6- Analise o oscilograma referente a oscilação de uma particula. a) Determine a Amplitude e o periodo da oscilação. b) Calcula a frequência da oscilação. c) Calcula a frequência angular. d) Quantas oscilações a partícula réuliza por minuto?

The amplitude is the maximum displacement from the equilibrium position.
From the graph, the maximum displacement $y_{max}$ is $0.8 \text{ dm}$.

$$A = 0.8 \text{ dm}$$

The period is the time taken for one complete oscillation.
One complete cycle (e.g., from $t=0$ to the next point where the wave repeats its pattern) occurs at $t = 0.4 \text{ s}$.

$$T = 0.4 \text{ s}$$

The amplitude is $\boxed{0.8 \text{ dm}}$ and the period is $\boxed{0.4 \text{ s}}$.

b) Calcula a frequência da oscilação.

$$f = \frac{1}{0.4 \text{ s}}$$ $$f = 2.5 \text{ Hz}$$

The frequency is $\boxed{2.5 \text{ Hz}}$.

c) Calcula a frequência angular.

$$\omega = 2\pi (2.5 \text{ Hz})$$ $$\omega = 5\pi \text{ rad/s}$$ $$\omega \approx 5 \times 3.14159 \text{ rad/s}$$ $$\omega \approx 15.71 \text{ rad/s}$$

The angular frequency is $\boxed{5\pi \text{ rad/s}}$ or approximately $\boxed{15.71 \text{ rad/s}}$.

d) Quantas oscilações a partícula realiza por minuto?

$$\text{Oscillations per minute} = 2.5 \text{ oscillations/s} \times 60 \text{ s/min}$$ $$\text{Oscillations per minute} = 150 \text{ oscillations/min}$$

The particle performs $\boxed{150 \text{ oscillations per minute}}$.

: Electrical Circuit Analysis

a) A resistência total.

$$R_2 = 6 \Omega$$ $$R_3 = 3 \Omega$$ $$\frac{1}{R_{23}} = \frac{1}{R_2} + \frac{1}{R_3}$$ $$\frac{1}{R_{23}} = \frac{1}{6 \Omega} + \frac{1}{3 \Omega}$$ $$\frac{1}{R_{23}} = \frac{1}{6 \Omega} + \frac{2}{6 \Omega}$$ $$\frac{1}{R_{23}} = \frac{3}{6 \Omega}$$ $$\frac{1}{R_{23}} = \frac{1}{2 \Omega}$$ $$R_{23} = 2 \Omega$$

$$R_1 = 3 \Omega$$ $$R_{total} = R_1 + R_{23}$$ $$R_{total} = 3 \Omega + 2 \Omega$$ $$R_{total} = 5 \Omega$$

The total resistance is $\boxed{5 \Omega}$.

b) A intensidade da corrente que percorre o circuito.

$$V_{total} = 15 \text{ V}$$ $$I_{total} = \frac{V_{total}}{R_{total}}$$

$$I_{total} = \frac{15 \text{ V}}{5 \Omega}$$ $$I_{total} = 3 \text{ A}$$

The total current is $\boxed{3 \text{ A}}$.

c) As correntes que percorrem as resistências R2 e R3.

This voltage is the same for $R_2$ and $R_3$.

$$V_{23} = I_{total} \times R_{23}$$ $$V_{23} = 3 \text{ A} \times 2 \Omega$$ $$V_{23} = 6 \text{ V}$$

$$I_2 = \frac{V_{23}}{R_2}$$ $$I_2 = \frac{6 \text{ V}}{6 \Omega}$$ $$I_2 = 1 \text{ A}$$

$$I_3 = \frac{V_{23}}{R_3}$$ $$I_3 = \frac{6 \text{ V}}{3 \Omega}$$ $$I_3 = 2 \text{ A}$$

The current through $R_2$ is $\boxed{1 \text{ A}}$ and the current through $R_3$ is $\boxed{2 \text{ A}}$.