Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

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.

Step-by-Step Solution

Answer

The tension in each wire is \(25\sqrt{2}\) pounds.

Step 1: Understanding the Problem

We are given a 100-pound chandelier suspended by four wires, and each wire makes a 45-degree angle with the horizontal. The task is to find the tension in each wire.

Step 2: Calculate Vertical Component of Tension

Each wire provides a vertical component of tension to counteract the weight of the chandelier. The vertical component of tension in each wire can be calculated as \(T \cdot \cos(45^{\circ})\), where \(T\) is the tension in the wire.

Step 3: Sum of Vertical Forces

Since the chandelier is in equilibrium, the sum of all vertical components of tension must be equal to the weight of the chandelier. We have 4 wires, so the equation is: \(4 \cdot T \cdot \cos(45^{\circ}) = 100\).

Step 4: Solve for Tension

We know \(\cos(45^{\circ}) = \frac{\sqrt{2}}{2}\). Substituting into the equation gives: \(4 \cdot T \cdot \frac{\sqrt{2}}{2} = 100\). Simplifying gives: \(2\sqrt{2}T = 100\). Divide both sides by \(2\sqrt{2}\) to find \(T\): \(T = \frac{100}{2\sqrt{2}}\).

Step 5: Rationalize the Denominator

To simplify, multiply numerator and denominator by \(\sqrt{2}\) to get \(T = \frac{100\sqrt{2}}{4}\). Simplify further: \(T = 25\sqrt{2}\).

Key Concepts

Equilibrium in Physics

In physics, equilibrium refers to the state where all the forces acting upon an object are balanced, resulting in no net force and, therefore, no movement. In the context of our chandelier problem, equilibrium is defined by the balance of forces in both the vertical and horizontal directions.
For our specific problem, we focus on vertical equilibrium. The chandelier does not move because the upward tension forces from the wires counterbalance the downward gravitational force (weight) of the chandelier. This leads to the equation where the sum of the vertical components of tension equals the weight of the chandelier, ensuring it stays in place without moving.
To summarize:

• Equilibrium is when opposing forces are balanced.
• In our problem, the sum of the upward forces (from tension) equals the downward force (weight).
• Thus, equilibrium keeps the chandelier stationary and safely hanging from the ceiling.

Tension in Physics

Tension is a force conducted through a string, rope, wire, or similar object that transmits force. In our exercise, tension is the force within the four wires holding the chandelier in place. Each wire undergoes tension from both the weight of the chandelier and from each wire's angle to the ceiling.
Here's how tension works:

• The tension in each wire provides a vertical force component that helps counterbalance the chandelier's weight.
• This force is calculated from the equation involving the cosine of the angle the wire makes with the horizontal. For instance, each wire's vertical force contribution is expressed as \(T \cdot \cos(45^{\circ})\).
• The task involves ensuring the sum of all these vertical components equals the chandelier’s weight, ensuring stability.

Understanding tension and how it operates through angles is crucial in diverse physics applications, from bridge construction to sports mechanics.

Trigonometry in Physics

Trigonometry is a math branch that studies the properties and applications of triangles. It plays a crucial role in physics by allowing precise calculation of forces, angles, and distances.
In our chandelier problem, trigonometry comes into play with the angles made by the wires with respect to the horizontal.
When we use an angle of \(45^{\circ}\), trigonometry lets us calculate the vertical component of the tension using the cosine function. The equation \(T \cdot \cos(45^{\circ})\) shows how much of the tension in each wire contributes to holding the chandelier up against gravity. Here, the cosine of \(45^{\circ}\) is \(\frac{\sqrt{2}}{2}\), reflecting the relationship between the angle and the forces in our triangle-like setup.

• Trigonometry helps split the diagonal forces into vertical and horizontal components.
• This splitting is essential for calculating how much force is needed in each direction to maintain equilibrium.
• Understanding trigonometric functions is crucial for any physics student, as it bridges the gap between mathematical theory and physical application.