Lectures 20-22 Central Forces, Kepler's Laws etc Flashcards
study for the final!!! :)
What are the three possibilities for the effective radial force?
Repulsive, strongly attractive, and moderately attractive
What is the condition for the effective force to be repulsive?
F(r) = -dU/dr > 0, making the effective force positive
What is the L^2/mr^3 term called?
The centrifugal force
Why is the centrifugal force also known as the angular momentum barrier?
Because it prevents objects from getting too close to the origin
The centrifugal force is always positive. Does it correspond to an attractive or a repulsive force?
Repulsive
What is the condition for a strongly attractive force?
F(r) < 0 and it’s strong enough to overcome the centrifugal force term everywhere
When the effective force is repulsive, what can you say about the trajectory?
It’s unbounded
How do we describe a force that leads to orbits around the origin?
Moderately attractive force
Circular orbits are special cases of which of the three forces?
Moderately attractive
How do circular orbits come about?
The centrifugal and the central forces are exactly balanced, so the effective force is zero
Suppose we have a circular orbit due to an attractive potential. What will happen if we perturb the orbit?
There will be SHM in r
What is the frequency of radial oscillations near the minimum of Ueff?
sqrt(U’‘eff (at r=r0) /m)
How can you tell if the orbit is closed when you have radial oscillations?
If the orbital period is commensurate with the radial oscillation period
Kepler’s first law
The orbit of a planet is an ellipse with the sun at one of the two foci
Kepler’s second law
A line segment joining a planet and the sun sweeps out equal areas during equal intervals of time
Kepler’s third law
The square of a planet’s orbital period is proportional to the cube of the length of the semi-major axis of its orbit
True or false: kepler’s second law, which is about sweeping out equal areas in equal times, applies to any central force
True
How do you derive Kepler’s second law?
Using conservation of angular momentum
To show that Kepler’s first law is true, we are looking for the shape of the trajectories, so we are only interested in spatial information. so are we looking for r(theta), r(t), or theta(t)?
r(theta) because we want to eliminate time from all of our equations
