1. **Problem statement:** A particle moves vertically on a spring with position given by $$s(t) = 4 + \frac{1}{25} \sin(100\pi t)$$ where $s(t)$ is in cm and $t$ in seconds. (a) Find the time for one complete vibration (period). (b) Find the velocity at $t=1$, $t=1.005$, $t=1.01$, and $t=1.015$ seconds. --- 2. **Formula and explanation:** The function is sinusoidal: $$s(t) = A + B \sin(\omega t)$$ where $A=4$, $B=\frac{1}{25}$, and angular frequency $\omega = 100\pi$. - The period $T$ of a sinusoidal function is given by $$T = \frac{2\pi}{\omega}$$. - Velocity $v(t)$ is the derivative of position: $$v(t) = \frac{ds}{dt}$$. --- 3. **Calculate the period:** $$T = \frac{2\pi}{100\pi} = \frac{2\cancel{\pi}}{100\cancel{\pi}} = \frac{2}{100} = 0.02 \text{ seconds}$$ So, one complete vibration takes $0.02$ seconds. --- 4. **Find velocity function:** $$v(t) = \frac{d}{dt} \left(4 + \frac{1}{25} \sin(100\pi t)\right) = 0 + \frac{1}{25} \cdot 100\pi \cos(100\pi t) = \frac{100\pi}{25} \cos(100\pi t) = 4\pi \cos(100\pi t)$$ --- 5. **Evaluate velocity at given times:** - At $t=1$: $$v(1) = 4\pi \cos(100\pi \times 1) = 4\pi \cos(100\pi)$$ Since $\cos(n\pi) = (-1)^n$ and $100$ is even, $\cos(100\pi) = 1$. $$v(1) = 4\pi \times 1 = 4\pi \approx 12.566$$ - At $t=1.005$: $$v(1.005) = 4\pi \cos(100\pi \times 1.005) = 4\pi \cos(100\pi + 0.5\pi)$$ Using $\cos(a + b) = \cos a \cos b - \sin a \sin b$ and $\cos(100\pi) = 1$, $\sin(100\pi) = 0$: $$v(1.005) = 4\pi (1 \cdot \cos 0.5\pi - 0 \cdot \sin 0.5\pi) = 4\pi \times 0 = 0$$ - At $t=1.01$: $$v(1.01) = 4\pi \cos(100\pi \times 1.01) = 4\pi \cos(100\pi + \pi)$$ Since $\cos(\theta + \pi) = -\cos \theta$ and $\cos(100\pi) = 1$: $$v(1.01) = 4\pi \times (-1) = -4\pi \approx -12.566$$ - At $t=1.015$: $$v(1.015) = 4\pi \cos(100\pi \times 1.015) = 4\pi \cos(100\pi + 1.5\pi)$$ Using $\cos(\theta + 1.5\pi) = \cos(\theta + \pi + 0.5\pi) = -\cos(\theta + 0.5\pi)$ and from above $\cos(100\pi + 0.5\pi) = 0$: $$v(1.015) = 4\pi \times 0 = 0$$ --- **Final answers:** - (a) Period $T = 0.02$ seconds - (b) Velocities: - $v(1) = 4\pi \approx 12.566$ cm/s - $v(1.005) = 0$ cm/s - $v(1.01) = -4\pi \approx -12.566$ cm/s - $v(1.015) = 0$ cm/s