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 25b

Chapter 10, Problem 25b

FIGURE EX10.25 is the potential-energy diagram for a 20 g particle that is released from rest at x = 1.0 m. What is the particle's maximum speed? At what position does it have this speed?

Verified step by step guidance

Step 1: Understand the problem. The particle is released from rest at x = 1.0 m, and we need to determine its maximum speed and the position where this occurs. The potential-energy diagram shows U(x) (potential energy) as a function of position x.

Step 2: Recall the relationship between kinetic energy (K), potential energy (U), and total mechanical energy (E). The total mechanical energy is conserved and given by E = K + U. At the release point (x = 1.0 m), the particle has no kinetic energy (K = 0), so E = U(x = 1.0 m).

Step 3: Determine the total mechanical energy E. From the graph, at x = 1.0 m, U = 2 J. Therefore, E = 2 J. This total energy remains constant throughout the motion.

Step 4: Find the position where the particle's kinetic energy is maximum. Kinetic energy is maximum when potential energy is minimum, because K = E - U. From the graph, the minimum potential energy occurs at x = 8.0 m, where U = 0 J.

Step 5: Use the relationship between kinetic energy and speed. Kinetic energy is given by K = (1/2)mv², where m is the mass of the particle and v is its speed. At x = 8.0 m, K = E = 2 J. Solve for v using the particle's mass (m = 20 g = 0.020 kg).

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Key Concepts

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

Potential Energy

Potential energy (U) is the energy stored in an object due to its position in a force field, such as gravitational or elastic fields. In the context of the diagram, it represents the energy of the 20 g particle at various positions along the x-axis. The particle's potential energy decreases as it moves to lower positions, converting into kinetic energy as it accelerates.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. For the particle in the potential energy diagram, the sum of its potential energy and kinetic energy at any point is constant. As the particle moves from its initial position, potential energy is converted into kinetic energy, allowing us to determine its maximum speed.

Kinetic Energy and Speed

Kinetic energy (KE) is the energy of an object due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. The maximum speed of the particle occurs when its potential energy is at a minimum, and all the energy has been converted to kinetic energy. By analyzing the potential energy diagram, we can find the position where this maximum speed occurs.

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FIGURE EX10.28 shows the potential-energy diagram for a 500 g particle as it moves along the x-axis. Suppose the particle's mechanical energy is 12 J. Where are the particle's turning points?

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In FIGURE EX10.27, what is the maximum speed of a 2.0 g particle that oscillates between x = 2.0 mm and x = 8.0 mm

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In FIGURE EX10.26, What minimum speed does a 100 g particle need at point B to reach point A?

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As a 15,000 kg jet plane lands on an aircraft carrier, its tail hook snags a cable to slow it down. The cable is attached to a spring with spring constant 60,000 N/m. If the spring stretches 30 m to stop the plane, what was the plane's landing speed?

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FIGURE EX10.24 is the potential-energy diagram for a 500 g particle that is released from rest at A. What are the particle's speeds at B, C, and D?

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