Outside a spherically symmetric mass, where can its mass be treated as concentrated for gravitational-field calculations?

What is the gravitational field magnitude at the exact center of a uniform spherical mass?

A uniform sphere has surface field magnitude \(g_s\). Find the field magnitude at radius \(R/3\) inside it.

A uniform sphere has surface field \(g_s\). Find the field magnitude at \(r=R/4\) and at \(r=3R\).

A uniform spherical planet has radius \(3.0\times10^6\,\mathrm{m}\) and surface field \(5.0\,\mathrm{N\,kg^{-1}}\). Estimate its density.

A tunnel is drilled through a uniform spherical planet. In the ideal model, where along the tunnel is the gravitational field zero, and why?

A uniform sphere has \(M=6.0\times10^{24}\,\mathrm{kg}\) and \(R=6.0\times10^6\,\mathrm{m}\). Find \(g\) at \(r=3.0\times10^6\,\mathrm{m}\) and at \(r=1.2\times10^7\,\mathrm{m}\).

For a uniform sphere, explain why the field magnitude is largest at the surface rather than at the center or far outside.

A uniform planet has density \(\rho\). Derive the surface field \(g_s\) in terms of \(\rho\), \(R\), and \(G\).

Inside a uniform spherical body, measurements show \(g(r)=Ar\). Derive the body's density and its total mass in terms of \(A\), \(R\), and \(G\).