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The problem describes a hydraulic system with a gate controlled by a cylinder (AB) that extends at a constant velocity of 0.3 m/s. The objective is to determine the angular velocity \(\omega\) and angular acceleration \(\alpha\) of the gate at the instant when \(\theta = 45^\circ\). The given data includes:

- \(BO = 2 \, \text{m}\)
- \(OA = 4 \, \text{m}\)
- The velocity of the cylinder, \(v_{AB} = 0.3 \, \text{m/s}\).

Let's break down the steps needed to find \(\omega\) and \(\alpha\).

### 1. Relation between the velocity of the hydraulic cylinder and angular velocity \(\omega\)
Since the point \(A\) is moving due to the extension of the cylinder AB, we need to relate the velocity of point \(A\) to the angular velocity of the gate. Using kinematic relationships for rotational motion, we'll express the velocity of point \(A\) and relate it to \(\omega\).

### 2. Calculating the position of point B
From the geometry of the problem, given the lengths \(BO = 2 \, \text{m}\) and \(OA = 4 \, \text{m}\), we can calculate the instantaneous lengths of segments using trigonometry, particularly for when \(\theta = 45^\circ\).

I will now calculate the angular velocity and acceleration based on the provided data.The length of the hydraulic cylinder \(AB\) when \(\theta = 45^\circ\) is approximately \(2.95 \, \text{m}\).

Next, I will compute the angular velocity \(\omega\) and angular acceleration \(\alpha\) using the relationships between the linear velocity of point \(A\) (determined by the extension of the cylinder) and the angular velocity of the gate.

Let me proceed with these calculations.The angular velocity of the gate when \(\theta = 45^\circ\) is approximately \(\omega = 0.102 \, \text{rad/s}\).

Since the velocity of the hydraulic cylinder is constant, the angular acceleration \(\alpha\) is zero at this instant.

Would you like to explore more details or ask questions about the solution?

Here are 5 related questions to deepen your understanding:
1. How would the angular velocity change if the velocity of the hydraulic cylinder increased?
2. What is the significance of the angle \(\theta\) in determining the movement of the gate?
3. How would you find the angular acceleration if the hydraulic cylinder were accelerating?
4. How would the problem change if the cylinder extended at a varying rate instead of a constant velocity?
5. How does the relationship between the lengths \(OA\), \(BO\), and \(AB\) affect the gate's motion?

**Tip:** Angular velocity is always linked to the rate at which a point moves along a circular path and is inversely proportional to the radius at that point.

La compuerta de la figura se cierra por medio de un cilindro hidráulico AB. Si éste se extiende a una velocidad constante de 0.3 m/s, determine la velocidad angular 'ω' y aceleración angular 'α' de la compuerta en el instante θ = 45°. Datos: BO= 2m; OA= 4m

Calculate the angular velocity (ω) and angular acceleration (α) of a gate controlled by a hydraulic cylinder. The cylinder extends at a constant velocity of 0.3 m/s, with θ = 45°, BO = 2 m, and OA = 4 m. Understand the concepts of kinematics and rotational motion to solve this problem.

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