Question

Problem 3 (25 points) Consider an infinitesimally thin circular disk of radius $R$ and mass $M$ centered at the origin sitting in the xy-plane. The disk has non-uniform surface mass density $\sigma$ which varies linearly with radius, i.e., $\sigma = cr$. (a) Determine the constant $c$ in terms of the mass $M$ and the radius $R$. (b) Determine the gravitational field $\vec{g}(\vec{r})$ at the point $(0, 0, z)$ above the disk. (c) Determine the gravitational potential $\Phi(\vec{r})$ at the point $(0, 0, z)$ above the disk. (d) In the limit of large $z$, determine the form for gravitational potential and field.

          Problem 3 (25 points)
Consider an infinitesimally thin circular disk of radius $R$ and mass $M$ centered at the origin
sitting in the xy-plane. The disk has non-uniform surface mass density $\sigma$ which varies
linearly with radius, i.e., $\sigma = cr$.
(a) Determine the constant $c$ in terms of the mass $M$ and the radius $R$.
(b) Determine the gravitational field $\vec{g}(\vec{r})$ at the point $(0, 0, z)$ above the disk.
(c) Determine the gravitational potential $\Phi(\vec{r})$ at the point $(0, 0, z)$ above the disk.
(d) In the limit of large $z$, determine the form for gravitational potential and field.

Problem 3 (25 points)
Consider an infinitesimally thin circular disk of radius R and mass M centered at the origin
sitting in the xy-plane. The disk has non-uniform surface mass density σ which varies
linearly with radius, i.e., σ = cr.
(a) Determine the constant c in terms of the mass M and the radius R.
(b) Determine the gravitational field g⃗(r⃗) at the point (0, 0, z) above the disk.
(c) Determine the gravitational potential Φ(r⃗) at the point (0, 0, z) above the disk.
(d) In the limit of large z, determine the form for gravitational potential and field.

Added by Angela S.

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