Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

An object weighing \(258.5\) pounds is held in equilibrium by two ropes that make angles of \(27.34^{\circ}\) and \(39.22^{\circ}\), respectively, with the vertical. Find the magnitude of the force exerted on the object by each rope.

Step-by-Step Solution

Answer

The force exerted by the first rope is approximately 151.8 lb, and the force exerted by the second rope is approximately 219.9 lb.

Step 1: Understand the Problem

We need to find the force exerted by each rope holding a 258.5 lb object in equilibrium, given the angles each rope makes with the vertical.

Step 2: Analyze the Forces

In equilibrium, the vertical components of the forces exerted by the ropes must sum up to 258.5 lb and the horizontal components must cancel each other out.

Step 3: Set Up the Force Equations

For Rope 1 with angle \(\theta_1 = 27.34^{\circ}\) and force \(F_1\), the vertical component is \(F_1 \cos \theta_1\) and the horizontal component is \(F_1 \sin \theta_1\). For Rope 2 with angle \(\theta_2 = 39.22^{\circ}\) and force \(F_2\), the vertical component is \(F_2 \cos \theta_2\) and the horizontal component is \(F_2 \sin \theta_2\).

Step 4: Write the Equilibrium Equations

Vertical equilibrium: \(F_1 \cos \theta_1 + F_2 \cos \theta_2 = 258.5\). Horizontal equilibrium: \(F_1 \sin \theta_1 = F_2 \sin \theta_2\).

Step 5: Solve for the Forces

Solve the horizontal equation for one of the forces: \(F_1 = \frac{F_2 \sin \theta_2}{\sin \theta_1}\). Substitute this expression in the vertical equation and solve for \(F_2\).

Step 6: Calculate \(F_2\)

Substitute \(\theta_1 = 27.34^{\circ}\) and \(\theta_2 = 39.22^{\circ}\) into the equations and solve: \[F_2 = \frac{258.5 \cos 27.34^{\circ}}{\cos 39.22^{\circ} + \frac{\sin 39.22^{\circ}}{\sin 27.34^{\circ}}}\].

Step 7: Calculate \(F_1\)

Using the value of \(F_2\) found in Step 6, plug it into the equation from Step 5 to find \(F_1\): \(F_1 = \frac{F_2 \sin 39.22^{\circ}}{\sin 27.34^{\circ}}\).

Step 8: Compute the Numeric Values

Calculate the numeric values for \(F_1\) and \(F_2\) using a calculator to find the exact forces exerted by the ropes.

Key Concepts

Trigonometric Components

When dealing with forces, especially those in equilibrium, understanding trigonometric components is vital. Every force exerted by a rope can be split into vertical and horizontal components using trigonometrical functions: sine and cosine. Such decomposition is crucial as it allows us to analyze different directions separately.

We identify a force's vertical component using the cosine of the angle it makes with the vertical axis. The horizontal component, in turn, utilizes the sine of that angle.

• Vertical Component: Calculated using the cosine function. For Rope 1, it’s expressed as \(F_1 \cos \theta_1\), and for Rope 2, \(F_2 \cos \theta_2\).
• Horizontal Component: Calculated using the sine function. It appears as \(F_1 \sin \theta_1\) for Rope 1, and \(F_2 \sin \theta_2\) for Rope 2.

Understanding these components helps us to set up equilibrium equations by separating the forces into manageable parts. This makes solving the problem step-by-step more straightforward as we pair off vertical and horizontal elements separately.

Vertical and Horizontal Equilibrium

In physics, when an object is in equilibrium, all forces acting on it balance out, resulting in no net force. For an object suspended by ropes, this translates to both its vertical and horizontal forces being in a state of equilibrium.

**Vertical Equilibrium**
Vertical equilibrium requires the sum of the vertical components of all forces to equal the object's weight. In our exercise, they must combine to support the 258.5 pound object:\(F_1 \cos \theta_1 + F_2 \cos \theta_2 = 258.5\) pounds.

**Horizontal Equilibrium**
Horizontal equilibrium means opposing horizontal forces must cancel each other out. This is because the object does not move sideways:\(F_1 \sin \theta_1 = F_2 \sin \theta_2\).

These conditions, derived from basic principles, guide how forces interact in balance and enable us to solve for unknown elements, such as the force each rope provides.

Force Vectors

A force vector is a representation of force that has both magnitude and direction. In the context of our problem, the forces from both ropes act as vectors holding the object in balance. Understanding vectors helps in visualizing the forces involved and predicting the required conditions for equilibrium.

Each force vector here is defined by two components based on its directional pull:

• *Magnitude*: This is what we calculate for each force. It tells us how strong the pull or push is.
• *Direction*: Here, it’s specified by the angle the rope makes with the vertical.

The crucial step in problems involving force vectors is setting up correct mathematical expressions for equilibrium, derived from the vector components. By ensuring that both the vertical and horizontal vectors align with equilibrium conditions, we determine the magnitude of the forces required to balance the object effectively.