The two ends of a 20-m rope are fastened at two points on the ceiling, 12 m from each other. If a load of 200 kg is placed at the center of the rope, find the tension in the rope.

To solve this problem, we can model the rope and load as a statics problem in physics. The tension in the rope is determined by the forces acting on the rope and the geometry of the situation.

### Step 1: Setup the problem
- Let’s assume the two ends of the rope are fastened at two points 12 meters apart.
- The length of the rope is 20 meters.
- A 200 kg load is hanging at the center of the rope.

We need to calculate the tension in the rope when the load is applied at the center. This problem can be simplified by breaking it into two parts, focusing on half the rope and using symmetry.

### Step 2: Determine the vertical displacement of the load
Because the rope forms a curve (often an approximation of a catenary), let's assume a simplified geometry by modeling the rope as two straight segments forming a triangle. The two ends of the rope are 12 meters apart horizontally, and the rope length is 20 meters. The midpoint of the rope will sag vertically under the weight of the load.

We can form a right triangle by:
- Half of the horizontal distance between the ceiling points = \( 6 \) meters (half of 12 m).
- Let \( h \) be the vertical sag (the distance the load drops down from the ceiling).

Using the Pythagorean theorem for one half of the rope:
\[
(6^2 + h^2) = 10^2
\]
\[
36 + h^2 = 100
\]
\[
h^2 = 64
\]
\[
h = 8 \, \text{m}
\]
So, the rope sags by 8 meters from the ceiling.

### Step 3: Calculate the forces
The forces involved are:
- The gravitational force from the load: \( F_g = mg \), where \( m = 200 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \).
\[
F_g = 200 \times 9.8 = 1960 \, \text{N}
\]
This force acts downward.

At the point where the load is attached, the tension in the two segments of the rope has both vertical and horizontal components. Let \( T \) be the tension in the rope. The vertical component of the tension in each side of the rope must balance half of the gravitational force, and the horizontal components must cancel each other out.

Using trigonometry, the vertical component of the tension is related to the total tension by the angle \( \theta \) formed with the horizontal. From the triangle, we know:
\[
\sin \theta = \frac{h}{10} = \frac{8}{10} = 0.8
\]
So the vertical component of the tension is:
\[
T_{\text{vertical}} = T \times \sin \theta
\]
Since the total downward force is 1960 N, and each half of the rope supports half the load:
\[
2T_{\text{vertical}} = 1960 \, \text{N}
\]
\[
2(T \times 0.8) = 1960
\]
\[
T \times 0.8 = 980
\]
\[
T = \frac{980}{0.8} = 1225 \, \text{N}
\]

Thus, the tension in the rope is **1225 N**.

### Summary:
The tension in each segment of the rope is **1225 N** when the 200 kg load is placed at the center.

Would you like more details or have any further questions?

Here are 5 related questions:
1. What if the rope length were 25 meters instead of 20 meters?
2. How does the angle between the rope and the horizontal affect the tension?
3. How would the tension change if the load were not placed at the center?
4. Can you derive the tension using a catenary equation instead of a triangle approximation?
5. What would happen to the tension if the distance between the ceiling points increased?

**Tip**: Always start by identifying the forces acting on the system and drawing a free-body diagram to visualize the problem.

Solve the problem of finding the tension in a 20-meter rope fastened 12 meters apart on the ceiling, with a 200 kg load at the center. This involves using the Pythagorean theorem, forces, and trigonometry to calculate tension. Suitable for Grades 10-12 students.

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