The table below shows four examples of pairs of objects, the masses of each objects in the pair, and the distances between the objects. In which example is a gravitational force of attraction between the two objects the greatest? Explain how you know this.
To answer this question, we need to consider the relationship between mass and the gravitational force of attraction. According to Newton's law of universal gravitation, the force of attraction between two objects is directly proportional to the product of their masses (m1 and m2) and inversely proportional to the square of the distance between them (r).
Example | Mass of object 1 (m1) | Mass of object 2 (m2) | Distance (r)
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1 | 5 kg | 1 kg | 2 meters
2 | 2 kg | 2 kg | 3 meters
3 | 3 kg | 4 kg | 4 meters
4 | 4 kg | 5 kg | 1 meter
To determine which example has the greatest gravitational force of attraction, we need to calculate and compare the forces using F = G(m1 * m2)/r^2, where G is the gravitational constant.
Let's compute the forces for each example:
Example 1: F1 = G * (5 kg * 1 kg) / (2 m)^2
Example 2: F2 = G * (2 kg * 2 kg) / (3 m)^2
Example 3: F3 = G * (3 kg * 4 kg) / (4 m)^2
Example 4: F4 = G * (4 kg * 5 kg) / (1 m)^2
Since G, the gravitational constant, is constant for all examples, we can compare the forces by comparing their product of masses divided by the square of the distances:
F1 = (5 kg * 1 kg) / (2 m)^2 = 2.5 kg/m^2
F2 = (2 kg * 2 kg) / (3 m)^2 ≈ 0.889 kg/m^2
F3 = (3 kg * 4 kg) / (4 m)^2 = 1.5 kg/m^2
F4 = (4 kg * 5 kg) / (1 m)^2 = 100 kg/m^2
Comparing the forces, we can see that the force in Example 4 is the greatest as it has the largest value of 100 kg/m^2. Therefore, the example with object masses of 4 kg and 5 kg separated by a distance of 1 meter has the greatest gravitational force of attraction.