Problem 1 - Basic Engineering Mathematics

Question

A force of \(4 \mathrm{~N}\) is inclined at an angle of \(45^{\circ}\) to a second force of \(7 \mathrm{~N}\), both forces acting at a point. Find the magnitude of the resultant of these two forces and the direction of the resultant with respect to the \(7 \mathrm{~N}\) force by both the 'triangle' and the 'parallelogram' methods.

Step-by-Step Solution

Answer

Resultant force: \(10.23\text{ N}\), Direction: \(16^{\circ}\) from the 7 N force.

Step 1: Understand the Problem

We need to find the resultant of two forces: one force is 4 N and the other is 7 N. These forces are at an angle of \( 45^{\circ} \) to each other. We will use both the triangle and parallelogram methods to find the magnitude and direction of the resultant force relative to the 7 N force.

Step 2: Use the Triangle Method

Using the triangle method, we apply the law of cosines to the vector triangle formed by the forces and the resultant. Let's denote the resultant as \( R \). According to the law of cosines: \[ R^2 = 4^2 + 7^2 - 2 \times 4 \times 7 \times \cos(135^{\circ}) \] Given that \( \cos(135^{\circ}) = -\frac{1}{\sqrt{2}} \) or \( -0.7071 \), substitute and solve for \( R \): \[ R^2 = 16 + 49 + 56 \times 0.7071 = 65 + 39.6 \approx 104.6 \] Thus, \( R = \sqrt{104.6} \approx 10.23 \text{ N} \).

Step 3: Direction Using Triangle Method

To find the direction, we need to use the law of sines. Let \( \theta \) be the angle between the 7 N force and the resultant: \[ \frac{\sin(\theta)}{4} = \frac{\sin(135^{\circ})}{R} \] \( \sin(135^{\circ}) = \frac{1}{\sqrt{2}} \) or \( 0.7071 \). Therefore \[ \sin(\theta) = \frac{4 \times 0.7071}{10.23} \approx 0.276 \]Thus, \( \theta = \arcsin(0.276) \approx 16^{\circ} \).

Step 4: Use the Parallelogram Method

For the parallelogram method, the resultant force \( R \) can also be calculated using the vector components. For two vectors, the magnitude of the resultant is given by:\[ R = \sqrt{4^2 + 7^2 + 2 \times 4 \times 7 \times \cos(45^{\circ})} \]Since \( \cos(45^{\circ}) = \frac{1}{\sqrt{2}} \): \[ R^2 = 16 + 49 + 56 \times \frac{1}{\sqrt{2}} = 65 + 39.6 \approx 104.6 \]Thus, \( R = \sqrt{104.6} \approx 10.23 \text{ N} \).

Step 5: Direction Using Parallelogram Method

The angle of the resultant \( \phi \) relative to the 7 N force can be found by calculating:\[ \tan(\phi) = \frac{4 \times \sin(45^{\circ})}{7 + 4 \times \cos(45^{\circ})} \]Substitute \( \sin(45^{\circ}) = \cos(45^{\circ}) = \frac{1}{\sqrt{2}} \):\[ \tan(\phi) = \frac{4 \times \frac{1}{\sqrt{2}}}{7 + 4 \times \frac{1}{\sqrt{2}}} \approx \frac{2.828}{9.828} \approx 0.288 \]\( \phi \approx 16^{\circ} \).

Step 6: Conclusion

Both the triangle and parallelogram methods give us the resultant magnitude as \( 10.23 \text{ N} \) and the direction as \( 16^{\circ} \) with respect to the 7 N force.

Key Concepts

Triangle Method

The triangle method is a straightforward way to find the resultant force when two forces act at a point. In this approach, the forces are represented as sides of a scaled vector triangle, with the resultant force forming the third side. By connecting the vectors tip to tail, a triangle is formed.

To find the magnitude of the resultant, the law of cosines is applied. The law allows us to calculate the length of the third side of a triangle if we know the lengths of two sides and the angle between them. This is particularly useful in our scenario where the forces are known and the angle is given.

• Form a vector triangle by drawing the given forces.
• Apply the law of cosines to calculate the magnitude of the resultant force.
• This method is useful when the angle is explicitly given or can be easily measured.

The triangle method provides a visual understanding, making it easier to grasp how forces add up.

Parallelogram Method

The parallelogram method is a widely used technique for finding the resultant of two forces acting at a point. In this method, the two vectors are drawn as adjacent sides of a parallelogram. The diagonal represents the resultant vector.

This method utilizes vector addition principles, where both vectors are added using a geometric approach. To solve it, follow these steps:

• Draw both vectors starting from a common point.
• Complete the parallelogram by drawing lines parallel to the other side of each vector.
• The diagonal of the parallelogram provides the magnitude and direction of the resultant force.

This method is particularly intuitive as it visually demonstrates how two non-parallel vectors can be combined to yield a resultant. It also allows using vector components, making it flexible in solving real-world force problems.

Law of Cosines

The law of cosines is a crucial formula in understanding vector addition in terms of magnitudes and angles. It's used in both the triangle and parallelogram methods to find the resultant force's magnitude. Specifically, it helps when knowing two sides of a triangle and the included angle, as follows: \[ c^2 = a^2 + b^2 - 2ab \cos(\theta) \]where \(c\) is the side opposite the angle \(\theta\), and \(a\) and \(b\) are the known sides.

In our resultant force calculation:

• Use the law of cosines when dealing with non-right angles between vectors.
• This allows calculation of the new vector's magnitude which represents forces combined at an angle.
• This approach is necessary when angles do not simplify to 90 degrees.

The law of cosines bridges the gap between known values and the unknown resultant, providing an accurate result for the angle's influence on combined vectors.

Law of Sines

The law of sines is another essential tool used when solving vector problems involving non-right triangles. It relates the lengths of the sides of a triangle to the sines of their opposite angles. The formula is:\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]where \(a\), \(b\), and \(c\) are sides of the triangle, and \(A\), \(B\), and \(C\) are the angles opposite these sides.

In force calculations, the law of sines is particularly useful for finding unknown angles and comparing sides when non-right triangles are involved:

• Apply it after using the law of cosines to find missing angles.
• Leverage it to find the direction of the resultant vector after calculating its magnitude.
• This method provides a way to relate different angles and provide a pathway to determine vector directions.

The law of sines simplifies the task of finding angles necessary to fully understand vector interactions at inclined angles.