To express [tex]\( y(t) = 2 \sin (4\pi t) + 5 \cos (4\pi t) \)[/tex] in the form [tex]\( y(t) = A \sin (\omega t + \phi) \)[/tex], we need to determine the amplitude [tex]\( A \)[/tex], the angular frequency [tex]\( \omega \)[/tex], and the phase shift [tex]\( \phi \)[/tex] of the spring motion.
Let's begin with the calculations:
1. Amplitude [tex]\( A \)[/tex]:
The amplitude [tex]\( A \)[/tex] is found using the formula:
[tex]\[
A = \sqrt{a^2 + b^2}
\][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the coefficients of the sine and cosine terms, respectively. Here, [tex]\( a = 2 \)[/tex] and [tex]\( b = 5 \)[/tex].
[tex]\[
A = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}
\][/tex]
Substituting gives us:
[tex]\[
A \approx 5.385
\][/tex]
2. Angular Frequency [tex]\( \omega \)[/tex]:
The angular frequency [tex]\( \omega \)[/tex] is the same for both sine and cosine functions present in the original equation. It is given by:
[tex]\[
\omega = 4\pi
\][/tex]
3. Phase Shift [tex]\( \phi \)[/tex]:
The phase shift [tex]\( \phi \)[/tex] can be calculated using the arctangent function:
[tex]\[
\phi = \arctan\left(\frac{b}{a}\right)
\][/tex]
where [tex]\( a = 2 \)[/tex] and [tex]\( b = 5 \)[/tex].
[tex]\[
\phi = \arctan\left(\frac{5}{2}\right)
\][/tex]
Substituting gives us:
[tex]\[
\phi \approx 1.190
\][/tex]
Now, we can write [tex]\( y(t) \)[/tex] in the desired form by substituting these values into the equation [tex]\( y(t) = A \sin (\omega t + \phi) \)[/tex]:
[tex]\[
y(t) = 5.385 \sin (4\pi t + 1.190)
\][/tex]
In summary, the amplitude [tex]\( A \)[/tex] is approximately [tex]\( 5.385 \)[/tex], the angular frequency [tex]\( \omega \)[/tex] is [tex]\( 4\pi \)[/tex], and the phase shift [tex]\( \phi \)[/tex] is approximately [tex]\( 1.190 \)[/tex].
