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According to Newton's law of gravitation, the gravitational attraction between two massive objects such as planets or asteroids is proportional to \(d^{-2}\), where \(d\) is the distance between the centers of the objects. Specifically, the gravitational force \(F\) between such objects is given by \(F=c d^{-2}\), where \(d\) is the distance between their centers. The value of the constant \(c\) depends on the masses of the two objects and on the universal gravitational constant. a. Suppose the force of gravity is causing two large asteroids to move toward each other. What is the effect on the gravitational force if the distance between their centers is halved? What is the effect on the gravitational force if the distance between their centers is reduced to one-quarter of its original value? b. Suppose that for a certain pair of asteroids whose centers are 300 kilometers apart, the gravitational force is \(2,000,000\) newtons. (One newton is about one-quarter of a pound.) What is the value of \(c\) ? Find the gravitational force if the distance between the centers of these asteroids is 800 kilometers. c. Using the value of \(c\) you found in part \(b\), make a graph of gravitational force versus distance between the centers of the asteroids for distances from 0 to 1000 kilometers. What happens to the gravitational force when the asteroids are close together? What happens to the gravitational force when the asteroids are far apart?

Step by step solution

01

Analyzing the effect of halving the distance

According to Newton's law of gravitation, the gravitational force is given by \(F = c d^{-2}\). If the distance \(d\) is halved, the new distance becomes \(d/2\). Substituting into the formula, the new force becomes \(F' = c (d/2)^{-2} = c (4/d^2) = 4F\). Therefore, halving the distance quadruples the gravitational force.

02

Effect of reducing distance to one-quarter

If the distance \(d\) is reduced to one-quarter, the new distance is \(d/4\). Substituting into the formula, the new force is \(F' = c (d/4)^{-2} = c (16/d^2) = 16F\). Thus, reducing the distance to one-quarter increases the gravitational force to 16 times the original value.

03

Calculate the constant c

We know \(F = 2000000\) newtons when \(d = 300\) kilometers. Using \(F = c d^{-2}\), we substitute these values to find \(c\): \(2000000 = c (300)^{-2}\). Rearranging gives \(c = 2000000 \times 300^2 = 180000000000\).

04

Gravitational force at 800 kilometers

Using \(c = 180000000000\) and \(d = 800\), the gravitational force is calculated by \(F = c d^{-2} = 180000000000 \times (800)^{-2} = 281250\) newtons.

05

Plotting the gravitational force vs. distance graph

Using \(c = 180000000000\), plot \(F = c d^{-2}\) for \(d\) ranging from 0 to 1000 kilometers. The plot will show an inverse square relationship, where the force increases significantly as distance decreases and diminishes as the distance increases. Close proximity leads to a sharp increase in force, indicated by the steep part of the curve, while increasing distance shows a gradual flattening of the curve.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Force

Gravitational force is a fundamental concept that explains how objects with mass attract each other. This force plays a crucial role in the motion of planets, moons, and other celestial bodies such as asteroids. According to Newton's Law of Gravitation, the gravitational force between two objects depends on their masses and the distance between them. The formula is expressed as: \[ F = c \cdot d^{-2} \]Here, \( F \) represents the gravitational force, \( c \) a constant that combines the universal gravitational constant and the masses of the objects, and \( d \) the distance between the centers of the masses. Understanding this force helps us predict how objects in space interact with each other over vast distances.

Distance and Force Relationship

The relationship between distance and gravitational force is a captivating aspect that governs many cosmic interactions. Essentially, as the distance between two objects changes, so does the gravitational pull they exert on each other. This relationship is inversely proportional, meaning:- If the distance \(d\) is halved, the gravitational force increases fourfold: \( F' = 4F \).- If the distance is reduced to one-quarter, the force escalates to sixteen times the original, expressed as \( F' = 16F \).This dynamics explains why objects closer to each other experience a significantly stronger pull, impacting their movement and orbits. It also illustrates the critical impact of distance on gravitational interactions in the universe.

Inverse Square Law

The inverse square law is a principle that describes how the strength of certain forces diminishes with distance. In the context of gravitational force, this law implies that as you double the distance between two objects, the gravitational force becomes one-fourth as strong.We express this mathematically as:\[ F = c \cdot d^{-2} \]This law isn't unique to gravity but applies to other phenomena, such as light and sound, which spread out over an area that's proportional to the square of the distance. The inverse square law is foundational in understanding celestial motion and the nature of forces like gravity that guide them.

Asteroids Interaction

Asteroids, like all objects with mass, are subject to gravitational forces. Their interactions with each other and other celestial bodies can lead to spectacular cosmic events. When two asteroids approach one another, their mutual gravitational attraction grows stronger as the distance decreases. This can cause: - **Collisions:** As in any bodies in space, reduced distance increases the likelihood of a collision by amplifying gravitational force. - **Gravitational Binding:** Close encounters may result in asteroids becoming gravitationally bound, orbiting each other as binary systems. Understanding these interactions is crucial for predicting potential impacts with Earth, managing space missions, and comprehending the dynamical behavior of celestial objects.