Question

QuestionDetails:

A spere of matter, of mass m andradius a, has a
concentric cavity of radius b, as shownbelow.
a.) sketch a curve of thegravitational force F exerted
by the sphere on a particle of massm, located at a distance r from
the center of the sphere,as afunction of r in the range 0 <= r
<= infinity. Consider r = 0, b, and and infinity in
particular.
b.) Sketch the corresponding curvefor the potential
energy U(r) of thesystem.

Answer 1

The sketch for the Gravitational force F and the potential energy U are attached to this answer.

Step-by-step explanation:

To obtain the gravitational force, we can consider the gravitational field GF(r) as:

`\vec{F_(g)}=m_(par) \cdot \vec{G_F(r)}`

To calculate the gravitational field we can use the Gauss theorem. By considering a homogeneous mass of the sphere (constant density) and the spherical symmetry, we can determinate than the gravitational field direction is
`-\hat{r}`.

Considering a constant density:

`Vol=\pi(4)/(3)(a^3-b^3)`

`\rho=(m_(sph))/(Vol) =(3m_(sph))/(4\pi(a^3-b^3))`

Applying a spherical gaussian surface for different radius r:

for R<b:

`\displaystyle\oint_(S) \vec{G_(f)}\, \vec{ds}=-4m_(int)G=0N/kg \rightarrow G_(f)=0N/kg \rightarrow F_(g)=0N`

for b<R<a:

`\displaystyle\oint_(S) \vec{G_(f)}\, \vec{ds}=-4\pi m_(int)G\\\displaystyle\oint_(S) -G_(f)\, ds=-4\pi G\displaystyle\int_{} \int_{} \int_(V) \rho\, dV\,\,\\\\-4\pi r^2 G_(f)(r)=-4\pi Gm_(sph)((r^3-b^3))/((a^3-b^3)) \\G_(f)(r)=Gm_(sph)((r^3-b^3))/(r^2(a^3-b^3)) \\F_g =Gm_(par)m_(sph)((r^3-b^3))/(r^2(a^3-b^3))(-\hat{r})`

for a<R:

`\displaystyle\oint_(S) \vec{G_(f)}\, \vec{ds}=-4\pi m_(int)G\\\\-4\pi r^2 G_(f)(r)=-4\pi Gm_(sph) \\    F_g =Gm_(par)m_(sph)(1)/(r^2)(-\hat{r})`

For the potential energy you can integrate the field to obtain the gravitational potential and the multiplying for the particle mass:

for a<R:

`U(r)=G(m_(sph)m_(par))/(r)`

for b≤R≤a:

`U(r)=G(m_(sph)m_(par))/(a) +G(m_(sph)m_(par))/((a^3-b^3)) ((a^(2))/(2)+(b^(2))/(a)-(r^(2))/(2)-(b^(2))/(r))`

for R≤b:

`U(r)=G(m_(sph)m_(par))/(a) +G(m_(sph)m_(par))/((a^3-b^3)) ((a^(2))/(2)+(b^(2))/(a)-(b^(2))/(2)-b)`