Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

Mark pushes on a post in the direction \(\mathrm{S} 30^{\circ} \mathrm{E}\left(30^{\circ}\right.\) east of south) with a force of 60 pounds. Dan pushes on the same post in the direction \(\mathrm{S} 60^{\circ} \mathrm{W}\) with a force of 80 pounds. What are the magnitude and direction of the resultant force?

Step-by-Step Solution

Answer

Resultant force is approximately 100 pounds at 23° west of south.

Step 1: Understanding the Problem

We have two forces acting on a post in different directions. Mark applies 60 pounds in the direction S 30° E, and Dan applies 80 pounds in the direction S 60° W. We need to find the magnitude and direction of the resultant vector from these two force vectors.

Step 2: Breaking Down Forces into Components

For Mark's force: - South component: \(60 \cos(30^\circ)\) - East component: \(60 \sin(30^\circ)\) For Dan's force: - South component: \(80 \cos(60^\circ)\) - West component: \(80 \sin(60^\circ)\) Convert angles to radians or use trigonometric values for more precise calculation.

Step 3: Calculate Components Using Trigonometry

Calculate the numerical values: - Mark's force components: - South: \(60 \times \sqrt{3}/2 = 60 \times 0.866 = 51.96\) pounds - East: \(60 \times 1/2 = 30\) pounds - Dan's force components: - South: \(80 \times 1/2 = 40\) pounds - West: \(80 \times \sqrt{3}/2 = 80 \times 0.866 = 69.28\) pounds

Step 4: Total Force Components

Since both have southward components, just add these components:- Total South: \(51.96 + 40 = 91.96\) poundsThe east-west components oppose each other:- East component: 30 pounds - West component: 69.28 pounds- Total East-West: \(30 - 69.28 = -39.28\) pounds which means 39.28 pounds west.

Step 5: Calculate Resultant Force Magnitude

Use Pythagorean theorem to find the magnitude of the resultant vector:\[R = \sqrt{(91.96)^2 + (39.28)^2}\]Calculate to find \(R\).

Step 6: Resultant Magnitude Calculation

Calculate the resultant:\[R = \sqrt{91.96^2 + 39.28^2} = \sqrt{8455.84 + 1542.38} = \sqrt{9998.22} \approx 100\] pounds.

Step 7: Find Direction of Resultant Force

Use trigonometry (tangent function) to calculate the angle:\[\theta = \arctan\left(\frac{39.28}{91.96}\right)\approx 23^\circ\]The angle \(\theta\) is measured west of south.

Step 8: Conclusion

Hence, the magnitude of the resultant force is approximately 100 pounds, and it acts in the direction of \(23^\circ\) west of south.

Key Concepts

Force Components

In physics, when calculating forces acting in different directions, breaking them into components is essential. Force components simplify understanding how each force acts. They help in adding different forces easily.

Let's take an example that's quite illustrative. Mark and Dan are both pushing on a post, but in different directions and with different magnitudes of force. To solve such a problem, we resolve each force into components parallel to south-north and east-west directions. This helps us see clearly how much of each force is acting in these directions separately.

• For Mark's direction (S 30° E), resolving his 60-pound force gives a component pointing south, using cosine: 60 cos(30°) = 51.96 pounds, and a component pointing east using sine: 60 sin(30°) = 30 pounds.
• Dan, on the other hand, with a force in the S 60° W direction, breaks down to 80 cos(60°) = 40 pounds towards south and 80 sin(60°) = 69.28 pounds towards west.

Understanding components simplifies the addition of forces acting in non-collinear directions, leading us smoothly to find the resultant force.

Resultant Force

The resultant force is essentially the single force that represents the combined effect of two or more individual forces acting on an object. In our scenario, after splitting Mark’s and Dan’s forces into components, the next step is to add them up.

When adding the southward components of forces acted by Mark and Dan, we simply sum them, resulting in a total of 91.96 pounds southward. For the east-west components, there’s a slight twist. Since these components are opposed (30 pounds east and 69.28 pounds west), subtraction is in order, leaving us with 39.28 pounds directed west.

To find the magnitude of the resultant force, we invoke the Pythagorean theorem. This stems from the two perpendicular and summed forces components to yield a clear resultant. Hence, the magnitude is calculated as \[ R = \sqrt{91.96^2 + 39.28^2} \approx 100 \] pounds. This resultant vector holds the key to simplifying and understanding how 'net force' acts in the given exercise.

Trigonometry in Physics

Trigonometry proves indispensable when tackling problems involving forces and vectors in physics. The exercise involving Mark and Dan shows how trigonometry helps unravel the intricate web of vector addition.

When forces aren't aligned, trigonometric functions such as sine and cosine are vital in determining the extent of each force along different axes. In this context, cosine aids in finding the south (vertical) component, while sine helps direct us to the east-west (horizontal) component.

• Sine and cosine with their angle counterparts (like Mark’s 30° and Dan's 60°) reveal how much of each force is going in a certain direction.
• Pythagorean theorem adds to the trigonometric toolkit, providing a method to compute the magnitude of combined forces.
• The arctan function steps in to furnish the angle of the resultant force vector, the direction of which is crucial to solving real-world physics problems, yielding angles like \( 23^\circ \) calculated in this task.

Trigonometry wraps the vectors into the neat results we derive, signifying direction and magnitude as central to understanding any force-related physics challenges.