The information in the table represents the effect of the mass of two objects on the gravitational force between the two objects.
| Mass of Object 1 (kg) | Mass of Object 2 (kg) | Distance between Objects 1 and 2 (m) | Gravitational Force between Objects 1 and 2 (N) |
|---|---|---|---|
| 1 | 1 | 1 | 4 |
| 2 | 1 | 1 | ? |
Which number should be in the cell with the question mark?
A. The number is two because when you double the mass of one of the objects, the force between the objects is halved.
B. The number is four because when you double the mass of one of the objects, the force between the objects remains the same.
C. The number is eight because when you double the mass of one of the objects, the force between the objects also doubles.
Answer (1)
The gravitational force is directly proportional to the product of the masses.
Determine the gravitational constant using the first set of values: G = 4 .
Calculate the new gravitational force when the mass of Object 1 is doubled: F ′ = 4 1 2 2 ⋅ 1 = 8 .
The number that should be in the cell with the question mark is 8 .
Explanation
Understanding the Problem We are given a table that shows the gravitational force between two objects with different masses. We need to find the gravitational force when the mass of one object is doubled, while the other mass and the distance between them remain constant.
Recalling the Gravitational Force Formula The gravitational force ($F[) between two objects is directly proportional to the product of their masses ($m_1[ and $m_2[) and inversely proportional to the square of the distance ($r[) between them. This relationship is expressed by the formula: F = G r 2 m 1 m 2 where $G[ is the gravitational constant.
Finding the Gravitational Constant From the first row of the table, we have: $m_1 = 1\text{ kg}[, $m_2 = 1\text{ kg}[, $r = 1\text{ m}[, and $F = 4\text{ N}[.
Plugging these values into the formula, we get: 4 = G 1 2 1 ⋅ 1 4 = G So, the gravitational constant $G = 4\text{ N} \cdot \text{m}^2/\text{kg}^2[ in this scenario.
Calculating the New Gravitational Force Now, we consider the second row of the table, where the mass of Object 1 is doubled: $m_1 = 2\text{ kg}[, $m_2 = 1\text{ kg}[, and $r = 1\text{ m}[.
We want to find the new gravitational force $F'[.
Using the formula with the value of $G[ we found: F ′ = 4 1 2 2 ⋅ 1 F ′ = 4 ⋅ 2 F ′ = 8 N
Final Answer Therefore, when the mass of Object 1 is doubled, the gravitational force between the objects also doubles. The number that should be in the cell with the question mark is 8.
Examples
Understanding how gravitational force changes with mass is crucial in space exploration. For example, when planning a mission to Mars, engineers need to calculate the gravitational forces between the spacecraft and the planets to accurately determine the trajectory and fuel requirements. If the mass of the spacecraft increases (due to additional equipment or cargo), the gravitational forces acting on it will also increase, requiring adjustments to the mission plan to ensure a successful landing and return.
