Skip to main content
Ch 12: Rotation of a Rigid Body
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 12: Rotation of a Rigid BodyProblema 36
Capitolo 12, Problema 36

A solid sphere of radius R is placed at a height of 30 cm on a 15° slope. It is released and rolls, without slipping, to the bottom. From what height should a circular hoop of radius R be released on the same slope in order to equal the sphere's speed at the bottom?

Guida verificata passo dopo passo
1
Step 1: Begin by analyzing the energy conservation principle for both the solid sphere and the circular hoop. The total mechanical energy at the top of the slope is purely potential energy, and at the bottom, it is converted into both translational kinetic energy and rotational kinetic energy.
Step 2: Write the energy conservation equation for the solid sphere. The potential energy at the top is \( mgh \), and the kinetic energy at the bottom is \( \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \), where \( I \) is the moment of inertia of the sphere and \( \omega \) is the angular velocity. For a solid sphere, \( I = \frac{2}{5}mR^2 \) and \( \omega = \frac{v}{R} \). Substitute these values into the equation.
Step 3: Write the energy conservation equation for the circular hoop. The potential energy at the top is \( mgh \), and the kinetic energy at the bottom is \( \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \). For a hoop, \( I = mR^2 \) and \( \omega = \frac{v}{R} \). Substitute these values into the equation.
Step 4: Compare the final speeds of the sphere and the hoop at the bottom of the slope. The speed of the sphere is determined by its energy conservation equation, and the speed of the hoop is determined by its own energy conservation equation. Set the speeds equal to each other to find the height \( h \) from which the hoop must be released.
Step 5: Solve the resulting equation for \( h \) in terms of the given height of the sphere (30 cm). This involves algebraic manipulation to isolate \( h \) for the hoop, ensuring that the speeds match at the bottom of the slope.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
18m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In this scenario, the potential energy of the objects at the top of the slope is converted into kinetic energy as they roll down. For both the solid sphere and the circular hoop, the initial gravitational potential energy must equal the sum of translational and rotational kinetic energy at the bottom.
Video consigliato:
Percorso guidato
06:24
Conservation Of Mechanical Energy

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotation. Different shapes have different moments of inertia, which affects how they roll down a slope. For a solid sphere, the moment of inertia is (2/5)MR², while for a circular hoop, it is MR². This difference influences the distribution of energy between translational and rotational motion.
Video consigliato:
Percorso guidato
11:47
Intro to Moment of Inertia

Rolling Without Slipping

Rolling without slipping occurs when an object rolls on a surface without sliding, meaning the point of contact with the surface is momentarily at rest. This condition relates the linear speed of the center of mass to the angular speed of the object. For both the sphere and the hoop, this relationship is crucial for determining their speeds at the bottom of the slope, as it affects how energy is distributed between translational and rotational forms.
Video consigliato:
Percorso guidato
08:47
Rolling Motion (Free Wheels)
Pratica correlata
Domanda del libro di testo

An 8.0-cm-diameter, 400 g solid sphere is released from rest at the top of a 2.1-m-long, 25 incline. It rolls, without slipping, to the bottom. What fraction of its kinetic energy is rotational?

2771
views
Domanda del libro di testo

A car tire is 60 cm in diameter. The car is traveling at a speed of 20 m/s. What is the speed of a point at the bottom edge of the tire?

1530
views
Domanda del libro di testo

Vector A ＝ 3î＋ĵ and vector B＝ 3î - 2ĵ ＋ 2k. What is the cross product A ✕ B?

1551
views
Domanda del libro di testo

An 8.0-cm-diameter, 400 g solid sphere is released from rest at the top of a 2.1-m-long, 25 incline. It rolls, without slipping, to the bottom. What is the sphere’s angular velocity at the bottom of the incline?

1310
views
Domanda del libro di testo

Evaluate the cross products and A ✕ B and C ✕ D.

1587
views
Domanda del libro di testo

Force F⃗=−10j^N\(\vec{F}\) = -10\(\hat{j}\) \, \(\text{N}\) is exerted on a particle at r⃗=(5i^+5j^)m\(\vec{r}\) = (5\(\hat{i}\) + 5\(\hat{j}\)) \, \(\text{m}\) . What is the torque on the particle about the origin?

2389
views