Comprehensive Study Notes on Universal Gravitation

Newton’s Law of Gravitation

Newton’s law of gravitation was formulated by Isaac Newton in 1665. It states that the force of attraction between two masses, $m_1$ and $m_2$, separated by a distance $r$, is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This is expressed by the formula $F = \frac{Gm_1m_2}{r^2}$. In this equation, $G$ is known as the universal gravitational constant and its value is approximately $6.67 \times 10^{-11}\,Nm^{-2}$.

Gravitational Field Intensity

Gravitational field intensity at a specific point is defined as the gravitational force per unit mass placed at that point. For the Earth, the gravitational field intensity is denoted as $g$. The mathematical representation for field intensity is $\vec{I} = \frac{\vec{F}}{m}$, where the test mass $m$ is considered to be exceptionally small. This ensures that the test mass itself does not significantly alter the gravitational field it is measuring.

Gravitational Potential ($V_g$)

Gravitational potential at a particular point is the amount of work done to bring a unit mass from infinity to that point under the influence of the gravitational field produced by a given mass $M$. The potential is represented by the formula $V_g = -\frac{GM}{r}$. The negative sign indicates that the work is being done by the gravitational field or that the potential represents an attractive system.

Potential and Field for Mass Distributions

For a system of discrete mass distributions, the total gravitational potential is the algebraic sum of individual potentials, given as $V = V_1 + V_2 + V_3 + \dots$, which can be expressed in summation notation as $V = \sum_{i=1}^N V_i$. Similarly, the total gravitational field intensity is the vector sum of individual intensities: $\vec{I} = \vec{I}_1 + \vec{I}_2 + \vec{I}_3 + \dots + \vec{I}_N$, or $\vec{I} = \sum_{i=1}^N \vec{I}_i$.

In the case of continuous mass distributions, the total potential and field intensity are determined through integration. The total potential is given by $V = \int dV$, where $(dV)$ is the potential due to an elementary mass $dM$. The total field intensity is given by $\vec{I} = \int d\vec{I}$, where $d\vec{I}$ is the field intensity generated by the elementary mass $dM$.

Gravitational Potential Energy ($U_g$)

The gravitational potential energy of a two-mass system is defined as the amount of work done to bring a mass $m$ from infinity to a point $P$ under the influence of the gravitational field of a primary mass $M$. It is calculated as $U_g = -\frac{GMm}{r}$. It is important to note the relationship between gravitational potential energy and gravitational potential: $U_g = V_g \times m$.

In general terms, the gravitational potential energy of a system is the work done against the gravitational force in order to assemble the system from its reference configuration. For mass systems, the standard reference configuration is one where there is infinite mutual separation between the masses.

Gravitational Field Intensity of a Ring

For a ring of radius $R$ and mass $M$, the gravitational field intensity at any point $P$ on the axial line at a distance $x$ from the centre of the ring is given by $E_g = \frac{GMx}{(R^2 + x^2)^{3/2}}$. This field is always directed towards the centre of the ring. At the centre of the ring, the field intensity is at its minimum, which is $E_g = 0$. The value of $E_g$ reaches its maximum when the distance is related to the radius by $x = \frac{R}{\sqrt{2}}$.

Relation Between Field and Potential

There is a direct mathematical relationship between the gravitational field intensity and the gravitational potential, expressed as the negative gradient of the potential. Specifically, $\vec{I} = -\frac{dV}{dr}$. In Cartesian coordinates, this field component is broken down as $\vec{I} = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$. Alternatively, the potential difference can be written in integral form as $dV = -\vec{I} \cdot d\vec{r}$.

Work Done and Gravitational Force

The work done against gravitational force to change the configuration of a system is equal to the difference between the potential energy in the final configuration and the potential energy in the initial configuration. The equation is $W_{\text{against gravitational force}} = U_2 - U_1$. Conversely, the work done by the gravitational force is the negative of this value: $W_{\text{by gravitational force}} = -(U_2 - U_1)$.

Variation of Gravity with Height ($h$)

The acceleration due to gravity changes as one moves away from the Earth's surface. The primary formula is $g' = g\left[\frac{R}{R + h}\right]^2$, which is applicable when the height $h$ is significant (specifically if $h > \frac{R}{10}$). For smaller heights where $h < \frac{R}{10}$, the approximation $g' = g\left(1 - \frac{2h}{R}\right)$ is used. It should be noted that the value of $g$ never actually becomes zero with height, but rather approaches zero ($g \rightarrow 0$) as height approaches infinity ($h \rightarrow \infty$).

Variation of Gravity with Depth ($d$)

As one moves below the surface of the Earth, the acceleration due to gravity changes according to the linear relationship $g' = g\left(1 - \frac{d}{R}\right)$. Here, $g$ represents the acceleration due to gravity at the Earth's surface. As the depth increases, the effective mass attracting the body decreases, leading to a decrease in $g'$.

Variation of Gravity with Rotation and Latitude

The rotation of the Earth affects the apparent acceleration due to gravity because of centrifugal forces. This variation is represented by the formula $g' = g\left(1 - \frac{R\omega^2 \cos^2(\lambda)}{g}\right)$. This effect means that the acceleration due to gravity is at its maximum at the poles and at its minimum at the equator.

