A hypothetical spherical planet of mass M and radius R is made of a uniform substance and has a gravitational field g at its surface. Suppose eight of these identical planets collide and combine into a new uniform, spherical planet. What will the gravitational field on the surface of this new planet be in terms of g ? A g B 2g C square D 8g

To solve this problem, we need to understand the relationship between the gravitational field and the mass and radius of the planet.<br /><br />The gravitational field g at the surface of a planet is given by the formula:<br /><br />g = GM/R^2<br /><br />where G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.<br /><br />Now, let's consider the collision and combination of the eight identical planets. When the planets collide and combine, the total mass of the new planet will be 8 times the mass of each individual planet. However, the radius of the new planet will also change.<br /><br />The radius of the new planet can be determined by the conservation of volume. Since the volume of the new planet must be equal to the combined volume of the eight individual planets, we have:<br /><br />V_new = 8 * V_old<br /><br />Since the volume of a sphere is given by V = (4/3)πR^3, we can write:<br /><br />(4/3)πR_new^3 = 8 * (4/3)πR_old^3<br /><br />Simplifying this equation, we get:<br /><br />R_new = 2 * R_old<br /><br />Now, we can substitute the new mass and radius into the formula for the gravitational field:<br /><br />g_new = G * (8M) / (2R_old)^2<br /><br />Simplifying this equation, we get:<br /><br />g_new = 2 * G * M / R_old^2<br /><br />Comparing this with the original gravitational field formula, we can see that:<br /><br />g_new = 2 * g<br /><br />Therefore, the correct answer is B) 2g.