Question

Problem III: When Swiss people heard the rumor that the moon was made of cheese, they of course got the idea that it may also contain holes. Let's assume that the moon does indeed contain a giant hole, and then consider how someone might be able to detect this hole via measuring the difference in gravitational acceleration at the moon's surface, near and far from the hole: Suppose the moon is an otherwise uniform density sphere which has an empty spherical cavity located halfway between the center and the surface, as shown below in the cross-sectional sketch below. Given a mass density, $\rho$, and the relative radii, $R_{moon} = 4 \times R_{cavity}$, what is the ratio of the acceleration due to the moon's gravity at the moon's surface nearest to (point A) and furthest from (point B) this hole? In terms of the parameters given above (and also Newton's gravitational constant), what are the escape velocities at these two positions at the moon surface, nearest to and furtherst from the hole? Note: here, please disregard the gravitational pulls of the earth and the sun. B $R_{moon}$ A $4 \times R_{cavity} = R_{moon}$ Ratio of accelerations ($g_{@A} / g_{@B}$)? Escape velocity at point A ? Escape velocity at point B?

          Problem III:
When Swiss people heard the rumor that the moon was
made of cheese, they of course got the idea that it may also
contain holes. Let's assume that the moon does indeed
contain a giant hole, and then consider how someone might
be able to detect this hole via measuring the difference in
gravitational acceleration at the moon's surface, near and far
from the hole:
Suppose the moon is an otherwise uniform density sphere
which has an empty spherical cavity located halfway
between the center and the surface, as shown below in the
cross-sectional sketch below. Given a mass density, $\rho$, and
the relative radii, $R_{moon} = 4 \times R_{cavity}$, what is the ratio of the
acceleration due to the moon's gravity at the moon's surface
nearest to (point A) and furthest from (point B) this hole?
In terms of the parameters given above (and also Newton's
gravitational constant), what are the escape velocities at
these two positions at the moon surface, nearest to and
furtherst from the hole?
Note: here, please disregard the gravitational pulls
of the earth and the sun.
B
$R_{moon}$
A
$4 \times R_{cavity} = R_{moon}$
Ratio of accelerations ($g_{@A} / g_{@B}$)?
Escape velocity at point A ?
Escape velocity at point B?

Problem III:
When Swiss people heard the rumor that the moon was
made of cheese, they of course got the idea that it may also
contain holes. Let's assume that the moon does indeed
contain a giant hole, and then consider how someone might
be able to detect this hole via measuring the difference in
gravitational acceleration at the moon's surface, near and far
from the hole:
Suppose the moon is an otherwise uniform density sphere
which has an empty spherical cavity located halfway
between the center and the surface, as shown below in the
cross-sectional sketch below. Given a mass density, ρ, and
the relative radii, Rmoon = 4 × Rcavity, what is the ratio of the
acceleration due to the moon's gravity at the moon's surface
nearest to (point A) and furthest from (point B) this hole?
In terms of the parameters given above (and also Newton's
gravitational constant), what are the escape velocities at
these two positions at the moon surface, nearest to and
furtherst from the hole?
Note: here, please disregard the gravitational pulls
of the earth and the sun.
B
Rmoon
A
4 × Rcavity = Rmoon
Ratio of accelerations (g@A / g@B)?
Escape velocity at point A ?
Escape velocity at point B?

Added by Valerie C.

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