Problem 4 A sphere of mass M and radius R, with uniform mass per unit volume, can roll without slipping along a plane inclined at an angle a with respect to a horizontal plane, as illustrated below. Gravity acts vertically, as usual. The sphere is initially at rest and then released, starting to roll down the inclined plane. (i) Determine the moment of inertia of the sphere with respect to an axis passing through its centre of mass. (ii) Determine the velocity of the centre of mass of the sphere after the sphere has travelled distance L along the plane. (iii) Consider now the case of a hollow sphere, where the mass is concentrated only on the external surface. The mass and the radius are M and R, respectively, as before. Determine the velocity of the centre of mass of this rigid body after it has travelled a distance L along the inclined plane. Which rigid body arrives with the largest velocity, the solid sphere or the hollow sphere? Hint: In principle, the translational and rotational motion of the sphere are unrelated. However, by saying that the sphere rolls down without slipping we are relating the modulus of the velocity of the centre of mass VCOM to the angular velocity w of the sphere about an instantaneous axis of rotation, as UCOм = Rw.
Problem 4 A sphere of mass M and radius R, with uniform mass per unit volume, can roll without slipping along a plane inclined at an angle a with respect to a horizontal plane, as illustrated below. Gravity acts vertically, as usual. The sphere is initially at rest and then released, starting to roll down the inclined plane. (i) Determine the moment of inertia of the sphere with respect to an axis passing through its centre of mass. (ii) Determine the velocity of the centre of mass of the sphere after the sphere has travelled distance L along the plane. (iii) Consider now the case of a hollow sphere, where the mass is concentrated only on the external surface. The mass and the radius are M and R, respectively, as before. Determine the velocity of the centre of mass of this rigid body after it has travelled a distance L along the inclined plane. Which rigid body arrives with the largest velocity, the solid sphere or the hollow sphere? Hint: In principle, the translational and rotational motion of the sphere are unrelated. However, by saying that the sphere rolls down without slipping we are relating the modulus of the velocity of the centre of mass VCOM to the angular velocity w of the sphere about an instantaneous axis of rotation, as UCOм = Rw.
Physics for Scientists and Engineers, Technology Update (No access codes included)
9th Edition
ISBN:9781305116399
Author:Raymond A. Serway, John W. Jewett
Publisher:Raymond A. Serway, John W. Jewett
Chapter11: Angular Momentum
Section: Chapter Questions
Problem 11.7OQ: Two ponies of equal mass are initially at diametrically opposite points on the rim of a large...
Related questions
Question
Transcribed Image Text:Problem 4
A sphere of mass M and radius R, with uniform mass per unit volume, can roll without slipping along a
plane inclined at an angle a with respect to a horizontal plane, as illustrated below. Gravity acts vertically,
as usual. The sphere is initially at rest and then released, starting to roll down the inclined plane.
(i) Determine the moment of inertia of the sphere with respect to an axis passing through its centre of
mass.
(ii) Determine the velocity of the centre of mass of the sphere after the sphere has travelled distance L
along the plane.
(iii) Consider now the case of a hollow sphere, where the mass is concentrated only on the external surface.
The mass and the radius are M and R, respectively, as before. Determine the velocity of the centre of
mass of this rigid body after it has travelled a distance L along the inclined plane. Which rigid body
arrives with the largest velocity, the solid sphere or the hollow sphere?
Hint: In principle, the translational and rotational motion of the sphere are unrelated. However, by saying
that the sphere rolls down without slipping we are relating the modulus of the velocity of the centre of mass
VCOM to the angular velocity w of the sphere about an instantaneous axis of rotation, as VCOM = Rw.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 5 steps with 5 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Recommended textbooks for you
Physics for Scientists and Engineers, Technology …
Physics
ISBN:
9781305116399
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
College Physics
Physics
ISBN:
9781285737027
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
Physics for Scientists and Engineers, Technology …
Physics
ISBN:
9781305116399
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
College Physics
Physics
ISBN:
9781285737027
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
Principles of Physics: A Calculus-Based Text
Physics
ISBN:
9781133104261
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Physics for Scientists and Engineers with Modern …
Physics
ISBN:
9781337553292
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Physics for Scientists and Engineers: Foundations…
Physics
ISBN:
9781133939146
Author:
Katz, Debora M.
Publisher:
Cengage Learning