Find Frequency of Combined Motion Help

[mmmboh]

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?

 

[mmmboh]

Help please...

 

[ehild]

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

 

[mmmboh]

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

Thanks.

 

[novop]

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?

 

[mmmboh]

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

 

[ehild]

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