Escape Velocity ($v_e$)

Escape velocity is the minimum velocity required for a mass to escape from the gravitational field of a planet or the Earth. If a mass is launched with a velocity greater than or equal to the escape velocity, it will never return to the planet. The formula for escape velocity is $v_e = \sqrt{\frac{2GM}{R}}$.

Planetary Motion and Orbital Velocity

Orbital velocity ($v_o$) is the speed at which a planet or satellite moves in its orbit. It is derived from the requirement that the gravitational force provides the necessary centripetal force: $\frac{GMm}{r^2} = \frac{mv_o^2}{r}$, which simplifies to $v_o = \sqrt{\frac{GM}{r}}$.

The orbital time period ($T$) is defined as $T = \frac{2\pi r}{v_o}$. Substituting the orbital velocity, the square of the time period can be expressed as $T^2 = \frac{4\pi^2 r^3}{GM}$.

The energy components of an orbiting body are as follows:

1. Kinetic Energy ($KE$): $KE = \frac{1}{2}m v_o^2 = \frac{GMm}{2r}$.
2. Potential Energy ($PE$): $PE = -\frac{GMm}{r}$.
3. Net Energy ($E$): $E = KE + PE = -\frac{GMm}{2r}$.

Kepler’s Laws of Planetary Motion

Kepler's First Law (Law of Orbits) states that all planets revolve around the Sun in elliptical orbits, with the Sun located at one of the two foci of the ellipse.

Kepler’s Second Law (Law of Areas) states that a radial line connecting a planet to the Sun sweeps out equal areas in equal intervals of time. This law is a direct consequence of the conservation of angular momentum because the torque about the axis of rotation is zero. The areal velocity is constant: $\frac{dA}{dt} = \frac{L}{2m} = \text{constant}$. This implies that $I_1\omega_1 = I_2\omega_2$, or $(mr_1^2)\frac{v_1}{r_1} = (mr_2^2)\frac{v_2}{r_2}$, simplifying to $v_1r_1 = v_2r_2$. Consequently, $\frac{v_{\text{perihelion}}}{v_{\text{aphelion}}} = \frac{r_{\text{aphelion}}}{r_{\text{perihelion}}}$. A planet moves faster when it is closer to the sun.

Kepler’s Third Law (Law of Periods) states that the square of the time period of a planet is proportional to the cube of the semi-major axis of its orbit ($T^2 \propto a^3$) or the cube of its mean orbital radius ($T^2 \propto r^3$).

If the eccentricity of the orbit is $e$, the relationship between the distance and the semi-major axis is given by:

• $r_{\text{aphelion}} = r_{\text{max}} = a(1 + e)$
• $r_{\text{perihelion}} = r_{\text{min}} = a(1 - e)$

Weightlessness in a Satellite

In a satellite, the net force towards the center is the resultant of gravitational force and the normal contact force ($N$). The equation of motion is $\frac{GMm}{r^2} - N = m\frac{v^2}{r}$. Since the orbital acceleration is provided solely by the gravitational field ($\frac{v^2}{r} = \frac{GM}{r^2}$), the equation becomes $\frac{GMm}{r^2} - N = m\left(\frac{GM}{r^2}\right)$. This results in $N = 0$, meaning the surface of the satellite exerts no force on the body, resulting in apparent weightlessness.

Gravitational Potential of Specific Shapes

For a ring of mass $M$ and radius $R$, the gravitational potential at any point on its axis at distance $x$ is $V = -\frac{GM}{\sqrt{R^2 + x^2}}$. For this ring, the potential at the center is $V = -\frac{GM}{R}$.

For a spherical shell (hollow sphere):

• Inside and on surface ($x \le R$): $V_{\text{in}} = V_{\text{sur}} = -\frac{GM}{R}$.
• Outside ($x > R$): $V_{\text{out}} = -\frac{GM}{x}$.

For a solid sphere of radius $R$:

• Inside ($0 \le x \le R$): $V_{\text{in}} = -\frac{GM(3R^2 - x^2)}{2R^3}$. Specifically, at the center ($x=0$), $V = -\frac{3GM}{2R}$.
• On the surface ($x = R$): $V_{\text{sur}} = -\frac{GM}{R}$.
• Outside ($x > R$): $V_{\text{out}} = -\frac{GM}{x}$.

Gravitational Field Intensity of Specific Shapes

For a disc of mass $M$ and radius $R$, the gravitational field intensity at a point $P$ on the axis at distance $x$ is $E = \frac{2GM}{R^2} \left[1 - \frac{x}{\sqrt{R^2 + x^2}}\right]$, which can also be written as $E = \frac{2GM}{R^2} (1 - \cos(\theta))$.

For a solid sphere:

• Inside ($x < R$): $E_{\text{in}} = \frac{GMx}{R^3}$.
• On surface ($x = R$): $E_{\text{sur}} = \frac{GM}{R^2}$.
• Outside ($x > R$): $E_{\text{out}} = \frac{GM}{x^2}$.

For a hollow sphere:

• Inside ($x < R$): $E_{\text{in}} = 0$.
• On surface ($x = R$): $E_{\text{sur}} = \frac{GM}{R^2}$.
• Outside ($x > R$): $E_{\text{out}} = \frac{GM}{x^2}$.