Understanding Gravitation: Laws, Forces, and Everyday Impact

Gravitation is a fundamental force responsible for the mutual attraction between all objects with mass in the universe. It plays a crucial role in governing the motion of planets, satellites, and various phenomena observed in astrophysics and everyday life. The study of gravitation involves mathematical laws, measurable constants, and practical applications relevant to JEE examinations.

Newton's Law of Universal Gravitation

Newton's law of universal gravitation states that every pair of point masses in the universe exerts an attractive force on each other. This force is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Mathematically, this is written as $F = G \dfrac{m_1 m_2}{r^2}$, where $F$ is the gravitational force, $G$ is the universal gravitational constant, $m_1$ and $m_2$ are the masses, and $r$ is the distance between their centers.

The force acts along the line joining the centers of the masses and is always attractive in nature. For spherical bodies like planets, $r$ denotes the distance between their centers.

Universal Gravitational Constant

The universal gravitational constant ($G$) is a fundamental quantity that determines the strength of gravitational force in the universe. Its value is the same everywhere and is unaffected by the nature or size of the bodies or the medium separating them.

For numerical problems on gravitation, using the correct value and units for $G$ is essential. Reference material such as Gravitation Revision Notes provides additional examples on its application.

Gravitational Field and Field Intensity

A gravitational field is a region around a mass where its gravitational influence is experienced by other masses. The gravitational field intensity at a point is defined as the force experienced by a unit mass placed at that point.

The gravitational field intensity produced by a mass $M$ at a distance $r$ is $I = \dfrac{GM}{r^2}$. The direction of the field is toward the mass creating it, and its strength decreases with increasing distance from the mass.

Gravitational field intensity is a vector quantity with dimensional formula $[M^0L^1T^{-2}]$ and is always directed towards the source mass, demonstrating the attractive nature of gravitation.

Acceleration Due to Gravity ($g$)

Acceleration due to gravity ($g$) is the acceleration experienced by a body falling freely under the influence of Earth's gravity. At the surface of the Earth, $g$ is given by $g = \dfrac{GM}{R^2}$, where $M$ is Earth's mass and $R$ is its radius.

The value of $g$ at Earth's surface is approximately $9.81$ m/s$^2$. It decreases with altitude and also varies slightly with latitude because of Earth's rotation and non-uniform density.

The variation of $g$ with height ($h$) above the Earth’s surface can be expressed as $g_h = g \left(1 - \dfrac{2h}{R}\right)$ for $h$ much smaller than $R$.

Gravitational Potential

Gravitational potential at a point is the work done by an external agent in bringing a unit mass from infinity to that point, without acceleration. It is denoted by $V$ and is given by $V = -\dfrac{GM}{r}$ for a point mass $M$, where the negative sign indicates the attractive force.

Gravitational potential is a scalar quantity and has SI unit of joule per kilogram (J/kg). For a spherical shell or sphere, the potential varies depending on the location: outside, on the surface, or inside the body.

Field intensity and potential are related by $I = - \dfrac{dV}{dr}$, emphasizing that the gravitational field points in the direction of decreasing potential.

Additional examples of potential calculations can be found in the Gravitation Important Questions.

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by a mass due to its position in a gravitational field. For two masses $m_1$ and $m_2$ separated by a distance $r$, the gravitational potential energy is $U = -G \dfrac{m_1 m_2}{r}$.

The negative value signifies that the system is bound; work must be done to separate the objects against the gravitational force. For a mass $m$ at Earth’s surface, $U = -G \dfrac{M m}{R}$.

For objects near Earth's surface, an approximate expression is $U = mgh$, valid only for heights much smaller than Earth's radius. For systems with more than two masses, total potential energy is the sum over all unique pairs.

Escape Velocity

Escape velocity is the minimum velocity required for an object to leave the gravitational influence of a celestial body without further propulsion. From the surface of Earth, escape velocity is $v_e = \sqrt{2gR} = \sqrt{\dfrac{2GM}{R}}$.

For Earth, $v_e$ is approximately $11.2$ km/s. Escape velocity is independent of the mass of the object and depends solely on the planet’s mass and radius.

Objects projected with velocities less than escape velocity fall back to the surface, whereas those with exact escape velocity reach infinity with zero final speed.

Solved problems on escape velocity are included in the Gravitation Practice Paper.

Binding Energy and Escape Energy

Binding energy is the minimum energy required to remove a particle from the gravitational field of a massive body. It is numerically equal to the magnitude of the gravitational potential energy at the body's surface.

• If total energy is negative, the object is bound to the gravitating body
• If total energy is zero or positive, the object can escape the gravitational field

Escape energy is supplied as kinetic energy to achieve escape velocity, thereby transforming the total mechanical energy to zero at infinity.

Kepler’s Laws of Planetary Motion

Kepler’s laws describe the motion of planets and satellites under gravitational force:

• The orbit of a planet is an ellipse with the Sun at one focus
• A line joining a planet to the Sun sweeps out equal areas in equal times
• The square of the orbital period is proportional to the cube of the semi-major axis

Mathematically, $T^2 \propto a^3$, where $T$ is the period and $a$ is the average radius of the orbit.

Satellite Motion and Orbital Velocity

A satellite is a body that revolves around a planet under the influence of the planet’s gravitational force. The motion of a satellite is stable if its orbit’s center coincides with that of the planet.

The required orbital velocity for a satellite at radius $r$ from the Earth's center is $v_o = \sqrt{\dfrac{GM}{r}}$. At the Earth’s surface, this value is around $8$ km/s.

The time period of revolution is given by $T = 2\pi \sqrt{\dfrac{r^3}{GM}}$. The total energy of a satellite in orbit is negative, indicating a bound system.

Practice questions on satellite motion can be found in Gravitation Mock Test.

Polar Satellite and Weightlessness

Polar satellites move in orbits which pass over Earth's poles and offer global coverage. The height of these satellites is typically 500–600 km above the Earth's surface, with an orbital period of about 100 minutes.

Objects inside satellites experience weightlessness since both satellite and object are in free fall, subject only to gravity. This results in apparent weight being zero.

Additional practice material is available in Gravitation Mock Test 1 and Gravitation Mock Test 2.

Common Features of Gravitational Force

• Always attractive and acts along the line joining centers
• Obeys the inverse-square law
• Central and conservative force
• Affects all objects with mass regardless of their state

Key Formulas and Units in Gravitation

For comprehensive practice, utilize resources like Gravitation Important Questions and Gravitation Practice Paper.