Find the period of the space shuttle in alternate universe

[hanburger]

Find the period T of the space shuttle

Homework Statement

(See image of problem statement for nice layout; the questions are stated below)
A space shuttle of mass m is in a circular orbit of radius r around a planet of mass M in an alternate universe.

In this alternate universe the laws of physics are exactly the same as in our universe, except the force of gravity between these two objects has magnitude

F=HMm/r³

where H is the alternate universe gravitational constant. The associated potential energy function is

U=−(1/2)HMm/r².

(a) Find the period T of the space shuttle's orbit.
(b) The astronauts want to launch a long-range probe from their shuttle. What is the minimum initial speed needed by this probe so that its trajectory will never return near the planet? (The probe does not have its own engines.)

Homework Equations

H is gravitational constant in this universe

T² = 4π²α³/HM
(where α is the semi-major axis)

F = Gm₁m₂/r² except on this planet it is F=HMm/r³

in this universe--> U=−(1/2)HMm/r²
E = KE - U = 0.5mv² − (1/2)HMm/r²

v_escape = sqrt(2M_planetG/r_planet)
not sure how escape velocity would change in this universe

The Attempt at a Solution

This question threw me off, and I'm not sure how to proceed with it. My first guess for T was just to
plug in the gravitational constant for this universe, H, into the orbital period relationship like so
T² = 4π²α³/HM
But that isn't right.

Will appreciate any guidance on getting started on this problem! I'm struggling a bit with the Gravity/Orbits section I'm on and will appreciate learning the correct approach.

 

[collinsmark]

Homework Statement

(See image of problem statement for nice layout; the questions are stated below)
A space shuttle of mass m is in a circular orbit of radius r around a planet of mass M in an alternate universe.

In this alternate universe the laws of physics are exactly the same as in our universe, except the force of gravity between these two objects has magnitude

F=HMm/r³

where H is the alternate universe gravitational constant. The associated potential energy function is

U=−(1/2)HMm/r².

(a) Find the period T of the space shuttle's orbit.
(b) The astronauts want to launch a long-range probe from their shuttle. What is the minimum initial speed needed by this probe so that its trajectory will never return near the planet? (The probe does not have its own engines.)

Homework Equations

H is gravitational constant in this universe

T² = 4π²α³/HM
(where α is the semi-major axis)

F = Gm₁m₂/r² except on this planet it is F=HMm/r³

in this universe--> U=−(1/2)HMm/r²
E = KE - U = 0.5mv² − (1/2)HMm/r²

v_escape = sqrt(2M_planetG/r_planet)
not sure how escape velocity would change in this universe

The Attempt at a Solution

This question threw me off, and I'm not sure how to proceed with it. My first guess for T was just to
plug in the gravitational constant for this universe, H, into the orbital period relationship like so
T² = 4π²α³/HM
But that isn't right.

Will appreciate any guidance on getting started on this problem! I'm struggling a bit with the Gravity/Orbits section I'm on and will appreciate learning the correct approach.

This is a weird question that involves some weird presumptions. But I'll give it a shot anyway.

Since, "In this alternate universe the laws of physics are exactly the same as in our universe, except the force of gravity... ," I think we can conclude that the centripetal force (and centripetal acceleration) and kinetic energy formulas are the same as in our universe.

Part (a):
What's the centripetal force formula? That force must balance out the force of gravity to produce a [STRIKE]stable[/STRIKE] circular orbit.

(Hint: the centripetal force must equal the gravitational force.)


Part (b):
What's the formula for kinetic energy?

(Hint: When the probe starts out, all its mechanical energy is kinetic energy. When it just barely manages to escape the planet, all that energy is converted to potential energy. [Edit: that is, potential energy relative to the orbital height of the shuttle.])

[Edit: (Special bonus hint: What is the difference in the potential energy of the probe between r = ∞ and r = orbital height of the shuttle?)]

[Edit: (Extra special bonus hint: You had the right idea in your relevant equations section when you said, E = KE - U. Keep in mind though that in this equation, the potential energy U is with respect to r = ∞. In other words, the potential energy is zero when r = ∞. When after the probe just barely escapes the planet, and r = ∞, and it's velocity has slowed to a standstill, what does that tell you about its total energy E? Now apply conservation of energy [i.e. E in one situation is equal to E in any other] and apply it to when r is the orbital height of the shuttle. Solve for v.)]