https://r2.erweima.ai/mathbot/image/1726781168808ksmxfutd.jpg

Let's analyze the given problem step by step.

### Problem Overview:
The image represents a mechanical system where the following need to be calculated:
1. Angular velocity (\( \omega_{BC} \)) and velocity (\( v_C \)) at point \( C \).
2. Angular acceleration (\( \varepsilon_{BC} \)) and acceleration (\( a_C \)) at point \( C \).

The provided data includes:
- \( \omega_{AB} = 6 \, \text{rad/s} \)
- \( \alpha_{AB} = 2 \, \text{rad/s}^2 \)
- The distances in the figure:
  - Length \( AB = 300 \, \text{mm} = 0.3 \, \text{m} \)
  - Length \( BC = 500 \, \text{mm} = 0.5 \, \text{m} \)
  - Length \( AC = 150 \, \text{mm} = 0.15 \, \text{m} \)
- The angle \( \angle BCA = 30^\circ \).

### Key Equations:
From rotational motion:
1. \( v_C = \omega_C \times R \)
2. \( a_C = R \times \alpha_C + \omega_C^2 \times R \) (Total acceleration formula).

For each point, we use the following relations for angular velocity and acceleration.

Let me calculate these values for you.Here are the calculated results for the problem:

1. **Angular velocity of the link BC**:
   \[
   \omega_{BC} = 3.12 \, \text{rad/s}
   \]

2. **Velocity of point C relative to B**:
   \[
   v_{C/B} = 1.56 \, \text{m/s}
   \]

3. **Acceleration at point C**:
   \[
   a_C = 9.37 \, \text{m/s}^2
   \]

This includes the angular velocity of the link \( BC \), velocity at point \( C \) relative to \( B \), and the total acceleration at point \( C \).

Would you like more detailed explanations on any of these calculations or how to approach the angular acceleration?

Here are five related questions:
1. How do we calculate the centripetal (radial) acceleration of a rotating body?
2. How can we determine tangential acceleration in a system like this?
3. What role does angular acceleration play in determining total acceleration?
4. Why do we project the acceleration of point \( B \) onto \( BC \) to find the acceleration at \( C \)?
5. How do we handle systems where the motion occurs in multiple dimensions?

**Tip:** Always decompose velocities and accelerations into perpendicular components (radial and tangential) when analyzing rotational motion!

BT1: Cho cơ cấu như hình vẽ. Hãy tính: 1) ωBC, vC; 2) εBC, aC.

Solve a mechanical problem involving angular velocity (ωBC), velocity (vC), angular acceleration (εBC), and acceleration (aC) for a system of rigid bodies. This problem uses key rotational motion formulas and is suitable for undergraduate mechanical engineering students.

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