To calculate the vertical reaction at the hinge ($R_y$), we apply the principles of static equilibrium, which state that:

1. The sum of all vertical forces is zero: $\sum F_y = 0$

2. The sum of all moments about any point is zero: $\sum \tau = 0$

Step-by-Step Solution:

Step 1: Forces acting on the system

1. Weight of the beam: $W_{\text{beam}} = m_{\text{beam}} \cdot g = 15 \cdot 9.8 = 147 \, \text{N}$ This force acts at the center of the beam, which is $8/2 = 4 \, \text{m}$ from the hinge.

2. Weight of the hanging mass: $W_{\text{mass}} = m_{\text{mass}} \cdot g = 61 \cdot 9.8 = 597.8 \, \text{N}$ This force acts $8 - 3 = 5 \, \text{m}$ from the hinge (to the left).

3. Tension in the rope ($T$):
   The tension acts at an angle of $11^\circ$ to the horizontal. This tension has both vertical and horizontal components:

   • Vertical component: $T_y = T \sin(11^\circ)$
   • Horizontal component: $T_x = T \cos(11^\circ)$

4. Reaction forces at the hinge: The hinge provides two reaction forces:

   • Horizontal reaction: $R_x$
   • Vertical reaction: $R_y$

Step 2: Write the equilibrium equations

1. Vertical forces equilibrium: $R_y + T_y = W_{\text{beam}} + W_{\text{mass}}$ $R_y + T \sin(11^\circ) = 147 + 597.8$ R_y + T \sin(11^\circ) = 744.8 \, \text{N} \tag{1}

2. Moments equilibrium about the hinge: Taking moments about the hinge ($\sum \tau_{\text{hinge}} = 0$):

   • Clockwise moments: $W_{\text{beam}} \cdot 4 + W_{\text{mass}} \cdot 5$
   • Counterclockwise moment: $T_y \cdot 8 = T \sin(11^\circ) \cdot 8$ Equating clockwise and counterclockwise moments: $T \sin(11^\circ) \cdot 8 = 147 \cdot 4 + 597.8 \cdot 5$ $T \sin(11^\circ) \cdot 8 = 588 + 2989$ $T \sin(11^\circ) \cdot 8 = 3577 \, \text{N} \cdot \text{m}$ $T \sin(11^\circ) = \frac{3577}{8} = 447.125 \, \text{N}$ $T = \frac{447.125}{\sin(11^\circ)}$ Using $\sin(11^\circ) \approx 0.1908$: $T = \frac{447.125}{0.1908} \approx 2343.6 \, \text{N}$

3. Substitute $T$ into the vertical force equation: Using $T_y = T \sin(11^\circ)$: $R_y + 447.125 = 744.8$ $R_y = 744.8 - 447.125$ $R_y \approx 297.675 \, \text{N}$

Final Answer:

The vertical reaction at the hinge is approximately: $\boxed{297.7 \, \text{N}}$

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Would you like a breakdown of the moment equation or any other part?
Here are some related questions for further understanding:

1. How is the tension in the rope distributed in its components?
2. What would happen if the angle of the rope were increased?
3. How is the horizontal reaction ($R_x$) at the hinge calculated?
4. Why does the beam's weight act at its center of mass?
5. What are some real-world applications of static equilibrium in physics?

Tip: Always resolve forces into their components to simplify equilibrium calculations!