the position of a weight attached to a spring is s(t)= - 3 cos 4πt inches after t seconds. complete parts…

the position of a weight attached to a spring is s(t)= - 3 cos 4πt inches after t seconds. complete parts a)-d) below.\nb) what are the frequency and period?\nthe frequency of the oscillation is 2 cycles per sec.\n(type an integer or a fraction. type an exact answer, using π as needed.)\nthe period of the oscillation is $\frac{1}{2}$ sec.\n(type an integer or a fraction. type an exact answer, using π as needed.)\nc) when does the weight first reach its maximum height?\nafter $\frac{1}{4}$ sec\n(type an integer or a fraction. type an exact answer, using π as needed.)\nd) calculate and interpret s(1.2)\ns(1.2)= (round to the nearest integer as needed.)

the position of a weight attached to a spring is s(t)= - 3 cos 4πt inches after t seconds. complete parts a)-d) below.\nb) what are the frequency and period?\nthe frequency of the oscillation is 2 cycles per sec.\n(type an integer or a fraction. type an exact answer, using π as needed.)\nthe period of the oscillation is $\frac{1}{2}$ sec.\n(type an integer or a fraction. type an exact answer, using π as needed.)\nc) when does the weight first reach its maximum height?\nafter $\frac{1}{4}$ sec\n(type an integer or a fraction. type an exact answer, using π as needed.)\nd) calculate and interpret s(1.2)\ns(1.2)= (round to the nearest integer as needed.)

Answer

Explanation:

Step1: Recall the cosine - function formula

The general form of a cosine - function for harmonic motion is $s(t)=A\cos(\omega t+\varphi)$, where $A$ is the amplitude, $\omega$ is the angular frequency, and $\varphi$ is the phase shift. In the given function $s(t)= - 3\cos(4\pi t)$, $A = 3$, $\omega=4\pi$, and $\varphi = 0$.

Step2: Calculate the period $T$

The formula for the period of a cosine - function in harmonic motion is $T=\frac{2\pi}{\omega}$. Substituting $\omega = 4\pi$ into the formula, we get $T=\frac{2\pi}{4\pi}=\frac{1}{2}$ seconds.

Step3: Calculate the frequency $f$

The frequency $f$ and the period $T$ are related by the formula $f=\frac{1}{T}$. Since $T=\frac{1}{2}$, then $f = 2$ cycles per second.

Step4: Find when the weight reaches its maximum height

The cosine function $y=\cos(x)$ has a maximum value of 1 when $x = 2k\pi$, $k\in\mathbb{Z}$. For $s(t)=-3\cos(4\pi t)$, we want to find $t$ when $\cos(4\pi t)=- 1$ (because of the negative sign in front of the cosine function). So $4\pi t=(2k + 1)\pi$. Solving for $t$, we get $t=\frac{2k + 1}{4}$. When $k = 0$, the weight first reaches its maximum height at $t=\frac{1}{4}$ seconds.

Step5: Calculate $s(1.2)$

Substitute $t = 1.2$ into $s(t)=-3\cos(4\pi t)$. Then $s(1.2)=-3\cos(4\pi\times1.2)=-3\cos(4.8\pi)$. Since $\cos(4.8\pi)=\cos(0.8\pi)\approx - 0.309$, then $s(1.2)=-3\times(-0.309)=0.927\approx1$.

Answer:

1