Newton developed a universal law of gravitation that can be used under most circumstances, not just for objects on the surface of the Earth. Here is the law:
[tex]F=G \frac{m_1 m_2}{r^2}[/tex] where [tex]G[/tex] is a constant.
Think about what happens when you increase and decrease each of the variables in Newton's equation for the gravitational force between two objects of mass [tex]m_1[/tex] and [tex]m_2[/tex], a distance [tex]r[/tex] from one another.
Given your result, what is the likely cause of Kepler's observation that planets travel faster when they are closer to the Sun?
Choose one:
A. The force of gravity is weaker closer to the Sun.
B. The mass of the Sun is greater when a planet is closer to it.
C. The force of gravity is stronger closer to the Sun.
D. The mass of the planet is greater when it is closer to the Sun.
Answer (1)
Newton's law of gravitation states that F = G "." m 1 m 2 / r 2 .
The gravitational force F is inversely proportional to the square of the distance r .
When a planet is closer to the Sun, the distance r decreases, and the gravitational force F increases.
Therefore, the likely cause of Kepler's observation is that the force of gravity is stronger closer to the Sun: C .
Explanation
Understanding the Problem We are given Newton's law of gravitation: F = G "." m 1 m 2 / r 2 , where F is the gravitational force, G is the gravitational constant, m 1 and m 2 are the masses of the two objects, and r is the distance between them. We need to determine the cause of Kepler's observation that planets travel faster when they are closer to the Sun.
Analyzing Newton's Law Newton's law tells us that the gravitational force F is directly proportional to the product of the masses m 1 and m 2 , and inversely proportional to the square of the distance r . This means that if we increase the distance r between the objects, the gravitational force F decreases. Conversely, if we decrease the distance r , the gravitational force F increases.
Evaluating the Options Now let's consider the given options:
A. The force of gravity is weaker closer to the Sun. This contradicts Newton's law, as the force of gravity should be stronger when the distance is smaller.
B. The mass of the Sun is greater when a planet is closer to it. The mass of the Sun does not change with the planet's position.
C. The force of gravity is stronger closer to the Sun. This aligns with Newton's law, as the force of gravity increases when the distance decreases.
D. The mass of the planet is greater when it is closer to the Sun. The mass of the planet does not change with its position.
Conclusion Based on our analysis, the correct answer is C. The force of gravity is stronger closer to the Sun. When a planet is closer to the Sun, the gravitational force between them is stronger. This stronger force causes the planet to accelerate more, resulting in a higher speed.
Examples
Imagine you're swinging a ball on a string. When you shorten the string (decreasing the distance), you have to exert more force to keep the ball moving at the same speed. Similarly, a planet closer to the Sun experiences a stronger gravitational pull, causing it to move faster to maintain its orbit. This principle is crucial in understanding celestial mechanics and the dynamics of orbiting bodies.
