Introduction
Gravitation is a fundamental force of nature that governs the motion of celestial bodies and objects on Earth. It is the force of attraction between two masses, and it plays a crucial role in various phenomena ranging from the falling of an apple to the orbits of planets around the Sun. In this study note, we will explore the concepts of gravitation as per the NEET Physics syllabus, breaking down complex ideas into simpler sections.
Universal Law of Gravitation
Statement of the Law
Sir Isaac Newton formulated the Universal Law of Gravitation, which 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 their centers.
Mathematically, the gravitational force ($F$) between two masses $m_1$ and $m_2$ separated by a distance $r$ is given by:
$$ F = G \frac{m_1 m_2}{r^2} $$
where $G$ is the gravitational constant, approximately equal to $6.674 \times 10^{-11} , \text{Nm}^2/\text{kg}^2$.
TipRemember that the gravitational force is always attractive and acts along the line joining the centers of the two masses.
Characteristics of Gravitational Force
- Mutual Force: Both masses experience the same magnitude of force but in opposite directions.
- Central Force: The force acts along the line joining the centers of the two masses.
- Inverse Square Law: The force decreases with the square of the distance between the masses.
Examples
ExampleConsider two masses, $m_1 = 5 , \text{kg}$ and $m_2 = 10 , \text{kg}$, separated by a distance of $2 , \text{m}$. The gravitational force between them is:
$$ F = G \frac{m_1 m_2}{r^2} = 6.674 \times 10^{-11} \frac{5 \times 10}{2^2} = 8.3425 \times 10^{-10} , \text{N} $$
Gravitational Field
Definition
The gravitational field at a point in space is defined as the gravitational force experienced by a unit mass placed at that point. It is a vector quantity.
Gravitational Field Strength
The gravitational field strength ($g$) at a distance $r$ from a mass $M$ is given by:
$$ g = \frac{GM}{r^2} $$
Gravitational Field due to a Point Mass
For a point mass $M$, the gravitational field at a distance $r$ is:
$$ \vec{g} = -G \frac{M}{r^2} \hat{r} $$
NoteThe negative sign indicates that the gravitational field is directed towards the mass $M$.
Gravitational Potential Energy
Definition
Gravitational potential energy ($U$) is the energy possessed by an object due to its position in a gravitational field.
Formula
The gravitational potential energy of a mass $m$ at a distance $r$ from another mass $M$ is:
$$ U = -G \frac{mM}{r} $$
Common MistakeDo not confuse gravitational potential energy with gravitational potential. Potential energy is a scalar quantity and depends on the mass of the object, while gravitational potential is the potential energy per unit mass.
Examples
ExampleCalculate the gravitational potential energy of a $2 , \text{kg}$ mass at a distance of $4 , \text{m}$ from a $5 , \text{kg}$ mass.
$$ U = -G \frac{mM}{r} = -6.674 \times 10^{-11} \frac{2 \times 5}{4} = -1.6685 \times 10^{-10} , \text{J} $$
Gravitational Potential
Definition
Gravitational potential ($V$) at a point is defined as the work done in bringing a unit mass from infinity to that point in the gravitational field.
Formula
The gravitational potential at a distance $r$ from a mass $M$ is:
$$ V = -G \frac{M}{r} $$
Relation to Gravitational Field
The gravitational field strength is the negative gradient of the gravitational potential:
$$ \vec{g} = -\nabla V $$
Kepler's Laws of Planetary Motion
First Law: Law of Orbits
Every planet moves in an elliptical orbit with the Sun at one of the two foci.
Second Law: Law of Areas
A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
Third Law: Law of Periods
The square of the orbital period ($T$) of a planet is directly proportional to the cube of the semi-major axis ($a$) of its orbit:
$$ T^2 \propto a^3 $$
or
$$ \frac{T^2}{a^3} = \text{constant} $$
TipKepler's laws help in understanding the motion of planets and can be derived from Newton's law of gravitation.
Escape Velocity
Definition
Escape velocity is the minimum velocity required for an object to escape the gravitational influence of a celestial body without further propulsion.
Formula
The escape velocity ($v_e$) from the surface of a planet of mass $M$ and radius $R$ is:
$$ v_e = \sqrt{\frac{2GM}{R}} $$
Examples
ExampleCalculate the escape velocity from Earth (mass $M = 5.97 \times 10^{24} , \text{kg}$, radius $R = 6.37 \times 10^6 , \text{m}$).
$$ v_e = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 5.97 \times 10^{24}}{6.37 \times 10^6}} \approx 11.2 , \text{km/s} $$
Satellites
Types of Satellites
- Geostationary Satellites: Orbit the Earth at an altitude where their orbital period matches the Earth's rotational period (24 hours).
- Polar Satellites: Orbit the Earth in a path that passes over the poles, allowing them to scan the entire Earth's surface.
Orbital Velocity
The orbital velocity ($v_o$) of a satellite at a distance $r$ from the center of the Earth is:
$$ v_o = \sqrt{\frac{GM}{r}} $$
Time Period of Satellites
The time period ($T$) of a satellite in a circular orbit of radius $r$ is:
$$ T = 2\pi \sqrt{\frac{r^3}{GM}} $$
NoteFor a satellite in a low Earth orbit, the radius $r$ is approximately equal to the Earth's radius plus the altitude of the satellite.
Conclusion
Gravitation is a fundamental force that governs the motion of objects in the universe. Understanding the principles of gravitation, including the Universal Law of Gravitation, gravitational field, potential energy, and the behavior of satellites, is essential for mastering the concepts in NEET Physics. By breaking down these concepts and using examples, we can gain a deeper understanding of the gravitational phenomena that shape our world and beyond.
