Question

A 100 -pound chandelier is held in place by four wires attached to the ceiling at the four corners of a square. Each wire makes an angle of \(45^{\circ}\) with the horizontal. Find the magnitude of the tension in each wire.

Short answer

The tension in each wire is approximately 35.35 pounds.

Step-by-step solution

Understanding the Problem

We are given a chandelier weighing 100 pounds held by four wires situated at the corners of a square, making an angle of \(45^{\circ}\) with the horizontal. We need to find the tension in each wire.

Analyze the Forces

Since the system is in equilibrium, the sum of the vertical and horizontal components of the forces must equal zero. The vertical force component of each wire must balance the weight of the chandelier.

Resolve the Tension into Components

The tension force in each wire makes a \(45^{\circ}\) angle with the horizontal. We can resolve it into vertical and horizontal components. For any wire with tension \(T\), the vertical component is \(T \sin(45^{\circ})\) and the horizontal component is \(T \cos(45^{\circ})\).

Calculate the Total Vertical Component

The total vertical component of all wires must equal the weight of the chandelier. Since there are 4 wires, the total vertical force is \(4 \times T \sin(45^{\circ})=100\).

Solve for Tension in Each Wire

Using \(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\), the equation becomes \(4 \times T \times \frac{\sqrt{2}}{2} = 100\). Simplifying, we get \(2T \sqrt{2} = 100\), and solving for \(T\), we have \(T = \frac{100}{2\sqrt{2}} = \frac{100 \sqrt{2}}{4}\). Further simplifying gives \(T = 25 \sqrt{2}\).

Final Calculation

Calculate the approximate value of \(T = 25 \sqrt{2}\). Since \(\sqrt{2} \approx 1.414\), we find \(T \approx 25 \times 1.414 = 35.35\) pounds.

Key concepts

Equilibrium of Forces

When objects are held in place by multiple forces, they are said to be in equilibrium. This means that the sum of forces acting on the object is zero. In the case of the chandelier secured by four wires, equilibrium indicates that both vertical and horizontal force components balance each other out.
When the chandelier is stationary, all forces pulling it down, like its weight, must be counteracted by forces pulling up, which is provided by the wires. Thus, the total downward force is equal to the total upward force. If any part of this balance is disturbed, the chandelier would move, showing that equilibrium was lost.
To achieve equilibrium, we consider:

• The chandelier's weight providing the downward force, which is 100 pounds.
• The upward force produced by the wires' vertical components.

Since all forces horizontally must also equal zero because the chandelier doesn't move sideways, this is an example of equilibrium of forces.

Trigonometric Components

Trigonometry helps us understand how the tension in the wires divides into two components: vertical and horizontal. Since the wires are at a 45-degree angle with the horizontal, we use trigonometric functions (sine and cosine) to find these components.
The angle plays a crucial role here: when a force occurs at an angle, part of that force works in the direction of the motion (vertical) and part works perpendicular to it (horizontal).
For each wire:

• The vertical component of tension is calculated as the sine of the angle: \(T \times \sin(45^{\circ})\).
• The horizontal component is calculated using cosine: \(T \times \cos(45^{\circ})\).

Using the properties of a 45-degree angle (where both sine and cosine are equal to \(\frac{\sqrt{2}}{2}\)), simplifies our analysis greatly.

Resolving Forces

Resolving forces means breaking down the total force into its components. In this scenario, we see the tension in each wire being split into upward and horizontal forces.
This is because the tension isn't acting in a straight line up but rather at an angle. By resolving or "breaking down" the forces, we find how much of the tension is pushing the chandelier upward, counteracting its weight, and how much is pulling sideways.
This resolution is crucial because we need to balance only the vertical components with the weight of the chandelier for it to stay in equilibrium.
Once we determine the vertical component of each wire, we multiply it by 4 (since there are four wires), ensuring that it equals the 100-pound weight of the chandelier. This step demonstrates how resolving forces aids in understanding both the main direction and the ancillary influences of tensions in practical scenarios.

Force Balance Analysis

To perform a force balance analysis, it's essential to consider all acting forces and ensure they counterbalance each other. This process involves systematically breaking down each force into components and using equations to find unknown values.
In this exercise, by knowing that the chandelier's total weight is 100 pounds and equally supported by the four wires, we can set up an equation:

• The sum of the vertical components of all wires (\(4 \times T \sin(45^{\circ})\)) equals the chandelier's weight.

The simplification using \(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\) leads to solving for the tension \(T\).
The calculated value substantiates that each wire carries \(T = 25 \sqrt{2}\) pounds, or approximately \(35.35\) pounds. This tension supports the chandelier while maintaining horizontal equilibrium.