Gravitation

Universal Law of Gravitation and Gravitational Potential

The force that governs the motion of planets around the sun, the falling of an apple from a tree, and the very structure of galaxies is the force of gravitation. This fundamental force was first mathematically described by Sir Isaac Newton.

Newton's Universal Law of Gravitation

Newton's Universal 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 their centers.

Let's consider two point masses, m₁ and m₂, separated by a distance r. The gravitational force (F) between them can be expressed as:

F ∝ (m₁ * m₂) / r²

To convert this proportionality into an equation, we introduce a constant of proportionality, G, known as the Universal Gravitational Constant.

F = G * (m₁ * m₂) / r²

Here, G is a universal constant, meaning its value is the same everywhere in the universe, regardless of the nature of the masses or the medium between them.

Properties of the Gravitational Force

The gravitational force exhibits several key properties:

• Attractive: Gravitational force is always attractive; it pulls objects towards each other.
• Central Force: It acts along the line joining the centers of the two masses.
• Universal: It acts between any two objects with mass, anywhere in the universe.
• Weak Force: Compared to other fundamental forces like the electromagnetic force, gravity is very weak. However, it becomes significant when dealing with celestial bodies due to their enormous masses.
• Depends on Mass: The force is directly proportional to the product of the masses.
• Depends on Distance: The force is inversely proportional to the square of the distance between the centers of the masses (inverse square law).
• Action-Reaction Pair: The force exerted by m₁ on m₂ is equal in magnitude and opposite in direction to the force exerted by m₂ on m₁, obeying Newton's third law.

The Universal Gravitational Constant (G)

The value of G was experimentally determined by Henry Cavendish in 1798 using a torsion balance. This experiment is often referred to as "weighing the Earth" because it allowed for the calculation of Earth's mass.

The accepted value of G is approximately:

G = 6.674 × 10^-11 N m² kg^-2

The unit of G can be derived from the formula F = G * (m₁ * m₂) / r². Rearranging for G gives G = (F * r²) / (m₁ * m₂). Substituting the units: (N * m²) / (kg * kg) = N m² kg^-2.

Gravitational Field Strength

The gravitational field strength at a point is defined as the gravitational force experienced by a unit mass placed at that point. It is a vector quantity. For a mass M, the gravitational field strength (g) at a distance r from its center is given by:

g = F / m₀

where m₀ is a small test mass. Using Newton's law of gravitation, F = G * (M * m₀) / r².

Therefore, the gravitational field strength due to mass M at a distance r is:

g = [G * (M * m₀) / r²] / m₀ = G * M / r²

The unit of gravitational field strength is N/kg, which is equivalent to m/s² (the acceleration due to gravity).

Gravitational Potential

Gravitational potential is a scalar quantity that describes the amount of work done per unit mass to move an object from infinity to a specific point in a gravitational field. It is defined such that the potential is zero at an infinite distance from the source mass.

Consider a point mass M. The work done (W) in bringing a unit mass from infinity to a distance r from M is defined as the gravitational potential (V) at that point.

Gravitational force on a test mass m₀ at a distance x from M is F(x) = G * (M * m₀) / x².

To bring the test mass from infinity to distance r, we need to do work against this attractive force. The work done by an external agent is:

W = ∫_∞^r F_external dx

Since F_external = -F_{gravitational} (work done against the force),

W = ∫_∞^r -[G * (M * m₀) / x²] dx

W = -G * M * m₀ ∫_∞^r x^-2 dx

The integral of x^-2 is -x^-1.

W = -G * M * m₀ [-1/x]_∞^r

W = -G * M * m₀ [(-1/r) - (-1/∞)]

Since 1/∞ approaches 0,

W = -G * M * m₀ / r

The gravitational potential (V) at a point is the work done per unit mass.

V = W / m₀

Therefore, the gravitational potential at a distance r from a mass M is:

V(r) = -G * M / r

The unit of gravitational potential is Joules per kilogram (J/kg).

Key Characteristics of Gravitational Potential

• Scalar Quantity: It has magnitude but no direction.
• Negative Value: Gravitational potential is always negative for any finite distance from the source mass. This indicates that the system is bound. To escape the gravitational pull, positive work must be done on the object.
• Zero at Infinity: By definition, the gravitational potential is zero at an infinite distance from the source mass.
• Relationship with Field Strength: Gravitational potential and gravitational field strength are related. The gravitational field strength is the negative gradient of the gravitational potential: g = -dV/dr.

