Homework Help is Here – Start Your Trial Now!
Science
Physics
Recall that gravitational potential energy is GMm U(r) = (1) In this homework, we'll derive some important results on gravity in classical mechanics. They are especially easier to find using the scalar potential energy, rather than the vector force. The figure below depicts a very thin spherical shell of radius R and mass M, uniformly distributed across the shell. At some point a distance r away from the centre of the shell, we have a smaller point mass m: Our shell can be constructed as the collection of an infinite number of very thin circular strips/rings, such as the shaded strip depicted above. (a) What is the infinitesimal mass dM contained inside each strip? (b) What is the infinitesimal potential energy dU provided by each strip onto our mass m? (c) We can find the total U provided by the shell onto our mass m by integrating over dU in (b). What is the total potential energy U? (d) Now consider when mass m is inside the shell, as depicted below: Raind Using a similar process as in (a) – (c), what is the total U provided by the shell onto a mass m inside the shell? (e) Draw a U(r) vs. r diagramme of our results in (c) and (d). Based on your diagramme, describe what the gravitational force on mass m will be, inside and outside the shell. BONUS: We can find similar results for a filled sphere of uniform mass distribution, by taking it as the sum of an infinite number of thin spherical shells. What is then the potential energy U(r) inside and outside the sphere? What does this imply for the motion of a particle dropped, from rest at the surface, through a hole that goes through the centre of the sphere and reaches the other side?

Recall that gravitational potential energy is GMm U(r) = (1) In this homework, we'll derive some important results on gravity in classical mechanics. They are especially easier to find using the scalar potential energy, rather than the vector force. The figure below depicts a very thin spherical shell of radius R and mass M, uniformly distributed across the shell. At some point a distance r away from the centre of the shell, we have a smaller point mass m: Our shell can be constructed as the collection of an infinite number of very thin circular strips/rings, such as the shaded strip depicted above. (a) What is the infinitesimal mass dM contained inside each strip? (b) What is the infinitesimal potential energy dU provided by each strip onto our mass m? (c) We can find the total U provided by the shell onto our mass m by integrating over dU in (b). What is the total potential energy U? (d) Now consider when mass m is inside the shell, as depicted below: Raind Using a similar process as in (a) – (c), what is the total U provided by the shell onto a mass m inside the shell? (e) Draw a U(r) vs. r diagramme of our results in (c) and (d). Based on your diagramme, describe what the gravitational force on mass m will be, inside and outside the shell. BONUS: We can find similar results for a filled sphere of uniform mass distribution, by taking it as the sum of an infinite number of thin spherical shells. What is then the potential energy U(r) inside and outside the sphere? What does this imply for the motion of a particle dropped, from rest at the surface, through a hole that goes through the centre of the sphere and reaches the other side?

Classical Dynamics of Particles and Systems
Classical Dynamics of Particles and Systems
5th Edition
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Stephen T. Thornton, Jerry B. Marion
Chapter7: Hamilton's Principle-lagrangian And Hamiltonian Dynamics
Section: Chapter Questions
Problem 7.36P
See similar textbooks
icon
Related questions
Question

NEed hellp only with the red mark question
it's a practice quesiton

Recall that gravitational potential energy is GMm U(r) = (1) In this homework, we'll derive some important results on gravity in classical mechanics. They are especially easier to find using the scalar potential energy, rather than the vector force. The figure below depicts a very thin spherical shell of radius R and mass M, uniformly distributed across the shell. At some point a distance r away from the centre of the shell, we have a smaller point mass m: Rd0 de Rsine Our shell can be constructed as the collection of an infinite number of very thin circular strips/rings, such as the shaded strip depicted above. (a) What is the infinitesimal mass dM contained inside each strip? (b) What is the infinitesimal potential energy dU provided by each strip onto our mass m? (c) We can find the total U provided by the shell onto our mass m by integrating over dU in (b). What is the total potential energy U? (d) Now consider when mass m is inside the shell, as depicted below: Raino Using a similar process as in (a) – (c), what is the total U provided by the shell onto a mass m inside the shell? (e) Draw a U(r) vs. r diagramme of our results in (c) and (d). Based on your diagramme, describe what the gravitational force on mass m will be, inside and outside the shell. BONUS: We can find similar results for a filled sphere of uniform mass distribution, by taking it as the sum of an infinite number of thin spherical shells. What is then the potential energy U(r) inside and outside the sphere? What does this imply for the motion of a particle dropped, from rest at the surface, through a hole that goes through the centre of the sphere and reaches the other side?
Transcribed Image Text:Recall that gravitational potential energy is GMm U(r) = (1) In this homework, we'll derive some important results on gravity in classical mechanics. They are especially easier to find using the scalar potential energy, rather than the vector force. The figure below depicts a very thin spherical shell of radius R and mass M, uniformly distributed across the shell. At some point a distance r away from the centre of the shell, we have a smaller point mass m: Rd0 de Rsine Our shell can be constructed as the collection of an infinite number of very thin circular strips/rings, such as the shaded strip depicted above. (a) What is the infinitesimal mass dM contained inside each strip? (b) What is the infinitesimal potential energy dU provided by each strip onto our mass m? (c) We can find the total U provided by the shell onto our mass m by integrating over dU in (b). What is the total potential energy U? (d) Now consider when mass m is inside the shell, as depicted below: Raino Using a similar process as in (a) – (c), what is the total U provided by the shell onto a mass m inside the shell? (e) Draw a U(r) vs. r diagramme of our results in (c) and (d). Based on your diagramme, describe what the gravitational force on mass m will be, inside and outside the shell. BONUS: We can find similar results for a filled sphere of uniform mass distribution, by taking it as the sum of an infinite number of thin spherical shells. What is then the potential energy U(r) inside and outside the sphere? What does this imply for the motion of a particle dropped, from rest at the surface, through a hole that goes through the centre of the sphere and reaches the other side?
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Space-time
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.
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Classical Dynamics of Particles and Systems
Classical Dynamics of Particles and Systems
Physics
ISBN:
9780534408961
Author:
Stephen T. Thornton, Jerry B. Marion
Publisher:
Cengage Learning
Modern Physics
Modern Physics
Physics
ISBN:
9781111794378
Author:
Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher:
Cengage Learning
University Physics Volume 3
University Physics Volume 3
Physics
ISBN:
9781938168185
Author:
William Moebs, Jeff Sanny
Publisher:
OpenStax
SEE MORE TEXTBOOKS