Ch 10: Interactions and Potential Energy

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 10: Interactions and Potential Energy

Problem 36b

Chapter 10, Problem 36b

A particle moves from A to D in FIGURE EX10.36 while experiencing force F = (6i + 8j) N. How much work does the force do if the particle follows path ACD. Is this a conservative force? Explain.

Verified step by step guidance

Step 1: Understand the problem. The particle moves along path ACD under the influence of a force F = (6i + 8j) N. The goal is to calculate the work done by the force along this path and determine if the force is conservative.

Step 2: Recall the formula for work done by a force: W = ∫ F · dr, where F is the force vector and dr is the displacement vector. For a conservative force, the work done depends only on the initial and final positions, not the path taken.

Step 3: Break the path ACD into segments AC and CD. For segment AC, calculate the displacement vector by subtracting the coordinates of A from C. For segment CD, calculate the displacement vector by subtracting the coordinates of C from D.

Step 4: Compute the dot product F · dr for each segment. For segment AC, use the displacement vector of AC and the force vector F. Similarly, for segment CD, use the displacement vector of CD and the force vector F.

Step 5: Add the work done along each segment to find the total work done along path ACD. To determine if the force is conservative, check if the work done is the same for all paths between the same initial and final points.

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Key Concepts

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

Work Done by a Force

Work is defined as the product of the force applied to an object and the displacement of that object in the direction of the force. Mathematically, it is expressed as W = F · d, where W is work, F is the force vector, and d is the displacement vector. In this case, the work done by the force F as the particle moves along the path ACD can be calculated by integrating the force along the specified path.

Path Dependence of Work

The work done by a force can depend on the path taken by the object when moving from one point to another. For non-conservative forces, the work done can vary based on the specific trajectory. In contrast, conservative forces, such as gravitational force, yield the same work regardless of the path taken, depending only on the initial and final positions.

Conservative Forces

A force is considered conservative if the work done by the force on an object moving between two points is independent of the path taken. Examples include gravitational and electrostatic forces. To determine if the force F in the question is conservative, one can check if the work done along different paths (like ACD and others) yields the same result, or if the force can be derived from a potential energy function.

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Related Practice

Textbook Question

A system in which only one particle moves has the potential energy shown in FIGURE EX10.31. What is the x-component of the force on the particle at x = 5, 15, and 25 cm?

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Textbook Question

A particle moving along the y-axis is in a system with potential energy U = 4y³ J, where y is in m. What is the y-component of the force on the particle at y = 0 m, 1 m, and 2 m?

Textbook Question

A particle moving along the x-axis is in a system that has potential energy U = x³ - 3x J, where x is in m. For each, is it a point of stable or unstable equilibrium?

Textbook Question

A 50 g mass is attached to a light, rigid, 75-cm-long rod. The other end of the rod is pivoted so that the mass can rotate in a vertical circle. What speed does the mass need at the bottom of the circle to barely make it over the top of the circle?

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Textbook Question

How much work is done by the environment in the process shown in FIGURE EX10.39? Is energy transferred from the environment to the system or from the system to the environment?

Textbook Question

A cable with 20.0 N of tension pulls straight up on a 1.50 kg block that is initially at rest. What is the block's speed after being lifted 2.00 m? Solve this problem using work and energy.