Gravitational Potential Energy

Gravitational potential energy (U) of an object of mass m at a distance r from a larger mass M is the work done to bring mass m from infinity to that point. It is given by the product of the gravitational potential at that point and the mass m.

U = m * V(r)

U = -G * M * m / r

This potential energy is negative, indicating that the two masses are bound together by gravity. To separate them infinitely, positive work must be done on the system.

Example: Calculating Gravitational Force and Potential

Consider two masses, m₁ = 1000 kg and m₂ = 500 kg, placed 10 meters apart. Calculate the gravitational force between them and the gravitational potential at the location of m₂ due to m₁.

Given:

• m₁ = 1000 kg
• m₂ = 500 kg
• r = 10 m
• G = 6.674 × 10^-11 N m² kg^-2

Calculation of Gravitational Force:

F = G * (m₁ * m₂) / r²

F = (6.674 × 10^-11 N m² kg^-2) * (1000 kg * 500 kg) / (10 m)²

F = (6.674 × 10^-11) * (500,000 kg²) / (100 m²)

F = (6.674 × 10^-11) * 5000 N

F = 3.337 × 10^-7 N

The gravitational force between the two masses is 3.337 × 10^-7 Newtons. This is a very small force, as expected for moderate masses.

Calculation of Gravitational Potential at m₂ due to m₁:

V = -G * m₁ / r

V = -(6.674 × 10^-11 N m² kg^-2) * (1000 kg) / (10 m)

V = -(6.674 × 10^-11) * 100 J/kg

V = -6.674 × 10^-9 J/kg

The gravitational potential at the location of m₂ due to m₁ is -6.674 × 10^-9 Joules per kilogram.

Relation between Gravitational Field and Potential

The gravitational field strength (a vector) is related to the gravitational potential (a scalar) through the gradient. In one dimension, for a spherically symmetric mass distribution, the field strength at a distance r is given by the negative derivative of the potential with respect to distance:

g(r) = - dV(r) / dr

Substituting V(r) = -GM/r:

g(r) = - d/dr (-GM/r)

g(r) = GM * d/dr (1/r)

The derivative of 1/r (or r^-1) with respect to r is -r^-2, which is -1/r².

g(r) = GM * (-1/r²)

g(r) = -GM/r²

The negative sign indicates that the gravitational field points radially inward towards the source mass, which is consistent with the attractive nature of gravity. The magnitude of the field is GM/r², as we derived earlier.

Work Done in Moving a Mass in a Gravitational Field

The work done by an external agent in moving a mass m from point A to point B in a gravitational field is given by the change in its gravitational potential energy:

W_external = ΔU = U_B - U_A

W_external = (-GMm/r_B) - (-GMm/r_A)

W_external = GMm * (1/r_A - 1/r_B)

If moving from a larger radius (r_A) to a smaller radius (r_B), r_B < r_A. The term (1/r_A - 1/r_B) will be negative. This implies that the external agent has to do negative work (or the gravitational field does positive work) to move the mass towards the source. This makes sense because the force is attractive, and moving the mass closer means the field is pulling it, doing work.

Conversely, if moving from a smaller radius (r_A) to a larger radius (r_B), r_B > r_A. The term (1/r_A - 1/r_B) will be positive. This implies that the external agent has to do positive work against the gravitational force to move the mass away from the source.

Example: Work Done to Move Earth Away from Sun

Consider the work required to move the Earth from its current orbit to an orbit twice as far from the Sun.

Let M be the mass of the Sun, and m be the mass of the Earth. Let r₁ be the current orbital radius of the Earth, and r₂ = 2r₁ be the new orbital radius.

Work done W = GMm * (1/r₁ - 1/r₂)

W = GMm * (1/r₁ - 1/(2r₁))

W = GMm * (2/2r₁ - 1/2r₁)

W = GMm * (1/2r₁)

W = (GMm) / (2r₁)

This positive work done indicates that energy must be supplied to the Earth-Sun system to move the Earth further away from the Sun.

