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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?

University Physics Volume 3
University Physics Volume 3
17th Edition
ISBN:9781938168185
Author:William Moebs, Jeff Sanny
Publisher:William Moebs, Jeff Sanny
Chapter7: Quantum Mechanics
Section: Chapter Questions
Problem 12CQ: Explain the difference between time-dependent and independent SchrÖdinger's equations.
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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?
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