AP DSC School Assistant Math Model Test 2

Concept:The centre of gravity of a composite body is the point where the total weight acts. For two uniform solid spheres of the same material, the mass is directly proportional to the volume, so we use volume in place of mass in the centre‑of‑gravity formula.Explanation:Let the centre of the larger sphere (radius $r_1 = 6$ cm) be taken as the origin. The distance between the centres of the two spheres is $6 + 3 = 9$ cm, so the centre of the smaller sphere is at $x = 9$ cm from the origin. Since both spheres have the same density, their masses are proportional to their volumes $V = \frac{4}{3}\pi r^3$. The centre of gravity from the origin is given by:$$x_{CG} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} = \frac{V_1 x_1 + V_2 x_2}{V_1 + V_2}$$Here $x_1 = 0$ (centre of larger sphere) and $x_2 = 9$ cm. Substituting volumes:$$x_{CG} = \frac{\frac{4}{3}\pi r_1^3 \cdot 0 + \frac{4}{3}\pi r_2^3 \cdot 9}{\frac{4}{3}\pi r_1^3 + \frac{4}{3}\pi r_2^3} = \frac{r_2^3 \cdot 9}{r_1^3 + r_2^3}$$Plug in $r_1 = 6$ cm, $r_2 = 3$ cm:$$x_{CG} = \frac{3^3 \cdot 9}{6^3 + 3^3} = \frac{27 \cdot 9}{216 + 27} = \frac{243}{243} = 1 \text{ cm}$$Thus the centre of gravity of the combined body is located 1 cm from the centre of the larger sphere (towards the smaller sphere).Answer:1 cm (Option A)