- #1
catkin
- 218
- 0
[SOLVED] 3 explanations of comet velocity. OK?
This is from Advanced Physics by Adams and Allday, spread 3.31.
Explain why comets in very eccentric orbits move very slowly far from the Sun and very fast close to it. Give three explanations, one in terms of energy, one in terms of angular momentum, and one in terms of forces acting on the comet around its orbit.
Rotational kinetic energy: [itex]R.K.E. = 0.5 I {\omega}^{2}[/itex]
Moment of inertia of point mass: [itex]I = m r^{2}[/itex]
Angular momentum: [itex]L = I \omega[/itex]
Centripetal acceleration: [itex]a = r {\omega}^{2}[/itex]
Gravitational force: [itex]F = G m_1 m_2 / r^{2}[/itex]
I would be confident this was right if I could understand and accept the implication.
In each case I answered the question by showing that [itex]r^{a} {\omega}^{b}[/itex] is constant so [itex]\omega[/itex] must become small as r becomes big and vice versa.
For energy, conservation of energy yields [itex]0.5 m r^{2} {\omega}^{2}[/itex] is constant so [itex]r^{2} {\omega}^{2}[/itex] is constant.
For angular momentum, conservation of angular momentum yields [itex] m r^{2} {\omega}[/itex] is constant so [itex]r^{2} \omega[/itex] is constant.
For forces, centripetal acceleration and gravity yield [itex]G m_1 m_2 / r^{2} = m r {\omega}^{2}[/itex] so [itex]r^3 {\omega}^{2}[/itex] is constant.
Is it correct that [itex]r^{2} {\omega}^{2}[/itex], [itex]r^{2} {\omega}[/itex], and [itex]r^3 {\omega}^{2}[/itex] are all constant? It feels unlikely -- very constrained. If it is true then what is its significance, physically? Am I being naieve -- are these simply properties of an ellipse?
Homework Statement
This is from Advanced Physics by Adams and Allday, spread 3.31.
Explain why comets in very eccentric orbits move very slowly far from the Sun and very fast close to it. Give three explanations, one in terms of energy, one in terms of angular momentum, and one in terms of forces acting on the comet around its orbit.
Homework Equations
Rotational kinetic energy: [itex]R.K.E. = 0.5 I {\omega}^{2}[/itex]
Moment of inertia of point mass: [itex]I = m r^{2}[/itex]
Angular momentum: [itex]L = I \omega[/itex]
Centripetal acceleration: [itex]a = r {\omega}^{2}[/itex]
Gravitational force: [itex]F = G m_1 m_2 / r^{2}[/itex]
The Attempt at a Solution
I would be confident this was right if I could understand and accept the implication.
In each case I answered the question by showing that [itex]r^{a} {\omega}^{b}[/itex] is constant so [itex]\omega[/itex] must become small as r becomes big and vice versa.
For energy, conservation of energy yields [itex]0.5 m r^{2} {\omega}^{2}[/itex] is constant so [itex]r^{2} {\omega}^{2}[/itex] is constant.
For angular momentum, conservation of angular momentum yields [itex] m r^{2} {\omega}[/itex] is constant so [itex]r^{2} \omega[/itex] is constant.
For forces, centripetal acceleration and gravity yield [itex]G m_1 m_2 / r^{2} = m r {\omega}^{2}[/itex] so [itex]r^3 {\omega}^{2}[/itex] is constant.
Is it correct that [itex]r^{2} {\omega}^{2}[/itex], [itex]r^{2} {\omega}[/itex], and [itex]r^3 {\omega}^{2}[/itex] are all constant? It feels unlikely -- very constrained. If it is true then what is its significance, physically? Am I being naieve -- are these simply properties of an ellipse?