Book cover for Applied Physics

Applied Physics

Dale Ewen, Neil Schurter, P. Erik Gundersen

ISBN #9780134159386

11th Edition

2,119 Questions

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18,063 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section develops the methods for analyzing forces in two dimensions, focusing on concurrent forces and equilibrium. Key techniques include vector addition using both triangle and parallelogram methods, resolving forces into components, and applying the Pythagorean theorem and trigonometric functions to determine magnitude and direction. The concepts extend to more complex scenarios such as rotational forces (torque) and real-world applications like cable-stayed bridges, where understanding compression, tension, and the center of gravity is crucial for structural stability.

Learning Objectives

1

Determine how to find the vector sum of concurrent forces acting at a single point.

2

Analyze equilibrium in one dimension and extend this analysis to two-dimensional force systems.

3

Interpret force diagrams and decompose forces into their x and y components.

4

Apply the principles of torque and rotational motion to solve equilibrium problems.

5

Understand and apply concepts of tension, compression, and center of gravity in real-world structures such as cable-stayed bridges.

Key Concepts

CONCEPT

DEFINITION

Concurrent Forces

Forces that act at the same point on a body. Their vector sum gives the resultant force.

Resultant Force

A single force that has the same effect as two or more forces acting together, found by vector addition.

Vector Components

The projections of a vector along the x and y axes, used to simplify vector addition.

Equilibrium

The state of a body when the net force acting on it is zero, meaning no acceleration.

Torque

A measure of the tendency of a force to rotate an object about an axis.

Tension and Compression

Types of forces where tension pulls apart and compression pushes together material or structures.

Center of Gravity

The average location of the weight distribution on an object, important in analyzing balance and stability.

Example Problems

Example 1

Find the sum of each set of forces acting at the same point in a straight line. $$355 \mathrm{~N} \text { (right) } ; 475 \mathrm{~N} \text { (right); } 245 \mathrm{~N} \text { (left); } 555 \mathrm{~N} \text { (left) }$$

Example 2

Find the sum of each set of forces acting at the same point in a straight line. $$703 \mathrm{~N} \text { (right); } 829 \mathrm{~N} \text { (left); } 125 \mathrm{~N} \text { (left); } 484 \mathrm{~N} \text { (left) }$$

Example 3

Find the sum of each set of forces acting at the same point in a straight line. Forces of $225 \mathrm{~N}$ and $175 \mathrm{~N}$ act at the same point. (a) What is the magnitude of the maximum net force the two forces can exert together? (b) What is the magnitude of the minimum net force the two forces can exert together?

Example 4

Find the sum of each set of forces acting at the same point in a straight line. Three forces with magnitudes of $225 \mathrm{~N}, 175 \mathrm{~N},$ and $125 \mathrm{~N}$ act at the same point. (a) What is the magnitude of the maximum net force the three forces can exert together? (b) What is the magnitude of the minimum net force the three forces can exert together?

Example 5

Find the sum of each set of vectors. Give angles in standard position.

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Step-by-Step Explanations

QUESTION

Given F1 = 525 N (vertical) and F2 = 763 N (horizontal) acting at a point, find the resultant force using the triangle method.

STEP-BY-STEP ANSWER:

Step 1: Resolve F1 and F2 into components. For F1: x-component = 0 N, y-component = 525 N; for F2: x-component = 763 N, y-component = 0 N.
Step 2: Add the components: Rx = 0 + 763 = 763 N, Ry = 525 + 0 = 525 N.
Step 3: Calculate the magnitude of the resultant using the Pythagorean theorem: R = √(763² + 525²) ≈ 926 N.
Step 4: Determine the direction (angle) using arctan(Ry/Rx): angle = arctan(525/763) ≈ 34.5°.
Final Answer: The resultant force is approximately 926 N at an angle of 34.5° above the horizontal.

Vector Addition using the Triangle Method

QUESTION

For F1 = 525 N at 40.0° above the horizontal and F2 = 763 N horizontally, determine the resultant force.

STEP-BY-STEP ANSWER:

Step 1: Compute F1 components: x-component = 525 cos(40.0°) ≈ 402 N, y-component = 525 sin(40.0°) ≈ 337 N.
Step 2: For F2, since it is horizontal: x-component = 763 N, y-component = 0 N.
Step 3: Add the components to find the resultant: Rx = 402 + 763 = 1165 N, Ry = 337 + 0 = 337 N.
Step 4: Find the angle of the resultant: angle = arctan(337/1165) ≈ 16.1° above the horizontal.
Step 5: Compute the magnitude: R = √(1165² + 337²) ≈ 1210 N.
Final Answer: The resultant vector is approximately 1210 N at 16.1° above the horizontal.

Vector Addition using the Parallelogram Method

QUESTION

For three forces F1 = 375 N, F2 = 575 N, and F3 = 975 N applied at the same point with angles between them (F1 and F2 at 60°, F2 and F3 at 80°), find the resultant force.

STEP-BY-STEP ANSWER:

Step 1: Place the point of application at the origin and align one force with the x-axis.
Step 2: Resolve each vector into x and y components. For example, use F2’s angle (60°) for F1 and appropriate adjustments for F3 (note F3 may require subtracting angles to obtain its x and y components).
Step 3: Sum up the x-components (note F3’s x-component may be negative depending on its direction) and the y-components separately.
Step 4: Compute the magnitude of the resultant using R = √(FRx² + FRy²).
Step 5: Determine the direction using arctan(|FRy|/|FRx|) and adjust for the correct quadrant. In the example, FR was found to be 1130 N at 94.3°.
Final Answer: The resultant force is approximately 1130 N directed at 94.3° from the reference force.

Resultant of Three Concurrent Forces

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Common Mistakes

  • Failing to resolve forces correctly into their x and y components, leading to incorrect vector sums.
  • Confusing the signs of force components, which can result in errors in quadrant determination.
  • Ignoring the importance of using consistent units and proper angle measurement (degrees vs. radians).
  • Assuming that all forces cause motion, without recognizing cases of equilibrium where net force is zero.
  • Overlooking the impact of rotational forces and torque when forces are not concurrent.