Question

did you guys get this : The x-component of motion of an object is given by x(t) = Axcos(ωxt + φx) and the y-component of motion of the object is given by y(t) = Aycos(ωyt + φy). What relationships between the A,ω, and φ parameters must be true so that the motion of the object is on a circle?

Answer 1

Aₓ = A_y = A

wₓ = w_y = w

Фₓ = Ф_y = Ф

Step-by-step explanation:

For a movement to be circular it must meet the exception of the circle

R² = x² + y²

in the exercise indicate the expressions of the movement in the two axes

x (t) = Aₓ cos (wₓ t + Фₓ)

y (t) = A_y cos (w_y t + Ф_y)

we substitute

R² = Aₓ² cos² (wₓ t + Фₓ) + A_y² sin² w_y t + Ф_y)

for this expression to be a circle it must meet

Aₓ = A_y = A

wₓ = w_y = w

Фₓ = Ф_y = Ф

with these expressions

R² = A² [cos² (w t + Ф) + sin² (wₓ t + Фₓ) ]

use the trigonometry relationship

cos² θ + sin² θ = 1

R² = A²

Therefore, it is fulfilled that it is a circle whose radius is equal to the amplitude of the movement