We are asked to find the force $F_{T,1}$ in the cord that supports the picture frame. Let's break down the forces and torques acting on the system.

Step 1: Define forces and torques

• The system is in equilibrium, so the sum of forces and the sum of torques must both be zero.
• Forces acting on the frame:
  1. $F_{T,1}$ at an angle of 50°, acting along the cord on the right.
  2. $F_{T,2}$, the tension force acting vertically on the left cord.
  3. The weight of the picture frame, which is 11.4 N, acting downwards at the center of the frame.

Step 2: Force equilibrium

In the horizontal and vertical directions, the sum of forces must be zero.

Horizontal direction:

$\sum F_x = 0 \quad \Rightarrow \quad F_{T,1} \cos(50^\circ) - F = 0$ So, $F = F_{T,1} \cos(50^\circ)$

Vertical direction:

$\sum F_y = 0 \quad \Rightarrow \quad F_{T,1} \sin(50^\circ) + F_{T,2} - 11.4 = 0$ $F_{T,2} = 11.4 - F_{T,1} \sin(50^\circ)$

Step 3: Torque equilibrium

We can take torques about point P (the lower-right corner) to avoid needing to include the unknown horizontal force $F$.

The sum of torques must be zero: $\sum \tau_P = 0$ Torque contributions:

• The weight (11.4 N) creates a counterclockwise torque: $\tau_{\text{weight}} = 11.4 \, \text{N} \times 10 \, \text{cm} = 1.14 \, \text{N}\cdot\text{m}$
• $F_{T,2}$ creates a counterclockwise torque: $\tau_{F_{T,2}} = F_{T,2} \times 20 \, \text{cm} = F_{T,2} \times 0.2 \, \text{m}$
• $F_{T,1}$ creates a clockwise torque, and the vertical component of $F_{T,1}$ contributes to this: $\tau_{F_{T,1}} = F_{T,1} \sin(50^\circ) \times 20 \, \text{cm} = F_{T,1} \sin(50^\circ) \times 0.2 \, \text{m}$

Setting the sum of torques to zero: $1.14 = F_{T,1} \sin(50^\circ) \times 0.2$ $F_{T,1} = \frac{1.14}{0.2 \times \sin(50^\circ)}$ $F_{T,1} = \frac{1.14}{0.153} \approx 7.45 \, \text{N}$

Final Answer:

The force in the cord $F_{T,1}$ is approximately 7.45 N.

Would you like further details or clarification on this solution?

Related Questions:

1. How do we calculate the force $F$ based on the value of $F_{T,1}$?
2. How does changing the angle of the cord affect the tension force $F_{T,1}$?
3. What is the value of $F_{T,2}$ using the equilibrium conditions?
4. How would the torque balance change if the picture frame's weight increased?
5. What is the significance of choosing point P for the torque calculation?

Tip:

Always select a pivot point where unknown forces pass through, simplifying your torque calculations!