Differential Equations

2nd Order Equations

Newton’s Law of Universal Gravitation

Page 1
Theory
Page 2
Problem 1

The Force of Gravity and Gravitational Potential

The law of universal gravitation was formulated by Isaac Newton \(\left(1643-1727\right)\) and published in \(1687.\) In accordance with this law, two point masses attract each other with a force that is directly proportional to the masses of these bodies \({m_1}\) and \({m_2},\) and inversely proportional to the square of the distance between them:

\[F = G\frac{{{m_1}{m_2}}}{{{r^2}}}.\]
Isaac Newton (1643−1727)

Figure 1.

Here, \(r\) is the distance between the centers of mass of the bodies, \(G\) is the gravitational constant, whose value found by experiment is \(G =\) \( 6,67 \times {10^{ – 11}}{\large\frac{{{\text{m}^3}}}{{\text{kg} \cdot {\text{s}^2}}}\normalsize}.\)

The force of gravitational attraction is a central force, i.e., directed along a line passing through the centers of the interacting bodies.

The attraction force between two galaxies

Figure 2.

In a system of two bodies (Figure \(2\text{),}\) the attraction force \({\mathbf{F}_{12}}\) of the second body acts on the first body of mass \({m_1}.\) Similarly, the attraction force \({\mathbf{F}_{21}}\) of the first body acts on the second body of mass \({m_2}.\) Both the forces \({\mathbf{F}_{12}}\) and \({\mathbf{F}_{21}}\) are equal and directed along \(\mathbf{r},\) where

\[\mathbf{r} = {\mathbf{r}_2} – {\mathbf{r}_1}.\]

Using the Newton’s second law we can write the following differential equations describing the motion of each body:

\[
{{m_1}\frac{{{d^2}{\mathbf{r}_1}}}{{d{t^2}}} = G\frac{{{m_1}{m_2}}}{{{r^3}}}\mathbf{r},\;\;\;}\kern-0.3pt
{{m_2}\frac{{{d^2}{\mathbf{r}_2}}}{{d{t^2}}} = – G\frac{{{m_1}{m_2}}}{{{r^3}}}\mathbf{r}}
\]

or

\[
{\frac{{{d^2}{\mathbf{r}_1}}}{{d{t^2}}} = G\frac{{{m_2}}}{{{r^3}}}\mathbf{r},\;\;\;}\kern-0.3pt
{\frac{{{d^2}{\mathbf{r}_2}}}{{d{t^2}}} = – G\frac{{{m_1}}}{{{r^3}}}\mathbf{r}.}
\]

It follows from the last two equations that

\[
{\frac{{{d^2}{\mathbf{r}_1}}}{{d{t^2}}} – \frac{{{d^2}{\mathbf{r}_2}}}{{d{t^2}}} }={ G\frac{{{m_2}}}{{{r^3}}}\mathbf{r} + G\frac{{{m_1}}}{{{r^3}}}\mathbf{r},\;\;}\Rightarrow
{\frac{{{d^2}\mathbf{r}}}{{d{t^2}}} = -G\frac{{{m_1} + {m_2}}}{{{r^3}}}\mathbf{r}.}
\]

This differential equation describes the change of the vector \(\mathbf{r}\left( t \right),\) i.e. the relative motion of two bodies under the force of gravitational attraction.

With a large difference in mass of the bodies, we can neglect the smaller body mass in the right side of this equation. For example, the mass of the Sun is \(333,000\) times greater than the mass of the Earth. In this case, the differential equation can be written in a simpler form:

\[\frac{{{d^2}\mathbf{r}}}{{d{t^2}}} = – G\frac{{{M_\text{C}}}}{{{r^3}}}\mathbf{r},\]

where \({M_\text{S}}\) is the mass of the Sun.

The gravitational interaction of bodies takes place through a gravitational field, which can be described by a scalar potential \(\varphi.\) The force acting on a body of mass \(m,\) placed in a field with potential \(\varphi,\) is equal to

\[{\mathbf{F} = m\mathbf{a} }={ – m\,\mathbf{\text{grad}}\,\varphi .}\]

In the case of a point mass \(M,\) the potential of the gravitational field is given by

\[\varphi = – \frac{{GM}}{r}.\]

The latter formula is also valid for distributed bodies with central symmetry, such as a planet or star.

Kepler’s Laws

The basic laws of planetary motion were established by Johannes Kepler \(\left(1571-1630\right)\) based on the analysis of astronomical observations of Tycho Brahe \(\left(1546-1601\right)\). In \(1609,\) Kepler formulated the first two laws. The third law was discovered in \(1619.\) Later, in the late \(17\)th century, Isaac Newton proved mathematically that all three laws of Kepler are a consequence of the law of universal gravitation.

Kepler’s First Law

The orbit of each planet in the solar system is an ellipse, one focus of which is the Sun (Figure \(3\)).

Kepler's First Law

Figure 3.

Kepler's Second Law

Figure 4.

Kepler’s Second Law

The radius vector connecting the Sun and the planet describes equal areas in equal intervals of time. Figure \(4\) shows the two sectors of the ellipse corresponding to the same time intervals. According to Kepler’s second law, the areas of these sectors are equal.

Kepler’s Third Law

The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit:

\[{T^2} \propto {a^3}.\]

The proportionality coefficient is the same for all planets in the solar system. Therefore, for any two planets, one can write the relationship

\[\frac{{T_2^2}}{{T_1^2}} = \frac{{a_2^3}}{{a_1^3}}.\]

Solved Problems

Click on problem description to see solution.

 Example 1

A small cosmic body starts to fall to Earth from rest under the action of gravitational force. The initial distance to the centre of the Earth is equal to \(L.\) Determine the velocity and time of the drop.

Page 1
Theory
Page 2
Problem 1