Problem 37 - Essential Calculus Early Transcendentals

Question

Find the work done by a force \(\mathbf{F}=8 \mathbf{i}-6 \mathbf{j}+9 \mathbf{k}\) that moves an object from the point \((0,10,8)\) to the point \((6,12,20)\) along a straight line. The distance is measured in meters and the force in newtons.

Step-by-Step Solution

Answer

144 joules of work is done by the force.

Step 1: Understanding the Equation for Work

The work done by a force moving an object along a straight path can be calculated using the dot product of the force vector and the displacement vector. The formula is given by:\[ W = \mathbf{F} \cdot \mathbf{d} \]where \(\mathbf{F}\) is the force vector and \(\mathbf{d}\) is the displacement vector.

Step 2: Calculating the Displacement Vector

To find the displacement vector \(\mathbf{d}\), subtract the initial position \((0,10,8)\) from the final position \((6,12,20)\):\[ \mathbf{d} = (6-0)\mathbf{i} + (12-10)\mathbf{j} + (20-8)\mathbf{k} \]\[ = 6\mathbf{i} + 2\mathbf{j} + 12\mathbf{k} \]

Step 3: Performing the Dot Product

Compute the dot product of the force vector \(\mathbf{F}=8\mathbf{i}-6\mathbf{j}+9\mathbf{k}\) and the displacement vector \(\mathbf{d}=6\mathbf{i}+2\mathbf{j}+12\mathbf{k}\):\[ W = (8\mathbf{i} - 6\mathbf{j} + 9\mathbf{k}) \cdot (6\mathbf{i} + 2\mathbf{j} + 12\mathbf{k}) \]Calculate each component:- \(8 \times 6 = 48\)- \(-6 \times 2 = -12\)- \(9 \times 12 = 108\)Add these results: \[ 48 - 12 + 108 = 144 \]

Step 4: Conclusion

The total work done by the force \(\mathbf{F}\) on the object is 144 joules.

Key Concepts

Dot Product

The dot product is an essential concept in calculating the work done by a force when moving an object. It is a mathematical operation that takes two equal-length sequences of numbers and returns a single number. In terms of vectors, this operation helps determine how much one vector goes in the direction of another.

To perform a dot product of two vectors, you multiply their corresponding components and then add up the results. For example, given two vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \)and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \),the dot product is calculated as:\(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3.\)

• This operation helps us find how much of one vector aligns with another, essentially measuring their parallelism.
• If the vectors are perpendicular, their dot product is zero since there is no overlap.
• If they are parallel, the dot product is maximal.

In the context of work, the dot product is used to find how effectively a force contributes to the movement along the displacement path.

Displacement Vector

The displacement vector represents the difference between the initial and final positions of an object, and it effectively "points" from the start location to the end location. It is a crucial element in determining the work done by a force because it shows the directional path on which the force acts.

To find the displacement vector, subtract the initial position vector from the final position vector. For instance, if an object moves from point \((x_1, y_1, z_1)\) to \((x_2, y_2, z_2)\), the displacement vector \( \mathbf{d} \) is calculated as:\[\mathbf{d} = (x_2 - x_1)\mathbf{i} + (y_2 - y_1)\mathbf{j} + (z_2 - z_1)\mathbf{k}.\]

• It describes both the distance and direction of the move.
• The displacement vector is crucial when performing the dot product with the force vector to compute work.
• It ignores the path taken by the object and only considers the start and end points.

In the example provided, computing the displacement vector gives clarity on how the object has actually shifted through space.

Force Vector

A force vector defines the force applied in a particular direction and is typically expressed in component form, such as \( \mathbf{F} = f_1\mathbf{i} + f_2\mathbf{j} + f_3\mathbf{k} \).The force vector conveys both magnitude and direction, which are crucial in calculating work over a displacement vector.

• The direction of the force vector indicates where the force is applied relative to the object's direction.
• The components of the vector, labeled systematically as \( f_1, f_2, \) and \( f_3 \), represent its projection along each of the coordinate axes.
• Magnitude can be found using the formula \( |\mathbf{F}| = \sqrt{f_1^2 + f_2^2 + f_3^2} \).
  

In the context of calculating work, only the component of the force that acts along the direction of the displacement vector does any work. This is why the dot product, which considers directional alignment, is crucial to determine how much of the force contributes to moving the object.