The question asks to identify the "universal law of gravitation" from the given options. This requires knowledge of fundamental laws in physics and astronomy.
The law that states every particle attracts every other particle 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 is known as Newton's Law of Universal Gravitation.
Mathematically, it is expressed as:
\[ F = G \frac{m_1 m_2}{r^2} \]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 their centers.
C) Newton's law of gravitation — This law describes the attractive force between any two objects with mass in the universe, making it a universal law. It is precisely what the question refers to.
This question tests understanding of Newton's Law of Universal Gravitation, specifically how the gravitational force changes with the distance between two objects. The key is to recall the inverse square relationship between gravitational force and distance.
A) It becomes four times weaker. As derived from Newton's Law of Universal Gravitation, when the distance is doubled, the force becomes \(1/(2^2) = 1/4\) of the original force, meaning it is four times weaker.
This question asks about the effect of doubling the mass of one object on the gravitational force between two objects. We need to recall Newton's Law of Universal Gravitation, which describes the relationship between gravitational force, masses of the objects, and the distance between them.
Recall Newton's Law of Universal Gravitation: The gravitational force (\(F\)) between two objects with masses \(m_1\) and \(m_2\), separated by a distance \(r\), is given by the formula:
\[ F = G \frac{m_1 m_2}{r^2} \]where \(G\) is the gravitational constant.
Identify the initial conditions: Let the initial masses be \(m_1\) and \(m_2\), and the initial force be \(F_1\). So,
\[ F_1 = G \frac{m_1 m_2}{r^2} \]Apply the change: The problem states that the mass of one of the objects is doubled. Let's assume \(m_1\) is doubled, so the new mass is \(m_1' = 2m_1\). The other mass \(m_2\) and the distance \(r\) remain unchanged.
Calculate the new gravitational force: Let the new force be \(F_2\). Substitute the new mass into the formula:
\[ F_2 = G \frac{(2m_1) m_2}{r^2} \]Compare the new force with the initial force: We can rewrite \(F_2\) as:
\[ F_2 = 2 \left( G \frac{m_1 m_2}{r^2} \right) \]Since \(F_1 = G \frac{m_1 m_2}{r^2}\), we can see that:
\[ F_2 = 2 F_1 \]This shows that the new gravitational force is twice the original gravitational force.
A) It is doubled — As derived from Newton's Law of Universal Gravitation, if one mass is doubled, the gravitational force between the objects also doubles, assuming the other mass and the distance remain constant.
B) It is halved — This would happen if one of the masses was halved, or if the distance between the objects was increased by a factor of \(\sqrt{2}\) (since force is inversely proportional to the square of the distance).
C) It is squared — This would imply a non-linear relationship where the force increases much more drastically, which is not the case for a linear change in mass as per Newton's law.
D) It remains unchanged — The gravitational force is directly proportional to the product of the masses. Therefore, changing one of the masses will directly affect the force; it cannot remain unchanged.
The question asks to identify the fundamental force responsible for keeping planets in orbit around the Sun. This is a classic example of celestial mechanics, governed by one of the four fundamental forces of nature.
Correct Option: C) Gravitational force is the attractive force between any two objects with mass. It is this force that pulls the planets towards the Sun, providing the centripetal force required to maintain their orbits.
To determine how a satellite's orbital speed changes as it moves closer to Earth, we need to consider the principles of conservation of angular momentum and gravitational potential energy, as well as the relationship between gravitational force and centripetal force.
A) It increases. As explained by the principles of conservation of angular momentum, conservation of mechanical energy, and the direct relationship derived from gravitational and centripetal forces, a satellite's orbital speed increases as it moves closer to the Earth.
The question asks about the nature of gravitational potential energy between two masses. We need to recall the formula for gravitational potential energy and analyze its components to determine its proportionality to distance and its sign.
C) It is always negative. As derived from the formula \(U = -G \frac{m_1 m_2}{r}\), where \(G\), \(m_1\), \(m_2\), and \(r\) are all positive, the gravitational potential energy is inherently negative.