Vector Form of Newton’s Law of Gravitation
Prove that Newton’s Law of Gravitation follows Newton’s Third Law of Motion
Newton’s law of gravitation in vector form states that the gravitational forces acting between two masses are equal in magnitude and opposite in direction, forming an action-reaction pair.
Mathematical Representation
The vector form of Newton’s law of gravitation signifies that the gravitational forces acting between the two particles form action-reaction pair.
From the above figure, it can be seen that the two particles of masses are placed at a distance.
\(\begin{array}{l}\vec{r_{21}}=(\vec{r_{2}}-\vec{r_{1}})\end{array} \)
The direction of the vector is from M1 to M2
Therefore, the force applied on M2 by M1 is
\(\begin{array}{l}\vec{F_{21}}=-\frac{GM_{1}M_{2}}{r^{2}_{21}}\hat{{r_{21}}}\end{array} \)
The negative sign indicates the attractive nature of the force.
Similarly, force on M1 and M2
\(\begin{array}{l}\vec{F_{12}}=-\frac{GM_{1}M_{2}}{r^{2}_{12}}\hat{{r_{12}}}\end{array} \)
Since,
\(\begin{array}{l}\hat{r_{12}}= -\hat{r_{21}}\end{array} \)
\(\begin{array}{l}\vec{F_{12}}=-\frac{GM_{1}M_{2}}{(-r_{21})^{2}}[\hat{-{r_{21}}}]\end{array} \)
\(\begin{array}{l}\vec{F_{12}}=\frac{GM_{1}M_{2}}{(r_{21})^{2}}[\hat{{r_{21}}}]\end{array} \)
\(\begin{array}{l}=-\vec{F_{21}}\end{array} \)
Thus, the forces are equal in magnitude but opposite in direction, satisfying Newton’s third law of motion.
Frequently Asked Questions (FAQs)
Q1. What is Newton’s law of gravitation?
A. Newton’s law of gravitation states that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Q2. Why is there a negative sign in the force equation?
A. The negative sign signifies that the gravitational force is attractive in nature.
Q3. Does Newton’s law of gravitation apply in a vacuum?
A. Yes, gravity acts in a vacuum since it does not require a medium to propagate.
Q4. What is the unit of gravitational force?
A. The SI unit of gravitational force is Newton (N).
Multiple Choice Questions (MCQs)
1. The force of gravitation between two bodies depends on:
a) The sum of their masses
b) The difference of their masses
c) The product of their masses
d) The square of the sum of their masses
Answer: (c) The product of their masses
Explanation: Newton’s law states that gravitational force is directly proportional to the product of the two masses.
2. If the distance between two masses is doubled, the gravitational force will become:
a) Half
b) One-fourth
c) Double
d) Four times
Answer: (b) One-fourth
Explanation: Since gravitational force follows an inverse square law, doubling the distance reduces the force by a factor of four.
3. Newton’s law of gravitation follows:
a) Newton’s first law
b) Newton’s second law
c) Newton’s third law
d) None of these
Answer: (c) Newton’s third law
Explanation: The action-reaction pair in gravitational force satisfies Newton’s third law.
Test Your Knowledge
Newton’s Law of Gravitation Quiz
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