The exercise 1 from the document asks:

If the roller R of the inclined connecting rod moves to the left at a constant speed of $V_R = 1.2 \, \text{m/s}$, determine:

1. The angular velocity of the link.
2. The velocity of the roller L when the angle $\theta = 20^\circ$.

Given Data:

• $V_R = 1.2 \, \text{m/s}$
• $LR = 2.3 \, \text{m}$
• $\theta = 20^\circ$

Solution:

To solve this problem, we will use concepts of relative velocity in mechanisms, specifically in a 4-bar mechanism involving a sliding element.

Step 1: Express the velocity of point R

The velocity of point R is horizontal, as given. We know that the velocity of any point on a rigid body in plane motion can be expressed as:

$V_R = V_L + \omega \times LR$

Where:

• $V_L$ is the velocity of point L,
• $\omega$ is the angular velocity of the link,
• $LR$ is the distance between L and R.

Since $V_R$ is horizontal, we can project the velocities along the direction of motion.

Step 2: Break down into components

Let’s decompose the velocity and angular relationships, considering that $\theta = 20^\circ$ affects the angle of the link.

The relationship between the angular velocity $\omega$ and the linear velocity $V_R$ is:

$V_R = \omega \times LR \times \sin(\theta)$

Step 3: Solve for $\omega$

$1.2 \, \text{m/s} = \omega \times 2.3 \, \text{m} \times \sin(20^\circ)$

$\omega = \frac{1.2}{2.3 \times \sin(20^\circ)}$

Using $\sin(20^\circ) \approx 0.342$:

$\omega \approx \frac{1.2}{2.3 \times 0.342} \approx \frac{1.2}{0.787} \approx 1.525 \, \text{rad/s}$

Step 4: Find the velocity of roller L

The velocity of point L can be found using the same relationship:

$V_L = \omega \times LR \times \cos(\theta)$

$V_L = 1.525 \times 2.3 \times \cos(20^\circ)$

Using $\cos(20^\circ) \approx 0.94$:

$V_L \approx 1.525 \times 2.3 \times 0.94 \approx 3.3 \, \text{m/s}$

Final Answer:

1. The angular velocity $\omega$ of the link is approximately $1.525 \, \text{rad/s}$.
2. The velocity of the roller L is approximately $3.3 \, \text{m/s}$.

Let me know if you would like more details on any part of the process!

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Next questions:

1. How does the velocity analysis change if the angle $\theta$ is different?
2. What if the roller moves in the opposite direction?
3. How would angular acceleration affect the results?
4. What happens if $LR$ is increased?
5. How can we apply the relative velocity method for different configurations?

Tip: Understanding vector decomposition is essential when working with velocities in mechanisms.