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 56a

Chapter 10, Problem 56a

A sled starts from rest at the top of the frictionless, hemispherical, snow-covered hill shown in FIGURE P10.56. a. Find an expression for the sled's speed when it is at angle ϕ .

Verified step by step guidance

Step 1: Begin by identifying the key principles involved in the problem. Since the sled starts from rest and the hill is frictionless, we can use the conservation of mechanical energy to solve the problem. The total mechanical energy (potential energy + kinetic energy) remains constant throughout the motion.

Step 2: Write the expression for the total mechanical energy at the top of the hill. At the top, the sled has only gravitational potential energy, which is given by $ U = m g h $, where $ h $ is the height of the hill. Since the hill is hemispherical, $ h = R $, where $ R $ is the radius of the hemisphere.

Step 3: Write the expression for the total mechanical energy at the angle $ \phi $. At this point, the sled has both kinetic energy $ K = \frac{1}{2} m v^2 $ and gravitational potential energy $ U = m g h $. The height $ h $ at angle $ \phi $ can be expressed as $ h = R \cos \phi $.

Step 4: Apply the conservation of mechanical energy principle. Equate the total energy at the top of the hill to the total energy at angle $ \phi $: $ m g R = \frac{1}{2} m v^2 + m g R \cos \phi $.

Step 5: Solve for the sled's speed $ v $ at angle $ \phi $. Rearrange the equation to isolate $ v $: $ v = \sqrt{2 g R (1 - \cos \phi)} $. This is the expression for the sled's speed at angle $ \phi $.

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Key Concepts

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

Conservation of Energy

The principle of conservation of energy states that the total mechanical energy of an isolated system remains constant if only conservative forces are acting. In this scenario, as the sled descends the hill, its potential energy is converted into kinetic energy, allowing us to relate the sled's height at angle ϕ to its speed.

Kinetic and Potential Energy

Kinetic energy (KE) is the energy of an object due to its motion, given by the formula KE = 1/2 mv², where m is mass and v is velocity. Potential energy (PE), particularly gravitational potential energy, is the energy stored due to an object's position in a gravitational field, calculated as PE = mgh, where h is height. Understanding these forms of energy is crucial for analyzing the sled's motion.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path and is directed towards the center of the circle. In this problem, as the sled moves along the hemispherical hill, the gravitational force provides the necessary centripetal force to maintain its circular motion at angle ϕ, which is essential for determining the sled's speed.

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Textbook Question

A 1.0 kg mass that can move along the x-axis experiences the potential energy U = (x²−x) J, where x is in m. The mass has velocity v_x= 3.0 m/s at position x = 1.0 m. At what position has it slowed to 1.0 m/s?

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Textbook Question

A system has potential energy $U (x) = (10 J) [1 - \sin ((3.14 rad/m) x)]$ as a particle moves over the range $0 m is less than or equal to x is less than or equal to 3 m$ . For each, is it a point of stable or unstable equilibrium?

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Textbook Question

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Textbook Question

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Textbook Question

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