I think helps to see the period of the new signal.

ehild

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Hm so that means I am right, and the answers at the end of the book are wrong...

Thanks.

If I use the equation for the "beats" of two combined motions,

$$x=2Acos(\frac{\omega_1-\omega_2}{2}t)cos(\frac{\omega_1+\omega_2}{2}t)$$

and I concern myself only with the frequency of the envelope, I get the correct answer for both. Why is this?

Btw a) should be a)sin(12(pi)t)+cos(13(pi)t-pi/4)

ehild
Homework Helper
Beat frequency is not the same as the period of combined motion. You get beats in every case, even when

$$x=2Acos(\frac{\omega_1-\omega_2}{2}t)cos(\frac{\omega_1+\omega_2}{2}t)$$

is not periodic.

ehild

• yucheng