Table of contents

0. Math Review31m
- Math Review
  31m
1. Intro to Physics Units1h 29m
2. 1D Motion / Kinematics3h 56m
3. Vectors2h 43m
4. 2D Kinematics1h 42m
5. Projectile Motion3h 6m
6. Intro to Forces (Dynamics)3h 22m
7. Friction, Inclines, Systems2h 44m
8. Centripetal Forces & Gravitation7h 26m
9. Work & Energy1h 59m
10. Conservation of Energy2h 54m
11. Momentum & Impulse3h 40m
12. Rotational Kinematics2h 59m
13. Rotational Inertia & Energy7h 4m
14. Torque & Rotational Dynamics2h 5m
15. Rotational Equilibrium3h 39m
16. Angular Momentum3h 6m
17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
19. Fluid Mechanics4h 27m
20. Heat and Temperature3h 7m
21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
23. The Second Law of Thermodynamics3h 11m
24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
29. Sources of Magnetic Field2h 30m
30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

3. Vectors

Adding Vectors by Components

Physics

3. Vectors

Adding Vectors by Components

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6:02 minutes

Problem 43

Textbook Question

FIGURE P3.43 shows three ropes tied together in a knot. One of your friends pulls on a rope with 3.0 units of force and another pulls on a second rope with 5.0 units of force. How hard and in what direction must you pull on the third rope to keep the knot from moving? Give the direction as an angle below the negative x-axis.

Verified step by step guidance

Step 1: Break down the forces into their x and y components. For the first student pulling with 4.0 N at an angle of 60° above the positive x-axis, calculate the x-component using the formula Fₓ = F * cos(θ) and the y-component using Fᵧ = F * sin(θ). For the second student pulling with 6.0 N along the negative x-axis, the x-component is -6.0 N and the y-component is 0 N.

Step 2: Add the x-components of the forces from the first and second students to find the total x-component of the force acting on the knot. Similarly, add the y-components of the forces from the first and second students to find the total y-component of the force acting on the knot.

Step 3: To keep the knot stationary, the third student's force must exactly cancel out the resultant force from the first and second students. This means the third student's x-component must be equal in magnitude but opposite in sign to the total x-component, and the y-component must be equal in magnitude but opposite in sign to the total y-component.

Step 4: Use the Pythagorean theorem to calculate the magnitude of the third student's force. The formula is F = √(Fₓ² + Fᵧ²), where Fₓ and Fᵧ are the x and y components of the third student's force.

Step 5: Determine the direction of the third student's force. Use the formula θ = arctan(Fᵧ / Fₓ) to calculate the angle, and adjust the angle to be below the negative x-axis as specified in the problem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. This involves adding the magnitudes and directions of the vectors, typically using graphical methods or mathematical calculations. In this scenario, the forces exerted by the first two students must be combined to find the necessary force and direction for the third student to maintain equilibrium.

Equilibrium

Equilibrium occurs when the net force acting on an object is zero, meaning all forces balance each other out. In the context of this problem, the forces applied by the three students must sum to zero to keep the knot stationary. This principle is crucial for determining the magnitude and direction of the force the third student must apply.

Trigonometry in Physics

Trigonometry is essential in physics for analyzing forces, especially when dealing with angles. It allows us to resolve forces into their components along the x and y axes. In this problem, the angles given for the forces applied by the first two students will be used to calculate their components, which will then help in determining the required force and angle for the third student.

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Textbook Question

A postal employee drives a delivery truck over the route shown in Fig. E1.25. Use the method of components to determine the magnitude and direction of her resultant displacement. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.

For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of the vector difference B - A

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Textbook Question

For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of the vector difference A - B

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Textbook Question

For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of the vector sum A + B

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Textbook Question

A position vector in the first quadrant has an x-component of 6 m and a magnitude of 10 m. What is the value of its y-component?

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Textbook Question

A velocity vector 40 degrees below the positive x-axis has a y-component of -10 m/s. What is the value of its x-component?

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Textbook Question

The bacterium E. coli is a single-cell organism that lives in the gut of healthy animals, including humans. When grown in a uniform medium in the laboratory, these bacteria swim along zig-zag paths at a constant speed of 20 μm/s. FIGURE P3.42 shows the trajectory of an E. coli as it moves from point A to point E. What are the magnitude and direction of the bacterium's average velocity for the entire trip?

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Textbook Question

FIGURE P3.26 shows vectors A and B. Find D = 2A ＋B Write your answer in component form.

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