Question

Both the x and y coordinates of a point execute simple harmonic motion. The frequencies are the same but the amplitudes are different. The resulting orbit might be:

Answer 1

the orbit resulting an ELIPSE

Step-by-step explanation:

Harmonic motion is described by the expression

x = A cos (wt +Ф₁)

In this exercise it is established that in the y axis there is also a harmonic movement with the same frequency, so its equation is

y = B cos (wt + Ф₂)

the combined motion of the two bodies can be found using the Pythagorean theorem

R² = x² + y²

R² = [A² cos² (wt + Ф₁) + B² cos² (wt + Ф₂)]

to simplify we can assume that the phase in the two movements are equal

R = √(A² + B²) cos (wt + Ф)

If the two amplitudes are equal we have a circular motion, if the two amplitudes are different in elliptical motion, the amplitudes of the two motions are

circular R² = (A² + A²)

elliptical R² = (A² + B²)

We see from the last expression that the broadly the two axes is different, so the amplitude is an ellipse.

By which the orbit resulting an ELIPSE