Problem 85 - Precalculus

Question

Resultant Force. A person is walking two dogs fastened to separate leashes that meet in a connective hub, leading to a single leash that she is holding. Dog 1 applies a force \(\mathrm{N} 60^{\circ} \mathrm{W}\) with a magnitude of \(8,\) and Dog 2 applies a force of \(\mathrm{N} 45^{\circ} \mathrm{E}\) with a magnitude of \(6 .\) Find the magnitude and direction of the force \(w\) that the walker applies to the leash in order to counterbalance the total force exerted by the dogs.

Step-by-Step Solution

Answer

The walker applies a force with the same magnitude as the resultant of the dogs' forces, but in the opposite direction.

Step 1: Draw a Diagram

Visualize the problem by drawing a diagram with vectors to indicate the forces applied by the dogs. Dog 1's force is represented by an arrow at 60 degrees west of north, and Dog 2's force as another arrow at 45 degrees east of north.

Step 2: Resolve Forces into Components

Decompose each force into its northward and eastward components. For Dog 1 (60° W of N): Northward component is \(8 \cos(60°)\) and westward component is \(8 \sin(60°)\). For Dog 2 (45° E of N): Northward component is \(6 \cos(45°)\) and eastward component is \(6 \sin(45°)\).

Step 3: Calculate Northward Resultant

Sum the northward components of both forces: \(R_N = 8 \cos(60°) + 6 \cos(45°)\). Compute the values using \(\cos(60°) = 0.5\) and \(\cos(45°) = \frac{\sqrt{2}}{2}\).

Step 4: Calculate East-West Resultant

Sum the eastward and westward components: since Dog 1 pulls westward, this is negative, and Dog 2 eastward is positive. Calculate \(R_E = 6 \sin(45°) - 8 \sin(60°)\) using \(\sin(60°) = \frac{\sqrt{3}}{2}\) and \(\sin(45°) = \frac{\sqrt{2}}{2}\).

Step 5: Compute Magnitude of Resultant Force

Use the Pythagorean theorem to find the magnitude of the resultant force \(R\), given by \(R = \sqrt{R_N^2 + R_E^2}\). Substitute the calculated northward and eastward components.

Step 6: Determine Direction of Resultant Force

Find the angle \(\theta\) with respect to the north using \(\theta = \tan^{-1}\left(\frac{R_E}{R_N}\right)\). Calculate \(\theta\) to find the direction.

Step 7: Calculate Walker's Force

The walker applies a force equal in magnitude but opposite in direction to counterbalance the dogs' forces. Thus, the magnitude of the walker's force is equal to \(R\), and its direction is opposite to \theta.

Key Concepts

Resultant Force

The concept of resultant force is all about understanding the net effect of multiple forces acting on a single point or object. When multiple forces, such as those applied by the two dogs in our exercise, are exerted on an object, they can be combined into a single force. This single equivalent force is known as the 'resultant force.'

A resultant force serves to represent the total effect that multiple forces would have if they were replaced by a single vector. In simpler terms, it's like finding a shortcut through a maze by going directly from start to finish rather than following every turn.

To determine the resultant force, it is crucial to account for both the magnitude and the direction of the individual forces. It's not enough to just add up the numbers because forces are vector quantities. Vectors have both size and direction, so the overall impact depends on how these forces interact in space.

In our exercise, the forces from the two dogs need to be resolved and then combined to find the magnitude and the direction of the single force that could replace the effect of both dogs pulling at different angles.

Vector Components

Vectors are fascinating because they can be broken down into components. Rather than viewing a force as merely acting in a single line with a specific direction, you can think of it as comprising several smaller directional forces, like pieces of a puzzle. These are called vector components.

When we resolve a vector into components, we typically do so along two perpendicular axes: usually the north-south axis and the east-west axis. For the exercise involving the two dogs:

• Dog 1 exerts force at 60 degrees west of north. This force can be split into: a northward component and a westward component.
• Dog 2 exerts force at 45 degrees east of north, splitting into: a northward component and an eastward component.

Each of these components represents the amount of force in that particular direction. By calculating these components, we can better understand and manipulate how forces combine.

The primary reason to use components is their simplicity. It is easier to handle mathematical problems in physics when working with horizontal and vertical forces separately. As seen in our step-by-step solution, decomposing each dog's force into north-south and east-west components allows us to clearly calculate their combined effect.

Vector Resolution

Vector resolution is closely tied to vector components. In essence, it involves taking a vector with a given magnitude and direction and splitting it into parts, or components, along specified axes. This process is crucial because it simplifies the calculations necessary to find resultant forces.

In vector resolution, you're essentially breaking down a vector such that the original vector is equal to the sum of its components. For our exercise:

• The force exerted by Dog 1 is divided into a northward component and a westward component using trigonometric functions sine and cosine.
• Similarly, the force from Dog 2 is divided into a northward and an eastward component.

The practical utility of vector resolution cannot be overstressed. The task of calculating how multiple forces interact becomes far more manageable when each force is considered in basic directions.

Understanding how to resolve vectors allows you to decipher real-world problems where forces do not act merely up, down, left, or right, but in combinations. By using vector resolution, each force is expressed in terms of its influence in two perpendicular directions, making it easier to find the total effect, or resultant force.