Question

Write y(t)=2sin(4πt) + 5cos(4πt) in the form y(t)=A sin(wt+ Φ) and identify the amplitude, angular frequency, and the phase shift of the spring motion.

More background info:

y(t) = distance of weight from equilibrium position
w= angular frequency (measured in radians per second)
A = amplitude
Φ = phase (depends on initial conditions)
c1 = AsinΦ
c2 = AcosΦ

Answer 1

The given equation can be rewritten in the form y(t) = A sin(wt + Φ), where A is the amplitude, w is the angular frequency, and Φ is the phase shift.

Step-by-step explanation:

The given equation, y(t) = 2sin(4πt) + 5cos(4πt), can be rewritten in the form y(t) = A sin(wt + Φ) using trigonometric identities.

Amplitude (A) represents the maximum displacement from the equilibrium position. In this case, the amplitude is the square root of the sum of the squares of the coefficients of sine and cosine terms. So, amplitude = √(2^2 + 5^2) = √29.

Angular frequency (w) is equal to 2π times the frequency of the wave. In this case, the angular frequency is 4π.

Phase shift (Φ) can be determined by comparing the given equation with the general equation and solving for Φ.