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Ch 10: Interactions and Potential Energy
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 10: Interactions and Potential EnergyProblema 56a
Capitolo 10, Problema 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 .

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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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Concetti chiave

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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.
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Conservation Of Mechanical Energy

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.
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Gravitational Potential Energy

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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Intro to Centripetal Forces
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