Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The work \(W\) done by a constant force \(\mathbf{F}\) acting along the line of motion of an object is represented by a vector.

Step-by-Step Solution

Answer

The statement is false. Work, denoted by \(W\), is a scalar quantity, not a vector. This is because it is calculated as the dot product of force (a vector) and displacement (a scalar), and the dot product of a vector and a scalar is a scalar.

Step 1: Understanding Scalars and Vectors

In physics, there are two types of quantities: scalars and vectors. Scalars are quantities that are fully described by a magnitude (a numerical value) alone. Examples include time, temperature, and volume. On the other hand, vectors are quantities that are defined by both a magnitude and a direction. Examples of vector quantities are displacement, velocity, and force.

Step 2: Understanding Work

Work, denoted by \(W\), is a measure of the energy transferred by a force over a distance. According to the formula \(W = \mathbf{F} \cdot d \cdot \cos(\theta)\), where \(\mathbf{F}\) is the force, \(d\) is the distance, and \(\theta\) is the angle between the force and the direction of motion, work is the product of the force (a vector), the distance (a scalar), and the cosine of the angle (a scalar). The dot product of a vector variable with another vector variable is a scalar, which means that work is a scalar, not a vector.

Step 3: Final Deduction

Given that work (\(W\)) is the dot product of the force (a vector) and the distance (a scalar), and we know that the dot product of a vector and a scalar gives a scalar, it follows that work is a scalar quantity, not a vector. Therefore, the statement that work is represented by a vector is false.