Question

15 mark for review a two - object system consists of a solid sphere of mass $m_0$ and radius $r_0$ located at the center of a thin spherical shell of mass $m_0$ and outer radius $3r_0$ as shown. calculate the ratio of the gravitational field of the system at a point $2r_0$ from the center to the gravitational field of the system at point located $4r_0$ from the center. a 2:1 b 4:1 c 9:2 d 9:1

Explanation:

Step1: Recall gravitational - field formula

The gravitational field strength $g$ at a distance $r$ from a point - like mass $M$ is given by $g=\frac{GM}{r^{2}}$, and for a spherical mass distribution, for points outside the sphere or shell, the mass can be considered as concentrated at the center.

Step2: Calculate the gravitational field at $r = 2r_0$

At $r = 2r_0$, only the solid sphere of mass $m_0$ contributes to the gravitational field since the thin - spherical shell does not contribute to the gravitational field inside it. So, $g_1=\frac{Gm_0}{(2r_0)^{2}}=\frac{Gm_0}{4r_0^{2}}$.

Step3: Calculate the gravitational field at $r = 4r_0$

At $r = 4r_0$, both the solid sphere and the thin - spherical shell contribute to the gravitational field. The total mass $M_{total}=m_0 + m_0=2m_0$. So, $g_2=\frac{G(2m_0)}{(4r_0)^{2}}=\frac{2Gm_0}{16r_0^{2}}=\frac{Gm_0}{8r_0^{2}}$.

Step4: Calculate the ratio of the gravitational fields

The ratio $\frac{g_1}{g_2}=\frac{\frac{Gm_0}{4r_0^{2}}}{\frac{Gm_0}{8r_0^{2}}}$. Canceling out $G$, $m_0$, and $r_0^{2}$, we get $\frac{g_1}{g_2}=\frac{8}{4}=2:1$.

Answer:

A. $2:1$