Problem 36 - Precalculus Mathematics for Calculus

Question

Find the work done by the force \(\mathbf{F}\) in moving an object from \(P\) to \(Q\). $$\mathbf{F}=400 \mathbf{i}+50 \mathbf{j} ; \quad P(-1,1), Q(200,1)$$

Step-by-Step Solution

Answer

The work done by the force is 80400 units.

Step 1: Identify Force Vector and Points

The force vector is given as \( \mathbf{F} = 400\mathbf{i} + 50\mathbf{j} \). The initial point is \( P(-1,1) \) and the terminal point is \( Q(200,1) \).

Step 2: Calculate Displacement Vector

To find the displacement vector, subtract the coordinates of point \( P \) from point \( Q \): \( \mathbf{d} = (200 - (-1))\mathbf{i} + (1 - 1)\mathbf{j} = 201\mathbf{i} + 0\mathbf{j} \). Therefore, the displacement vector is \( \mathbf{d} = 201\mathbf{i} \).

Step 3: Find Dot Product of Force and Displacement

The work done by the force \( \mathbf{F} \) is the dot product of \( \mathbf{F} \) and \( \mathbf{d} \). Calculate the dot product: \( \mathbf{F} \cdot \mathbf{d} = (400\mathbf{i} + 50\mathbf{j}) \cdot (201\mathbf{i} + 0\mathbf{j}) \).

Step 4: Calculate Dot Product

Calculate the dot product: \( 400 \cdot 201 + 50 \cdot 0 = 80400 + 0 = 80400 \).

Step 5: Conclusion

Since the only contribution to the work is from the \( \mathbf{i} \)-component, the total work done by the force \( \mathbf{F} \) in moving the object from point \( P \) to point \( Q \) is 80400. The \( \mathbf{j} \)-component does not contribute as there is no movement in the \( y \)-direction.

Key Concepts

Force Vector

A force vector is a way to represent a force that acts on an object in terms of both magnitude and direction. In mathematical terms, it is expressed in component form, typically using the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), which represent the directions of the coordinate axes.
Here, the force vector \( \mathbf{F} = 400\mathbf{i} + 50\mathbf{j} \) consists of two components: 400 units in the x-direction and 50 units in the y-direction. This allows us to visualize the force as an arrow pointing from an initial point in space, with the direction and length of the arrow representing how the object will move under this force.

• The x-component (\(400\mathbf{i}\)) dictates movement along the horizontal, or x, axis.
• The y-component (\(50\mathbf{j}\)) dictates movement along the vertical, or y, axis.

Recognizing the components of a force vector helps in calculating the effect of forces and determining how they influence the motion of objects.

Displacement Vector

A displacement vector shows how an object moves from one point to another in space. It is calculated by subtracting the initial position from the final position. This vector is crucial in mechanics to understand how far and in which direction an object has traveled.
In this case, the object moves from point \( P(-1,1) \) to point \( Q(200,1) \). The displacement vector is then calculated as \( \mathbf{d} = 201\mathbf{i} + 0\mathbf{j} \).

• The x-component (\(201\mathbf{i}\)) indicates a travel of 201 units in the x-direction.
• There is no y-component (\(0\mathbf{j}\)), indicating no vertical or y-direction movement.

This simplifies our task of understanding how the object has physically moved: completely along the x-axis with no vertical change.

Dot Product

The dot product, also known as the scalar product, is an essential operation in vectors used to calculate how much one vector combines with another. When calculating work done by a force, the dot product helps in determining the contribution of a force vector in the direction of the displacement vector.
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:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \)

In our problem, the dot product \( \mathbf{F} \cdot \mathbf{d} \) simplifies to \( (400 \cdot 201) + (50 \cdot 0) = 80400 + 0 \).
This operation results in 80400, which tells us the work done by the force is solely in the direction known as \( \mathbf{i} \), or the x-direction.

Vectors in Physics

Vectors are fundamental tools in physics that allow us to describe quantities having both magnitude and direction. They are commonly used to model physical phenomena such as force, velocity, and displacement which naturally occur in our three-dimensional world.
In practical applications, understanding vectors helps us visualize and predict the effects of these quantities on an object's motion.

• Force Vectors: Represent forces acting on objects, showing how they affect motion.
• Displacement Vectors: Indicate changes in position and help calculate distance and direction.
• Dot Product: Assists in finding the work done by forces, especially when forces and movement are not perfectly aligned.

By decomposing vectors into components, calculating with vector operations like the dot product, or studying interactions, students not only solve textbook problems but also gather insights into the physics governing the real world.