Problem 39 - Trigonometry

Question

Find the work performed when the given force \(\mathbf{F}\) is applied to an object, whose resulting motion is represented by the displacement vector \(d\). Assume the force is in pounds and the displacement is measured in feet. \(\mathbf{F}=45 \mathbf{i}-12 \mathbf{j}, \mathbf{d}=170 \mathbf{i}+15 \mathbf{j}\)

Step-by-Step Solution

Answer

The work performed is 7470 foot-pounds.

Step 1: Understand the Dot Product

When a force is applied to an object resulting in a displacement, the work performed is given by the dot product of the force and displacement vectors. The formula for work done is: \[W = \mathbf{F} \cdot \mathbf{d}\] where \(W\) is the work, \(\mathbf{F}\) is the force vector, and \(\mathbf{d}\) is the displacement vector.

Step 2: Compute the Dot Product

The dot product of two vectors \(\mathbf{A} = a_1 \mathbf{i} + a_2 \mathbf{j}\) and \(\mathbf{B} = b_1 \mathbf{i} + b_2 \mathbf{j}\) is calculated as:\[\mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2\]Plugging in the given vectors \(\mathbf{F}=45 \mathbf{i}-12 \mathbf{j}\) and \(\mathbf{d}=170 \mathbf{i}+15 \mathbf{j}\), we get:\[W = (45 \mathbf{i}-12 \mathbf{j}) \cdot (170 \mathbf{i}+15 \mathbf{j})\]

Step 3: Multiply and Add the Components

Calculate the dot product of \(\mathbf{F}\) and \(\mathbf{d}\) by multiplying each component and summing the results:\[W = (45 \times 170) + (-12 \times 15)\]which results in:\[W = 7650 - 180 = 7470\]

Step 4: Conclude the Work Done

Based on the calculation, the work performed by the force \(\mathbf{F}\) over the displacement \(\mathbf{d}\) is 7470 foot-pounds. This means that 7470 units of work were performed when the object was displaced under the given force.

Key Concepts

Work Performed

When talking about work in physics, it’s important to understand that the concept is different from how we use the word in everyday language. In this context, work is about how much energy is transferred when a force moves an object over a distance.
Imagine you are pushing a shopping cart. If you apply a force and the cart moves in the direction of that force, you are doing work. Mathematically, work is calculated using the dot product of the force vector and the displacement vector. This means we analyze the components of these vectors to see how they contribute to moving the object.
The formula for calculating work is \[ W = \mathbf{F} \cdot \mathbf{d} \]where:

• \(W\) is the work done (in foot-pounds in this context)
• \(\mathbf{F}\) is the force vector
• \(\mathbf{d}\) is the displacement vector

Knowing how to calculate work is very useful in understanding how forces affect the movement of objects in our environment.

Force Vector

A force vector represents a push or pull exerted on an object. It has both a magnitude (how strong the force is) and a direction (the path along which the force is applied). In the real world, forces cause objects to start moving, stop moving, or change direction.
When writing force vectors, we often use unit vectors like \(\mathbf{i}\) and \(\mathbf{j}\) to indicate directional components. In the problem you're dealing with, the force vector is given as \(\mathbf{F}=45 \mathbf{i} - 12 \mathbf{j}\). This means:

• The force has a strong component in the positive \(\mathbf{i}\) direction (which we can think of as rightward)
• It also has a weaker component in the negative \(\mathbf{j}\) direction (which points downward)

Understanding force vectors is crucial because they help us predict and calculate how objects will move when subjected to different amounts and directions of force.

Displacement Vector

A displacement vector shows how far and in what direction an object has moved from its initial position. Like the force vector, it has both magnitude and direction. Displacement is not the same as distance, as it considers the object's overall change in position, rather than the path taken.
In vector form, displacement can be described using \(\mathbf{i}\) and \(\mathbf{j}\), similar to force vectors. In your exercise, the displacement vector is \(\mathbf{d}=170 \mathbf{i} + 15 \mathbf{j}\), indicating:

• The object has moved a significant amount in the positive \(\mathbf{i}\) direction
• It also has a smaller movement in the positive \(\mathbf{j}\) direction

Displacement vectors are essential in calculating work because they quantify the actual movement of an object under the influence of a force. This helps us understand the effectiveness of exerted forces in moving an object.