Problem 31 - Algebra and Trigonometry

Question

29-32 Find the work done by the force \(\mathbf{F}\) in moving an object from \(P\) to \(Q .\) $$\mathbf{F}=10 \mathbf{i}+3 \mathbf{j} ; \quad P(2,3), Q(6,-2)$$

Step-by-Step Solution

Answer

The work done by the force is 25.

Step 1: Determining the Vector from P to Q

To find the vector that represents the displacement from point \(P\) to point \(Q\), calculate the difference between their coordinates. We have \(P(2,3)\) and \(Q(6,-2)\). The vector from \(P\) to \(Q\) is given by \(\vec{PQ} = (6-2)\mathbf{i} + (-2-3)\mathbf{j} = 4\mathbf{i} - 5\mathbf{j}\).

Step 2: Calculating the Dot Product of \(\mathbf{F}\) and \(\vec{PQ}\)

The work done by a constant force in moving an object along a displacement is given by the dot product of the force vector and the displacement vector. The force is \(\mathbf{F} = 10\mathbf{i} + 3\mathbf{j}\), and we have found \(\vec{PQ} = 4\mathbf{i} - 5\mathbf{j}\). The dot product is \(\mathbf{F} \cdot \vec{PQ} = (10)(4) + (3)(-5) = 40 - 15 = 25\).

Step 3: Concluding the Work Done

The calculated dot product represents the work done by the force in moving the object. Thus, the work done by the force \(\mathbf{F}\) as the object moves from point \(P\) to point \(Q\) is \(25\).

Key Concepts

Vector Displacement

Vector displacement is a fundamental concept in physics that helps describe how an object moves from one point to another in space. Unlike distance, which is just a scalar and gives you the total path length traveled, vector displacement provides both the magnitude and the direction of that movement.
For our exercise, we determined the vector displacement from point \(P\) to point \(Q\), using their coordinates \((2,3)\) and \((6,-2)\) respectively.

• Subtract the starting point coordinates from the endpoint coordinates to find the components of the displacement vector.
• For instance \((6-2)\mathbf{i} + (-2-3)\mathbf{j}\) gives us the vector \(4\mathbf{i} - 5\mathbf{j}\).

This vector provides both direction and magnitude, and is important because it lays the foundation for calculating work, through operations like the dot product. Recognizing how objects shift from one point to another allows us to better understand the energy changes they experience during their path.

Dot Product

The dot product, also known as the scalar product, is a mathematical operation that takes two vectors and returns a single number, or scalar. This scalar can tell us several things, including the work done when a force moves an object.
In the given problem, the force vector \(\mathbf{F} = 10\mathbf{i} + 3\mathbf{j}\) and displacement vector \(\vec{PQ} = 4\mathbf{i} - 5\mathbf{j}\) were used to find work done.

• The dot product formula is \(\mathbf{F} \cdot \vec{PQ} = (F_x \cdot PQ_x) + (F_y \cdot PQ_y)\).
• Plugging in: \((10)(4) + (3)(-5) = 40 - 15 = 25\).

The dot product gives us a tangible metric of work done, based on how much of the force aligns with the direction of displacement. If these vectors were perpendicular, their dot product would be zero, indicating no work done in that direction.

Vectors in Physics

Vectors are essential to physics because they capture information about both magnitude and direction, unlike scalars which provide magnitude only. This is especially useful in physics, where both direction and size matter significantly, such as in forces, velocities, and displacements.
In our context, we have multiple vector quantities at play: the force vector \(\mathbf{F}\) and the displacement vector \(\vec{PQ}\).

• Force vectors like \(\mathbf{F} = 10\mathbf{i} + 3\mathbf{j}\) tell us the direction and magnitude of exerted forces.
• Displacement vectors like \(\vec{PQ} = 4\mathbf{i} - 5\mathbf{j}\) show how far and in what direction objects have moved.

Vectors also provide a way to easily perform operations such as the dot product or vector addition to solve more complex physics problems. Understanding vectors allows for more accurate predictions of how objects will move and react to different forces, making them crucial for anyone studying physics.