Problem 61 - College Trigonometry

Question

Two drunken college students have filled an empty beer keg with rocks and tied ropes to it in order to drag it down the street in the middle of the night. The stronger of the two students pulls with a force of 100 pounds at a heading of \(\mathrm{N} 77^{\circ} \mathrm{E}\) and the other pulls at a heading of \(\mathrm{S} 68^{\circ} \mathrm{E}\). What force should the weaker student apply to his rope so that the keg of rocks heads due east? What resultant force is applied to the keg? Round your answer to the nearest pound.

Step-by-Step Solution

Answer

The weaker student should apply approximately 25 pounds. The resultant force is approximately 106 pounds eastward.

Step 1: Understand the Vectors

The stronger student pulls with a force of 100 pounds at a heading of \(N 77^\circ E\) which translates to \(77^\circ\) east of north. The weaker student pulls at \(S 68^\circ E\), which translates to \(68^\circ\) east of south. We need to determine the forces in the east-west direction (x-axis) and the north-south direction (y-axis).

Step 2: Set Up the Equations for the Components

Using trigonometry, the stronger student's force can be broken down into components. The x-component is \(100\cos(13^\circ)\) and the y-component is \(100\sin(13^\circ)\). For the weaker student, the x-component is \(F\cos(68^\circ)\), and the y-component is \(F\sin(-68^\circ)\), where \(F\) is the force applied by the weaker student. The angle for the stronger student is computed using \(90^\circ - 77^\circ = 13^\circ\).

Step 3: Solve for the Weaker Student's Force

Since we want the resultant force to be purely eastward, the y-components should cancel each other out. Therefore, we set the equation \(100\sin(13^\circ) = -F\sin(68^\circ)\). Solving for \(F\), we have \(F = \frac{100\sin(13^\circ)}{-\sin(68^\circ)}\). Calculate \(F \approx 25.470\) pounds.

Step 4: Determine the Resultant Force

After finding \(F\), calculate the x-components of both forces and add them. The stronger student's x-component is \(100\cos(13^\circ)\) and the weaker student's x-component is \(25.470\cos(68^\circ)\). Compute the total x-component: \(96.775 + 9.513 \approx 106.288\) pounds, rounded to 106 pounds.

Key Concepts

Force Components

To understand how different forces affect an object, we need to break them into components. Think of these components as splitting a force into its vertical (y-axis) and horizontal (x-axis) parts.

• The horizontal (x) component is how much of the force pushes forward or backward.
• The vertical (y) component is how much of the force pushes up or down.

This breakdown helps you see how each force contributes to the overall direction and movement of the object. In our exercise, we're dealing with two forces from two students pulling a keg. By using trigonometry, the forces are divided based on angles with respect to the east-west and north-south directions, helping us to solve the problem more effectively.

Angle Conversion

Angles in navigation and geometry can sometimes be confusing because they refer to directions rather than ordinary degrees.

• The direction "N 77° E" means 77 degrees east of north, which we think of as turning 77 degrees clockwise from north.
• The direction "S 68° E" means 68 degrees east of south, which is turning 68 degrees clockwise from south.

When solving such problems, it's crucial to adjust angles into a format that's easy to work with in trigonometric formulas. For example, the 77° of the stronger student needs adjustment as it's measured starting from north. To convert, you might take it as 90° - 77° = 13° to find how it lies with respect to the easterly direction.

Resultant Force

The resultant force is the sum of all individual components of the forces acting on an object. It's the overall force that determines the movement and direction of the object. To find the resultant force after resolving force components:

• Add together all horizontal (x-axis) components to get the total horizontal force.
• For this exercise, since we want the keg to move due east, we ensure that the vertical forces (y-axis) cancel out.

Next, by computing the x-components from the stronger and weaker pulls, you sum them up for the total force applied to the keg. This is calculated as 96.775 pounds from the stronger student's pull plus 9.513 pounds from the weaker's, concluding in a total resultant force of approximately 106 pounds directed purely east.

Problem Solving Strategy

Solving problems involving trigonometric vectors requires a systematic approach to ensure accuracy. Here's a simple strategy to follow:

• **Understand the Problem:** Break down each force into its components using trigonometric functions (sine and cosine).
• **Use Angles Wisely:** Convert angles into easily workable degrees before using in formulas.
• **Solve for Unknowns:** Once the components are known, set equations to find unknown forces or angles, especially when conditions such as purely directional resultant forces are needed.
• **Combine and Conclude:** Add up the components to find the resultant force and ensure all objectives of the problem are met, like the keg moving due east in this case.

Following these steps not only makes calculations easier but also ensures an effective understanding of force interactions in various scenarios.