To solve this problem, we need to determine:

1. The angular velocity of the link $\omega$ (between points L and R).
2. The velocity of point $L$.

We know the following information:

• The velocity of point $R$ is given as $V_R = 1.2 \, \text{m/s}$ to the left.
• The length of the link $LR = 2.3 \, \text{m}$.
• The angle $\theta = 20^\circ$ between the link and the horizontal.

Step 1: Angular Velocity of the Link

The velocity at point $R$ is related to the angular velocity $\omega$ of the link. Using the relative velocity equation for rigid body rotation:

$V_R = V_L + \omega \times LR$

Since $V_L$ is the velocity of point $L$ and we are not given it explicitly, we will decompose the motion:

For point $R$, the velocity $V_R$ is purely horizontal. For point $L$, its velocity will have both horizontal and vertical components. However, the rotational relationship gives us the connection between the angular velocity $\omega$ and the linear velocity at $R$:

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

We can rearrange this to solve for the angular velocity $\omega$:

$\omega = \frac{V_R}{LR \cdot \sin(\theta)}$

Substitute the values:

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

Now calculate $\omega$:

$\omega = \frac{1.2}{2.3 \cdot 0.3420} = \frac{1.2}{0.7866} \approx 1.53 \, \text{rad/s}$

Step 2: Velocity of Point $L$

Once we know $\omega$, we can find the velocity of point $L$ using the rotational relationship. The velocity at point $L$ will be:

$V_L = \omega \cdot LR$

Substituting the values:

$V_L = 1.53 \, \text{rad/s} \cdot 2.3 \, \text{m} \approx 3.52 \, \text{m/s}$

Thus:

• The angular velocity $\omega$ is approximately $1.53 \, \text{rad/s}$.
• The velocity of point $L$ is approximately $3.52 \, \text{m/s}$.

Would you like a detailed breakdown of any of these steps or further clarification?

5 related questions:

1. How would the angular velocity change if the velocity of point $R$ were doubled?
2. What would happen to the velocity of point $L$ if the angle $\theta$ increased?
3. How can we calculate the components of the velocity of point $L$ in the x and y directions?
4. Can the link between points $L$ and $R$ experience angular acceleration in this scenario?
5. How is the direction of velocity related to the angular velocity in rigid body motion?

Tip: Always ensure that your angle units match your trigonometric functions—radians or degrees—as this can lead to calculation errors.