The three ropes are tied to a small, very light ring. Two of the ropes are anchored to walls at right angles, and the third rope pulls as shown: F=85 N, q=54°.

What is the magnitude of the tension force in the first rope

To find the magnitude of the tension force in the first rope, we can break down the given information and solve using trigonometry. Here's how you can approach this problem:

1. Draw a diagram: Visualize the scenario described in the problem, including the ring and the three ropes. Make sure to label any angles and forces mentioned.

2. Identify the known values: From the problem, we know that the force (F) acting on the ring is 85 N, and the angle (q) between the direction of this force and the first rope is 54°.

3. Find the horizontal and vertical components: Since the two ropes are anchored at right angles, the tension force in the first rope can be broken down into two components: horizontal and vertical.

- The horizontal component (T_x) of the tension force is given by T_x = T * cos(q), where T is the magnitude of the tension force and q is the angle between the force and the first rope.
- The vertical component (T_y) of the tension force is given by T_y = T * sin(q).

4. Use trigonometry: We can now use the trigonometric ratios (sine and cosine) to find the values of T_x and T_y.

- T_x = T * cos(q) = T * cos(54°)
- T_y = T * sin(q) = T * sin(54°)

5. Substitute known values: Substitute the given values and calculate T_x and T_y.

- T_x = T * cos(54°)
- T_y = T * sin(54°)

6. Find the magnitude of the tension force: The magnitude of the tension force (T) can be calculated using the Pythagorean theorem.

- T^2 = T_x^2 + T_y^2

7. Solve for T: Substitute the values of T_x and T_y into the equation and calculate T.

8. Evaluate the magnitude of the tension force: Take the square root of T to find the magnitude of the tension force in the first rope.

By following these steps, you should be able to determine the magnitude of the tension force in the first rope.