Problem 60 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the work done by a force of 100 newtons acting in the direction \(S 70^{\circ} \mathrm{E}\) in moving an object 30 meters east.

Step-by-Step Solution

Answer

The work done is approximately 2819.1 joules.

Step 1: Determine the Components of the Force

A force of 100 newtons is acting in the direction "South 70° East." This means that the angle is measured 70° East from the South direction line. To resolve this force along the East and South directions, we can consider this as 20° East from the East direction (since 90° - 70° = 20°). Thus, the Eastward component of the force is obtained by multiplying the force magnitude by the cosine of 20°. This gives us:\[ F_x = 100 \times \cos(20^{\circ}) \approx 100 \times 0.9397 = 93.97 \text{ N} \]

Step 2: Calculate the Work Done

Work done is calculated as the dot product of the force and displacement vectors. Here, only the component of the force in the direction of displacement (which is East) contributes to the work done. The displacement is given as 30 meters east.Therefore, the work done, \( W \), is:\[ W = F_x \times d = 93.97 \times 30 = 2819.1 \text{ joules} \]

Step 3: Conversion and Finalization

Since the units given and used are consistent (newtons for force, meters for distance), the work is calculated directly in joules. Confirming that all components align correctly with the direction of displacement ensures the proper computation.

Key Concepts

Force Components

When a force is applied at an angle, it can be broken down into components. These components represent the force's influence in specific directions. Think of it like having a flashlight beam that spreads out; you want to know how much light falls in each direction. In this problem, we have a force of 100 newtons with a direction, "South 70° East." This indicates how the force divides between "South" and "East." To calculate these components, we use basic trigonometry by thinking of 'East' as the horizontal component and 'South' as the vertical component.

• The force is resolved into two segments: one acting eastward and the other southward.
• The eastward component affects displacement since the object moves in that line.

This breakdown into components makes it easier to determine which part of the force directly contributes to movement in a given direction.

Trigonometric Functions

Trigonometric functions play a crucial role in breaking down forces into components. In this problem, we employ these functions to calculate the portion of force acting eastward.
Here's how it works:

• The angle given is "South 70° East." Trigonometry helps us reframe this angle with respect to the 'East' direction.
• This is done using the cosine function, as it helps us find the adjacent side of the right triangle formed by the force vector.
• By using cosine, we calculate the eastward component with this formula: \(F_x = 100 \times \cos(20^{\circ})\).

Using trigonometry ensures that we account for directional angles accurately, determining how effectively the force contributes to the horizontal movement. This approach clarifies the operation of forces in physical systems.

Dot Product

The dot product is a mathematical operation that helps calculate work done when a force causes movement. It incorporates both the magnitude and direction of vectors (force and displacement in this scenario).
Here’s why it’s essential:

• Work done is defined as the product of force in the direction of movement and the displacement.
• In mathematical terms, if both the force and displacement are in vector form, the dot product lets us compute this interaction.
• The formula used is \(W = F_x \times d\), where \(F_x\) is the force component in the direction of the displacement, and \(d\) is the distance moved.

The dot product helps integrate the physical directionality of the force, ensuring calculations reflect the real work done along the movement's direction.

Displacement

Displacement refers to the distance and direction of an object's movement. In this context, it's a vector, emphasizing its directional nature. When considering work, only displacement along the force's direction is crucial.
Here’s the focus in this scenario:

• The object moves 30 meters east, strictly in the horizontal plane.
• Since displacement is solely in the 'East' direction, only the eastward component of the force impacts the work done.
• Work combines both displacement and the aligned force component, making it direction-sensitive.

Understanding displacement as a vector object highlights how only directional matching forces contribute to the work exerted, ensuring precise calculation of energy transfer in the system.