Problem 86 - Precalculus

Question

Resultant Force. A person is walking three 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 \(8,\) Dog 2 applies a force of \(\mathrm{N} 45^{\circ} \mathrm{E}\) with a magnitude of \(6,\) and Dog 3 moves directly \(\mathrm{N}\) with a magnitude of \(12 .\) Find the magnitude and direction of the force \(\mathbf{w}\) 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 of 20.42 N, directed 82.43° E of S, to counterbalance the dogs' forces.

Step 1: Represent Forces in Cartesian Coordinates

Convert the forces applied by each dog into their Cartesian (x, y) components. For Dog 1 (60° W of N):- North component (y): \(8 \cos(60^\circ) = 4\)- West component (x, negative): \(-8 \sin(60^\circ) = -6.93\)For Dog 2 (45° E of N):- North component (y): \(6 \cos(45^\circ) = 4.24\)- East component (x): \(6 \sin(45^\circ) = 4.24\)For Dog 3 (directly N):- North component (y): \(12\)- West/East component (x): \(0\)

Step 2: Sum of Forces in Each Direction

Calculate the sum of forces in both x and y directions considering all the dogs. Total force in x-direction (East positive, West negative):\[ -6.93 + 4.24 + 0 = -2.69 \]Total force in y-direction (North positive):\[ 4 + 4.24 + 12 = 20.24 \]

Step 3: Determine Resultant Force

Calculate the resultant force \( \mathbf{R} \) using the x and y components obtained. Magnitude: \[ \|\mathbf{R}\| = \sqrt{(-2.69)^2 + (20.24)^2} \approx 20.42 \]Direction (angle \( \theta \) with the positive x-axis):\[ \theta = \tan^{-1}\left(\frac{20.24}{-2.69}\right) \approx 97.57^\circ \] with respect to the positive x-axis (East direction), which translates to 82.43° W of N.

Step 4: Counteracting Force by Walker

The force \( \mathbf{w} \) applied by the walker must be equal in magnitude and opposite in direction to the resultant force \( \mathbf{R} \) from the dogs to maintain equilibrium.Thus, \( \mathbf{w} \) has a magnitude of 20.42 and a direction of 82.43° E of S (opposite direction to \( \mathbf{R} \)).

Key Concepts

Resultant Force

The concept of resultant force is fundamental when dealing with vector addition. Here, multiple forces act on an object, and the resultant force is the single force that has the same effect as all the original forces combined. In the scenario with the dogs, each force has a magnitude and direction. These forces are combined vectorially to determine their overall effect on the person holding the leash.

When calculating the resultant force, we need to break each force down into its Cartesian components. This involves separating each force into x (horizontal) and y (vertical) components. By summing these components separately, you obtain the net force in the x-direction and y-direction. Use the Pythagorean theorem to calculate the magnitude of the resultant force:

• Magnitude: \[ |\mathbf{R}| = \sqrt{(\text{Total force in x})^2 + (\text{Total force in y})^2} \]

For its direction, use the arctangent function:

• Direction: \[ \theta = \tan^{-1}\left(\frac{\text{Total force in y}}{\text{Total force in x}}\right) \]

This angle can give you the directional reference with respect to the positive x-axis.

Cartesian Coordinates

To work with forces depicted in terms of direction and magnitude, it's essential to translate these into Cartesian coordinates. This conversion allows for easier calculation and visualization of vector addition. In Cartesian coordinates, a force vector is broken down into two components:

• x-component (horizontal)
• y-component (vertical)

For instance, a force given at an angle from the horizontal can be decomposed as:

• x-component: \[ F_x = F \cos(\theta) \]
• y-component: \[ F_y = F \sin(\theta) \]

Where \( \theta \) is the angle the force makes with the horizontal. In our dog walking scenario, each dog's force is converted into these two components based on the specified direction, allowing their effects to be added or countered precisely. This step is crucial because it standardizes the way vectors are represented and simplifies subsequent calculations.

Equilibrium

Equilibrium is achieved when all the forces acting on a system result in a net force of zero. This means that the object, in this case, the person walking the dogs, does not accelerate but remains in stable steadiness. In physical terms, equilibrium occurs when the total vector sum of the forces is zero.

For the walker to maintain equilibrium, the force they apply must perfectly counterbalance the resultant force from the dogs pulling in different directions. In mathematical terms, this equates to:

• Walker’s Force = \( - \mathbf{R} \)

Which indicates that the walker’s force should be equal in magnitude but opposite in direction to the resultant force of the dogs, ensuring stabilization.

In our example, because the dogs had a combined resultant force at 82.43° W of N, the walker must apply an equal force in the exact opposite direction to ensure they are not dragged off course, achieving a balanced and controlled walk. Understanding equilibrium in this context is key to comprehending how forces interact dynamically within a system.