Question

Find the acute angle that a constant unit force vector makes with the positive \(x\) -axis if the work done by the force in moving a particle from (0,0) to (4,0) equals 2 .

Short answer

The acute angle is \( 60^\text{o} \).

Step-by-step solution

Identify Given Information

Given a constant unit force vector \(\textbf{F}\) and the path of the particle moving from \((0,0)\) to \((4,0)\). The work done by the force is given as 2 units.

Use the Work Formula

The work done by a force is given by the dot product of the force vector \(\textbf{F}\) and the displacement vector \(\textbf{d}\) as \( W = \textbf{F} \bullet \textbf{d} \).

Express Displacement Vector

The displacement vector from \((0,0)\) to \((4,0)\) is \( \textbf{d} = 4 \textbf{i} \), where \(\textbf{i}\) is the unit vector along the x-axis.

Represent Force Vector

Assume the unit force vector as \( \textbf{F} = \text{cos} \theta \textbf{i} + \text{sin} \theta \textbf{j} \), where \( \theta \) is the angle between the force vector and the positive x-axis.

Calculate Dot Product

Compute the dot product \(\textbf{F} \bullet \textbf{d} = ( \text{cos} \theta \textbf{i} + \text{sin} \theta \textbf{j} ) \bullet ( 4 \textbf{i} )\). This results in \( \textbf{F} \bullet \textbf{d} = 4 \text{cos} \theta \).

Set Up Equation with Work Done

Set up the equation with the given work: \( 4 \text{cos} \theta = 2 \).

Solve for \( \theta \)

Solving \( 4 \text{cos} \theta = 2 \) for \( \text{cos} \theta \), we get \( \text{cos} \theta = \frac{1}{2} \). The acute angle \( \theta \) such that \( \text{cos} \theta = \frac{1}{2} \) is \( 60^\text{o} \).

Key concepts

dot product

When you talk about work in physics, you often have to deal with the dot product. The dot product is an operation that takes two vectors and returns a scalar. It's essential in calculating work.
To compute the dot product, you multiply the corresponding components of two vectors and add them up:
\(\textbf{A} \bullet \textbf{B} = A_xB_x + A_yB_y \).
In the context of work, the formula \(\textbf{F} \bullet \textbf{d} = W\) means you multiply the force vector by the displacement vector.

unit vector

Unit vectors are used to indicate direction. They have a magnitude of 1 and point in a specific direction. Commonly, unit vectors are labeled as \(\textbf{i}\) and \(\textbf{j}\) which represent directions along the x-axis and y-axis, respectively.
In our exercise, \(\textbf{i}\) is the unit vector along the x-axis. This helps simplify calculations and keeps the focus on direction rather than magnitude.

displacement vector

The displacement vector \(\textbf{d}\) represents the change in position of an object. It doesn't just tell you how far you've moved; it tells you the direction too.
In our example, the particle moves from \(0,0\) to \(4,0\), making the displacement vector \(\textbf{d} = 4\textbf{i}\). This indicates a movement of 4 units along the x-axis without any change in the y-axis.

trigonometric functions

Trigonometric functions such as cosine and sine are key to understanding angles and directions in physics. Using trigonometric functions, you can decompose vectors into their components.
For example, our force vector is given by \( \textbf{F} = \text{cos}\theta \textbf{i} + \text{sin}\theta \textbf{j} \), where \(\theta\) is the angle.
The cosine function helps us find the x-component of our vector, while the sine helps with the y-component.

angle calculation

To find the angle between two vectors, you often use trigonometric functions and the dot product. The dot product formula \(\textbf{F} \bullet \textbf{d} = W\) was rearranged to find \(\theta\).
With the work given as 2 units, the equation simplifies to \(4 \text{cos} \theta = 2\), making \( \text{cos} \theta = \frac{1}{2} \).
The angle \(\theta\) corresponding to \( \text{cos} \theta = \frac{1}{2} \) is an acute 60 degrees, as per trigonometric identities.