Problem 16 - Applied Mathematics

Question

Express a force \(\mathbf{F}\) in the form \(a \mathbf{i}+b \mathbf{j}\). (a) The magnitude of the force is \(5 \mathrm{~N}\). (b) The force is inclined at \(60^{\circ}\) to the horizontal. (c) \(\mathrm{j}\) is in the direction of the upward vertical.

Step-by-Step Solution

Answer

\[\mathbf{F} = 2.5 \mathbf{i} + \frac{5 \sqrt{3}}{2} \mathbf{j} \]

Step 1: Understand the problem

The task is to express the force vector \(\textbf{F}\) in its component form \(a \textbf{i} + b \textbf{j}\). The magnitude of the force is given as 5 N, and it is inclined at 60° to the horizontal.

Step 2: Decompose the force into horizontal and vertical components

Using trigonometric functions, calculate the components of the force. The horizontal component (a) can be found using \(a = F \cos(\theta)\), and the vertical component (b) can be found using \(b = F \sin(\theta)\).

Step 3: Calculate the horizontal component

The horizontal component is \(a = 5 \cos(60°)\). Since \(\text{cos}(60°) = 0.5\), \(a = 5 \times 0.5 = 2.5 \text{N}\).

Step 4: Calculate the vertical component

The vertical component is \(b = 5 \sin(60°)\). Since \(\text{sin}(60°) = \frac{\sqrt{3}}{2}\), \(b = 5 \times \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} \approx 4.33 \text{N}\).

Step 5: Write the force as a vector

Combine the components to express the force in \(a \textbf{i} + b \textbf{j}\) form: \[\mathbf{F} = 2.5 \textbf{i} + \frac{5 \sqrt{3}}{2} \textbf{j} \].

Key Concepts

Force Vector

In physics, a force vector is a way to represent force mathematically. This vector not only contains information about the magnitude of the force but also its direction. A force vector can be broken down into its horizontal and vertical components. This is often denoted in the form \(a \mathbf{i} + b \mathbf{j}\), where 'a' is the horizontal component, 'b' is the vertical component, \(\mathbf{i}\) represents the unit vector in the horizontal direction, and \(\mathbf{j}\) represents the unit vector in the vertical direction.

Understanding this concept is crucial for analyzing forces in various directions. When given the total force and its angle of inclination, you can use trigonometric functions to find the individual components of the force vector. This decomposition is necessary for problem-solving in fields like engineering, mechanics, and physics.

Trigonometric Functions

Trigonometric functions like cosine (\text{cos}) and sine (\text{sin}) are essential for breaking down vectors into their components. To find the components of a force vector, you can use:

• The horizontal component, which is calculated with the cosine function: \(a = F \cos(\theta)\).
• The vertical component, which is found using the sine function: \(b = F \sin(\theta)\).

For example, in our problem, the force \(\mathbf{F}\) has a magnitude of 5 N and is inclined at 60° to the horizontal. The cosine of 60° (\text{cos}(60°)) is 0.5 and the sine of 60° (\text{sin}(60°)) is \(\frac{\sqrt{3}}{2}\).

So, the calculations would be:

• Horizontal component: \(a = 5 \times 0.5 = 2.5 \text{N}\).
• Vertical component: \(b = 5 \times \frac{\sqrt{3}}{2} = \frac{5 \sqrt{3}}{2} \approx 4.33 \text{N}\).

Magnitude and Direction

The magnitude and direction of a vector give us a full picture of its influence. The magnitude is the size or strength of the vector, expressed as a single number. For our exercise, the magnitude of the force is 5 N. The direction is given as 60° to the horizontal, which tells us the angle at which the force is applied.

To represent this vector, we break it into two components aligned with standard axes: horizontal and vertical. Decomposing the vector helps in understanding how much of the force acts in each direction. You need both magnitude and direction to fully describe the vector and solve related physics problems efficiently.
If the angle or magnitude changes, the vector components will also change, affecting the force's impact and response in different directions. So, understanding how to decompose and combine vectors is vital for the study of force and motion.