Problem 8 - Trigonometry

Question

For Questions 1 through 8, fill in the blank with an appropriate word. If a constant force F is applied to an object, and the resulting movement of the object is represented by the displacement vector d, then the work performed by the force is given by the ____ ______ of F and d.

Step-by-Step Solution

Answer

The dot product.

Step 1: Understanding the Problem

We need to find a term that describes the mathematical operation performed between two vectors, specifically, the force vector \( F \) and the displacement vector \( d \), which results in the calculation of work.

Step 2: Revisiting the Work Formula

Work \( W \) calculated when a constant force is applied to an object can be found using the formula \( W = F \cdot d \). This formula involves the operation of taking the product of two vectors.

Step 3: Determining the Type of Product

In vector mathematics, the multiplication of two vectors results in either a scalar or another vector. When calculating work, a scalar quantity, the operation used is the dot product (also known as the scalar product).

Step 4: Filling in the Blank

Based on the formula \( W = F \cdot d \), it is evident that the work is given by the dot product of \( F \) and \( d \). We fill in the blank with 'dot product'.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is an essential operation in vector mathematics. It takes two vectors and returns a scalar, a single numerical value. This operation is a cornerstone in physics and engineering, particularly when assessing work done by a force.To perform a dot product, each component of one vector is multiplied by the corresponding component of the other vector. Then, these products are summed up to produce the final scalar result.Mathematically, if you have two vectors, \( \mathbf{A} = [a_1, a_2, a_3] \) and \( \mathbf{B} = [b_1, b_2, b_3] \), the dot product \( \mathbf{A} \cdot \mathbf{B} \) is calculated as:

• \( a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3 \)

This operation is simple yet powerful, revealing the directional alignment between the two vectors involved.

Work Formula

The work formula is crucial in understanding how forces cause displacement in physics. Work is a measure of energy transferred when an object is moved over a distance by a force.The formula for work is given by:

• \( W = F \cdot d \)

where \( W \) represents work, \( F \) stands for the force vector, and \( d \) is the displacement vector.It's important to note the dot product in the formula. It indicates that only the component of the force directed along the path of displacement contributes to work. If the force is perpendicular to the movement, the work done is zero.This equation highlights that the amount of work depends not just on the magnitudes of force and displacement but also their direction relative to each other.

Force and Displacement Vectors

Vectors are fundamental in describing physical quantities like force and displacement. A vector has both magnitude and direction, and understanding this helps when solving physics problems involving these quantities. The **force vector** describes the push or pull on an object, with both strength and direction. In contrast, the **displacement vector** represents the change in position of that object, also defined by size and direction. When calculating work, both these vectors are crucial. Work depends on how much of the force is directed along the object's path, a detail captured by the dot product. Visualize a force vector pointing to the right while the displacement vector points upwards. Although a force is active, no work affects displacement due to perpendicular directions, leading to zero work. Understanding vector orientations, therefore, is essential in solving work-related problems.