Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

Problem 44

Chapter 3, Problem 44

A crate, seen from above, is pulled with three ropes that have the tensions shown in FIGURE P3.44. Tension is a vector directed along the rope, measured in newtons (abbreviated N). Suppose the three ropes are replaced with a single rope that has exactly the same effect on the crate. What is the tension in this rope? Write your answer in component form using unit vectors.

Verified step by step guidance

Step 1: Identify the forces acting on the crate. The three forces are T₁ = 20 N directed along the positive y-axis, T₂ = 32 N at an angle of 25° above the positive x-axis, and T₃ = 30 N at an angle of 45° below the negative x-axis.

Step 2: Resolve each tension vector into its x and y components using trigonometric functions. For T₂ and T₃, use the formulas: x-component = T * cos(θ) and y-component = T * sin(θ).

Step 3: Calculate the x and y components of each tension vector: - T₁: x-component = 0, y-component = 20 N. - T₂: x-component = 32 * cos(25°), y-component = 32 * sin(25°). - T₃: x-component = -30 * cos(45°), y-component = -30 * sin(45°).

Step 4: Add the x-components of all three tensions to find the net x-component of the resultant tension vector. Similarly, add the y-components to find the net y-component of the resultant tension vector.

Step 5: Express the resultant tension vector in component form using unit vectors: T = (net x-component) î + (net y-component) ĵ.

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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 components of the vectors in each direction (x and y) separately. The resultant vector represents the cumulative effect of all the individual vectors acting together, which is crucial for solving problems involving forces, such as the tensions in the ropes.

Components of a Vector

A vector can be broken down into its components along the coordinate axes, typically the x-axis and y-axis. Each component represents the influence of the vector in that direction. For example, a tension vector can be expressed as T = T_x i + T_y j, where T_x and T_y are the components along the x and y axes, respectively, and i and j are the unit vectors in those directions.

Equilibrium of Forces

In physics, an object is in equilibrium when the net force acting on it is zero. This means that all the forces (or tensions, in this case) acting on the object must balance out. For the crate being pulled by the ropes, the sum of the x-components and the sum of the y-components of the tensions must equal zero, allowing us to find the equivalent single tension that would produce the same effect.

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Related Practice

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.

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Textbook Question

FIGURE P3.46 shows four electric charges located at the corners of a rectangle. Like charges, you will recall, repel each other while opposite charges attract. Charge B exerts a repulsive force (directly away from B) on charge A of 3.0 N. Charge C exerts an attractive force (directly toward C) on charge A of 6.0 N. Finally, charge D exerts an attractive force of 2.0 N on charge A. Assuming that forces are vectors, what are the magnitude and direction of the net force F_netexerted on charge A?

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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

The treasure map in FIGURE P3.41 gives the following directions to the buried treasure: 'Start at the old oak tree, walk due north for 500 paces, then due east for 100 paces. Dig.' But when you arrive, you find an angry dragon just north of the tree. To avoid the dragon, you set off along the yellow brick road at an angle 60° east of north. After walking 300 paces you see an opening through the woods. In which direction should you walk, as an angle west of north, and how far, to reach the treasure?

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Textbook Question

Four forces are exerted on the object shown in FIGURE $P 3.45$ . (Forces are measured in newtons, abbreviated $N$ .) The net force on the object is ${\vec{F}}_{net} = {\vec{F}}_{1} + {\vec{F}}_{2} + {\vec{F}}_{3} + {\vec{F}}_{4} = 4.0 \hat{i} N$ . What are (a) $F ⃗ sub 3$ and (b) $F ⃗ sub 4$ ? Give your answers in component form.

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