Problem 59 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the work done by the force \(\mathbf{F}=3 \mathbf{i}+10 \mathbf{j}\) newtons in moving an object 10 meters north (i.e., in the \(\mathbf{j}\) direction).

Step-by-Step Solution

Answer

100 joules of work is done.

Step 1: Understanding the Work Formula

The formula for calculating work when a force is applied is given by the dot product of the force vector \( \mathbf{F} \) and the displacement vector \( \mathbf{d} \): \[ \text{Work} = \mathbf{F} \cdot \mathbf{d} \].It calculates how much of the force is used in the direction of displacement.

Step 2: Identifying Vectors

Given the force vector \( \mathbf{F} = 3\mathbf{i} + 10\mathbf{j} \), and the object moves 10 meters north, the displacement vector is \( \mathbf{d} = 0\mathbf{i} + 10\mathbf{j} \) because only the \( \mathbf{j} \) component has magnitude.

Step 3: Calculating the Dot Product

The dot product \( \mathbf{F} \cdot \mathbf{d} \) is calculated by multiplying corresponding components and summing them: \[(3 \cdot 0) + (10 \cdot 10) = 0 + 100 = 100 \].This results in 100 joules.

Step 4: Conclusion

The work done by the force in moving the object is 100 joules, which represents how much force contributes to moving the object in the northward \( \mathbf{j} \) direction.

Key Concepts

Force Vector

In physics, a vector denotes a quantity with both magnitude and direction. A force vector specifically represents the force applied in a certain direction. This is crucial when calculating work, as it impacts how an object moves in a given direction. The original problem gives us a force vector \( \mathbf{F} = 3\mathbf{i} + 10\mathbf{j} \).

• \( \mathbf{i} \) represents the east-west horizontal direction.
• \( \mathbf{j} \) denotes the north-south vertical direction.

In this vector, the numbers 3 and 10 are the magnitudes of the force in these respective directions. Understanding the direction of the forces gives insight into which part of the force is effective in doing work during displacement.

Displacement Vector

The displacement vector indicates how far and in what direction an object has moved from its initial position. It's another crucial vector in work calculations because it determines the direction in which the force leads to movement. Given that the object moves 10 meters north, the displacement vector is \( \mathbf{d} = 0\mathbf{i} + 10\mathbf{j} \). Here:

• 0\mathbf{i} indicates no movement east or west.
• 10\mathbf{j} signifies a 10-meter movement towards the north.

This setup is perfect for illustrating how the components of force and displacement interact through the dot product.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (in this case, vectors) and returns a single number. This single number represents work when dealing with force and displacement vectors. Calculating the dot product involves:

• Multiplying the corresponding components of the vectors.
• Summing the results of these multiplications.

In our situation:\[ \mathbf{F} \cdot \mathbf{d} = (3 \cdot 0) + (10 \cdot 10) = 0 + 100 \]This calculation shows that 100 units of energy, or joules, are involved in moving the object north, effectively contributing to the work done by the force.

Joule

Joule is the standard unit of work or energy in the International System of Units (SI). It measures how much energy is transferred when a force of one newton moves an object one meter. This concept is pivotal in assessing how efficient a force is in performing work. In this exercise, the work calculated was 100 joules, meaning the force effectively moved the object 10 meters north with 100 units of energy. It quantifies the effectiveness of the force applied. Consider some common examples of joules:

• Lifting an apple one meter requires roughly one joule.
• 100 joules can heat a cup of coffee slightly.

In this context, joules help understand the practical impact of the calculated work.