Gravitational Potential due to a System of Masses

Since gravitational potential is a scalar quantity, the total gravitational potential at a point due to a system of several point masses is simply the algebraic sum of the potentials due to each individual mass.

If we have masses m₁, m₂, ..., m_n at distances r₁, r₂, ..., r_n from a point P, the total potential V at P is:

V = V₁ + V₂ + ... + V_n

V = (-Gm₁/r₁) + (-Gm₂/r₂) + ... + (-Gm_n/r_n)

V = -G * (Σ (m_i/r_i))

This principle of superposition for potential simplifies calculations involving multiple celestial bodies.

Gravitational Potential Energy of a System of Masses

The gravitational potential energy of a system of particles is defined as the work done to assemble the system by bringing the particles from infinity to their final positions. For a system of two particles, it's simply U = -GMm/r.

For a system of multiple particles, we consider the work done to bring each particle from infinity to its position, assuming the others are already in place. The total potential energy is the sum of the potential energies of all possible pairs of particles.

For three particles with masses m₁, m₂, and m₃, at distances r₁₂, r₁₃, and r₂₃ from each other, the total potential energy is:

U_total = U₁₂ + U₁₃ + U₂₃

U_total = (-Gm₁m₂/r₁₂) + (-Gm₁m₃/r₁₃) + (-Gm₂m₃/r₂₃)

U_total = -G * [ (m₁m₂/r₁₂) + (m₁m₃/r₁₃) + (m₂m₃/r₂₃) ]

This represents the total energy stored in the gravitational configuration of the system.

Escape Velocity

Escape velocity is the minimum speed an object must have at the surface of a celestial body (like a planet or moon) to overcome its gravitational pull and escape into space, without any further propulsion.

To escape, the object's total energy (kinetic + potential) must be at least zero (i.e., it reaches infinity with zero speed).

Kinetic Energy (KE) = 1/2 * m * v_e², where m is the object's mass and v_e is the escape velocity.

Potential Energy (PE) at the surface of a planet of mass M and radius R is PE = -GMm/R.

Total Energy = KE + PE = 0

1/2 * m * v_e² + (-GMm/R) = 0

1/2 * m * v_e² = GMm/R

v_e² = 2GM/R

v_e = √(2GM/R)

We also know that the acceleration due to gravity at the surface is g = GM/R². So, GM = gR².

Substituting GM in the escape velocity formula:

v_e = √(2(gR²)/R)

v_e = √(2gR)

The escape velocity is independent of the mass of the escaping object.

Orbital Velocity

Orbital velocity is the speed at which an object orbits a celestial body. For a circular orbit of radius r around a mass M, the gravitational force provides the centripetal force.

Gravitational Force = Centripetal Force

GMm/r² = mv_o²/r

Where m is the orbiting body's mass and v_o is its orbital velocity.

v_o² = GM/r

v_o = √(GM/r)

For an object orbiting at the surface of a planet (r=R), the orbital velocity is v_o = √(GM/R).

Comparing orbital velocity and escape velocity:

v_e = √(2GM/R) = √2 * √(GM/R) = √2 * v_o (at the surface)

This means the escape velocity from the surface is √2 times the orbital velocity at the surface.

Kepler's Laws of Planetary Motion

Johannes Kepler, using Tycho Brahe's meticulous observations, formulated three laws that describe the motion of planets around the Sun. Newton later showed that these laws are a direct consequence of his law of universal gravitation.

1. Kepler's First Law (Law of Orbits)

Every planet moves in an elliptical orbit, with the Sun at one of the foci of the ellipse.

2. Kepler's Second Law (Law of Areas)

A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This implies that the planet moves faster when it is closer to the Sun and slower when it is farther away. This is a consequence of the conservation of angular momentum.

Area swept per unit time = dA/dt = (1/2) * r * (r dθ/dt) = (1/2) * r² * ω = L / (2m), where L is the angular momentum and m is the planet's mass. Since L and m are constant, dA/dt is constant.

3. Kepler's 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. For a circular orbit, the semi-major axis is simply the radius (r).

T² ∝ a³

For elliptical orbits, T² = (4π² / GM) * a³

Where M is the mass of the central body (e.g., the Sun).

This law is fundamental for understanding the relationship between the size of an orbit and the time it takes to complete it. For planets orbiting the same central body, the constant (4π² / GM) is the same for all planets.