Ch. 07 - Work and Energy

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 07 - Work and Energy

Problem 39

Chapter 7, Problem 39

The net force exerted on a particle acts in the positive x direction. Its magnitude increases linearly from zero at x = 0, to 380 N at x = 3.0m. It remains constant at 380 N from x = 3.0m to x = 7.0m, and then decreases linearly to zero at x = 12.0m. Determine the work done to move the particle from x = 0 to x = 12.0m graphically, by determining the area under the Fₓ versus x graph.

Verified step by step guidance

Step 1: Understand the problem. The work done on a particle by a force is equal to the area under the force (Fₓ) versus position (x) graph. The force varies in three distinct regions: (1) linearly increasing from 0 to 380 N over x = 0 to x = 3.0 m, (2) constant at 380 N from x = 3.0 m to x = 7.0 m, and (3) linearly decreasing from 380 N to 0 over x = 7.0 m to x = 12.0 m.

Step 2: Divide the graph into three regions and calculate the area for each. For the first region (x = 0 to x = 3.0 m), the force increases linearly, forming a triangle. The area of a triangle is given by $ A = \frac{1}{2} \cdot \text{base} \cdot \text{height} $. Here, the base is 3.0 m, and the height is 380 N.

Step 3: For the second region (x = 3.0 m to x = 7.0 m), the force is constant at 380 N, forming a rectangle. The area of a rectangle is given by $ A = \text{base} \cdot \text{height} $. Here, the base is $ 7.0 \text{ m} - 3.0 \text{ m} = 4.0 \text{ m} $, and the height is 380 N.

Step 4: For the third region (x = 7.0 m to x = 12.0 m), the force decreases linearly from 380 N to 0, forming another triangle. The area of this triangle is given by $ A = \frac{1}{2} \cdot \text{base} \cdot \text{height} $. Here, the base is $ 12.0 \text{ m} - 7.0 \text{ m} = 5.0 \text{ m} $, and the height is 380 N.

Step 5: Add the areas of all three regions to find the total work done. The total work is the sum of the areas of the triangle from Step 2, the rectangle from Step 3, and the triangle from Step 4. This gives the total work done to move the particle from $ x = 0 $ to $ x = 12.0 \text{ m} $.

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Key Concepts

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

Net Force

Net force is the vector sum of all forces acting on an object. It determines the object's acceleration according to Newton's second law, F = ma, where F is the net force, m is mass, and a is acceleration. Understanding how net force varies with position is crucial for analyzing motion and calculating work done.

Work Done

Work done on an object is defined as the product of the force applied to the object and the distance over which that force is applied, in the direction of the force. Mathematically, it is expressed as W = F * d * cos(θ), where θ is the angle between the force and the direction of motion. In this scenario, work can be calculated as the area under the force versus position graph.

Area Under the Curve

The area under a graph of force versus position represents the work done on an object as it moves through that position range. For linear segments, this area can be calculated using geometric shapes such as rectangles and triangles. Understanding how to compute these areas is essential for determining the total work done in the given scenario.

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Calculating Work As Area Under F-x Graphs

Related Practice

Textbook Question

A 2800-kg space vehicle, initially at rest, falls vertically from a height of 2900 km above the Earth’s surface. Determine how much work is done by the force of gravity in bringing the vehicle to the Earth’s surface.

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Textbook Question

In a certain library the first shelf is 15.0 cm off the ground, and the remaining four shelves are each spaced 38.0 cm above the previous one. If the average book has a mass of 1.25 kg with a height of 22.0 cm, and an average shelf holds 28 books (standing vertically), how much work is required to fill all the shelves, assuming the books are all laying flat on the floor to start?

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Textbook Question

A grocery cart with mass of 16 kg is pushed at constant speed up a 12° ramp by a force F_P which acts at an angle of 17° below the horizontal. Find the work done by each of the forces (m $\vec{g}$ , ${\vec{F_{N}}}_{}$ , $\vec{F_{N}}$ ) on the cart if the ramp is 7.5 m long.

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Textbook Question

Consider a force F₁ = $A / \sqrt{x}$ which acts on an object during its journey along the x axis from x = 0.0 to x = 1.0m, where A = 3.0 Nm¹⸍². Show that during this journey, even though F₁ is infinite at x = 0.0, the work W done on the object by this force is finite, and determine W.

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Textbook Question

A 3.0-m-long steel chain is stretched out along the top level of a horizontal scaffold at a construction site, in such a way that 2.0 m of the chain remains on the top level and 1.0 m hangs vertically, Fig. 7–27. At this point, the force on the hanging segment is sufficient to pull the entire chain over the edge. Once the chain is moving, the kinetic friction is so small that it can be neglected. How much work is performed on the chain by the force of gravity as the chain falls from the point where 2.0 m remains on the scaffold to the point where the entire chain has left the scaffold? (Assume that the chain has a linear weight density of 24 N/m.)

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Textbook Question

If the hill in Example 7–2 (Fig. 7–4) was not an even slope but rather an irregular curve as in Fig. 7–23, show that the same result would be obtained as in Example 7–2: namely, that the work done by gravity depends only on the height of the hill and not on its shape or the path taken.

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