Imagine that two identical containers, one filled with marbles and the other filled with Styrofoam beads, are released simultaneously from rest from the same height - say, \(3 \mathrm{~m}\) -above Earth's surface. (a) Which container, if either, experiences the greater gravitational force? Justify your answer. (b) Which container, if either, experiences the greater acceleration from gravity? Again, justify your answer.

To compare the gravitational forces acting on both containers, we need to apply Newton's law of universal gravitation, which states that the force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is given by: \(F = G \frac{m_1 m_2}{r^2}\) where: \(F\) – Instantaneous gravitational force \(G\) – Gravitational constant (\(6.674 \times 10^{-11} \mathrm{Nm}^2\mathrm{kg}^{-2} \)) \(m_1\) and \(m_2\) – Mass of the two objects \(r\) – Distance between the two centers Let's compare the masses of the marbles and the Styrofoam beads. Since the containers are identical, their masses and the distance between their centers will be the same when they are released. As marbles are denser than Styrofoam beads, the container filled with marbles will have a greater mass (\(m_1\)) than the container filled with Styrofoam beads (\(m_2\)). Now, we can use the formula to compare the gravitational forces: \(F_M > F_S\) since: \(m_1 > m_2\) So, the container filled with marbles experiences a greater gravitational force.

To compare the gravitational accelerations of both containers, we need to use Newton's second law: \(F = ma\) where: \(F\) – Gravitational force \(m\) – Mass \(a\) – Acceleration due to gravity Since we have already established that the container filled with marbles experiences greater gravitational force, let us derive the acceleration due to gravity for both containers: \(F_M = m_M a_M\) \(F_S = m_S a_S\) Now, divide both equations: \(\frac{F_M}{F_S} = \frac{m_M a_M}{m_S a_S} \) We know that \(F_M > F_S\) and \(m_M > m_S\). So: \(\frac{a_M}{a_S} = \frac{F_M/m_M}{F_S/m_S}\) Since the gravitational force acting on an object is proportional to its mass, the ratio of gravitational force to mass will be constant: \(\frac{a_M}{a_S} = 1\) Hence: \(a_M = a_S\) Therefore, both containers, filled with marbles and Styrofoam beads, experience the same gravitational acceleration from gravity.