Question

Scientists use the following formula to calculate the force of gravity that two objects exert on each other: In the equation $$ F=\frac{G \times M \times m}{r^2} $$ \(F\) is the force of gravity; \(G\) is a constant; \(M\) is the mass of one of the objects; \(m\) is the mass of the second object; \(r\) is the distance between the centers of the objects. If an object with a given mass \(m\) is replaced by an object of half its mass, which of the following will increase the force of gravity? A. increasing mass \(M\) and doubling the distance between the objects B. reducing mass \(M\) and doubling the distance between the objects C. reducing mass \(M\) and halving the distance between the objects D. increasing mass \(M\) and halving the distance between the objects

Short answer

D. increasing mass \(M\) and halving the distance between the objects.

Step-by-step solution

Write down the given formula and consider the given mass change.

The formula to calculate the force of gravity between two objects is: \(F = \frac{G \times M \times m}{r^2}\). We are given that mass \(m\) is replaced by an object of half its mass, therefore the new mass is \(\frac{1}{2}m\).

Analyze the effects of changes in mass and distance on the gravitational force.

We will now analyze each scenario considering the changes in mass and distance: A. increasing mass \(M\) and doubling the distance between the objects: The new force would be \(F' = \frac{G \times M' \times (\frac{1}{2}m)}{(2r)^2}\), where \(M'\) is the increased mass. B. reducing mass \(M\) and doubling the distance between the objects: The new force would be \(F' = \frac{G \times M' \times (\frac{1}{2}m)}{(2r)^2}\), where \(M'\) is the reduced mass. C. reducing mass \(M\) and halving the distance between the objects: The new force would be \(F' = \frac{G \times M' \times (\frac{1}{2}m)}{(\frac{1}{2}r)^2}\), where \(M'\) is the reduced mass. D. increasing mass \(M\) and halving the distance between the objects: The new force would be \(F' = \frac{G \times M' \times (\frac{1}{2}m)}{(\frac{1}{2}r)^2}\), where \(M'\) is the increased mass.

Compare the new forces in each scenario to the original force.

We will now compare the new forces in each scenario to the original force to find out which one increases the force of gravity: A. \(F' = \frac{G \times M' \times (\frac{1}{2}m)}{(2r)^2} = \frac{M'}{4M}F\). If \(M'\) is increased, then \(\frac{M'}{4M} < 1\), so the force of gravity will decrease. B. \(F' = \frac{G \times M' \times (\frac{1}{2}m)}{(2r)^2} = \frac{M'}{4M}F\). If \(M'\) is reduced, then \(\frac{M'}{4M} < 1\), so the force of gravity will decrease. C. \(F' = \frac{G \times M' \times (\frac{1}{2}m)}{(\frac{1}{2}r)^2} = 4\frac{M'}{M}F\). If \(M'\) is reduced, then \(\frac{M'}{M} < 1\), so the force of gravity will increase. D. \(F' = \frac{G \times M' \times (\frac{1}{2}m)}{(\frac{1}{2}r)^2} = 4\frac{M'}{M}F\). If \(M'\) is increased, then \(\frac{M'}{M} > 1\), so the force of gravity will increase. The force of gravity will increase in options C and D. However, since C and D are not separate answer choices, we must eliminate one: In option C, reducing mass \(M\) is counteracted by the increased gravitational force due to halving the distance between the objects. However, since \(m\) is also reduced to half its original value, this may potentially lessen the overall increase in gravitational force compared to option D, where mass \(M\) is increased. Therefore, option D is the most plausible answer: #answer#D. increasing mass \(M\) and halving the distance between the objects.

Key concepts

Force of Gravity

Understanding the force of gravity is essential to grasp many principles of physics. It is the force that pulls two objects with mass towards each other. To conceptualize this pull, imagine when you drop an object, it falls to the ground rather than floating away. That's gravity in action.

The force of gravity between two objects is determined by their masses and the distance between them. If we look at the example provided, where an object with mass m is replaced with one half its mass, the gravitational force is dependent on the changes we apply according to the formula F = \( \frac{G \times M \times m}{r^2} \), where G is the gravitational constant, M is the mass of the first object, m is the mass of the second object (in this case, halved), and r is the distance between the objects.

For practical learning, remember that the force will increase if the mass of the objects increases, or if the distance between them decreases, and vice versa. Exploring variations of these parameters, like halving the mass or altering the distance, reveals the sensitivity of gravitation to these elements, which directly translates into a real-world understanding, such as why the Moon's orbit affects ocean tides.

Newton's Law of Universal Gravitation

Sir Isaac Newton defined the law of universal gravitation, which states that every point mass attracts every other point mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This law is encapsulated in the renowned equation F = \( \frac{G \times M \times m}{r^2} \).

Newton's groundbreaking insight ties the force of gravity to both the masses of objects and the space between them. The gravitational constant G is a key part of this equation, serving as a proportionality constant that makes the math work out. This constant provides consistency to gravitational calculations across the cosmos. From teaching students about planetary orbits to explaining why we're not launched into space when we jump, Newton's law puts into perspective how gravity governs the motion of celestial and terrestrial objects.

Mass and Distance Relationship

The mass and distance relationship in gravitational force is a vital concept that tells us how the gravitational force will respond to changes in these two parameters. As per the formula provided, the gravitational force is directly proportional to the masses of the objects and inversely proportional to the square of the distance between them.

What does this mean? If you increase the mass of an object, the gravitational force goes up. If you move two objects closer together, the gravitational pull between them also goes up. Conversely, decreasing mass or increasing distance will reduce the gravitational force. It's crucial to note the squared distance rule, meaning that the force changes significantly with even slight changes in distance.

For the student trying to solve problems related to gravitational force, tinkering with the mass and distance in the formula will help them see the big picture. For instance, when the exercise asks about halving one mass and altering the distance, working through the formula by making these adjustments yields clear answers about how gravitational force will behave. This understanding is foundational for fields like astronomy, engineering, and physics, where gravity plays a pivotal role.