Gravitation

Early in our lives, we become aware of the tendency of all material objects to be attracted towards the Earth. Anything thrown up falls down towards the earth, going uphill is lot more tiring than going downhill, raindrops from the clouds above fall towards the earth and there are many other such phenomena. Historically it was the Italian Physicist Galileo (1564-1642) who recognised the fact that all bodies, irrespective of their masses, are accelerated towards the earth with a constant acceleration. It is said that he made a public demonstration of this fact. To find the truth, he certainly did experiments with bodies rolling down inclined planes and arrived at a value of the acceleration due to gravity which is close to the more accurate value obtained later. A seemingly unrelated phenomenon, observation of stars, planets and their motion has been the subject of attention in many countries since the earliest of times. Observations since early times recognised stars which appeared in the sky with positions unchanged year after year. The more interesting objects are the planets which seem to have regular motions against the background of stars. The earliest recorded model for planetary motions proposed by Ptolemy about 2000 years ago was a ‘geocentric’ model in which all celestial objects, stars, the Sun and the planets, all revolved around the Earth. The only motion that was thought to be possible for celestial objects was motion in a circle. Complicated schemes of motion were put forward by Ptolemy in order to describe the observed motion of the planets. The planets were described as moving in circles with the centre of the circles themselves moving in larger circles. Similar theories were also advanced by Indian astronomers some 400 years later. However a more elegant model in which the Sun was the centre around which the planets revolved – the ‘heliocentric’ model – was already mentioned by Aryabhatta (5th century A.D.) in his treatise. A thousand years later, a Polish monk named Nicolas Copernicus (1473-1543)

Summary Points to ponder Exercises

proposed a definitive model in which the planets moved in circles around a fixed central Sun. His theory was discredited by the church, but notable amongst its supporters was Galileo who had to face prosecution from the state for his beliefs. It was around the same time as Galileo, a nobleman called Tycho Brahe (1546-1601) hailing from Denmark, spent his entire lifetime recording observations of the planets with the naked eye. His compiled data were analysed later by his assistant Johannes Kepler (1571-1640). He could extract from the data three elegant laws that now go by the name of Kepler’s laws. These laws were known to Newton and enabled him to make a great scientific leap in proposing his universal law of gravitation.

The three laws of Kepler can be stated as follows: 1. Law of orbits : All planets move in elliptical orbits with the Sun situated at one of the foci

of the ellipse (Fig. 7.1a). This law was a deviation from the Copernican model which allowed only circular orbits. The ellipse, of which the circle is a special case, is a closed curve which can be drawn very simply as follows. Select two points $F_1$ and $F_2$. Take a length of a string and fix its ends at $F_1$ and $F_2$ by pins. With the tip of a pencil stretch the string taut and then draw a curve by moving the pencil keeping the string taut throughout. (Fig. 7.1(b)) The closed curve you get is called an ellipse. Clearly for any point T on the ellipse, the sum of the distances from $F_1$ and $F_2$ is a constant. $F_1$, $F_2$ are called the focii. Join the points $F_1$ and $F_2$ and extend the line to intersect the ellipse at points P and A as shown in Fig. 7.1(b). The midpoint of the line PA is the centre of the ellipse O and the length PO = AO is called the semi-major axis of the ellipse. For a circle, the two focii merge into one and the semi-major axis becomes the radius of the circle. 2. Law of areas : The line that joins any planet to the Sun sweeps equal areas in equal intervals of time (Fig. 7.2). This law comes from the observations that planets appear to move slower when they are farther from the Sun than when they are nearer.

Planet a T Q

Mercury 5.79 0.24 2.95 Venus 10.8 0.615 3.00 Earth 15.0 1 2.96 Mars 22.8 1.88 2.98 Jupiter 77.8 11.9 3.01 Saturn 143 29.5 2.98 Uranus 287 84 2.98 Neptune 450 165 2.99

The law of areas can be understood as a consequence of conservation of angular momentum which is valid for any central force. A central force is such that the force on the planet is along the vector joining the Sun and the planet. Let the Sun be at the origin and let the position and momentum of the planet be denoted by $\vec{r}$ and $\vec{p}$ respectively. Then the area swept out by the planet of mass $m$ in time interval $\Delta t$ is (Fig. 7.2) $\Delta \vec{A}$ given by

$$\Delta \vec{A}=\frac{1}{2}(\vec{r}\times \vec{v}\Delta t)(7.1)$$ Hence

$$\Delta \vec{A}/\Delta t=\frac{1}{2}(\vec{r}\times \vec{p})/m,(\text{since }\vec{v}=\vec{p}/m)=\vec{L}/(2m)(7.2)$$ where $\vec{v}$ is the velocity, $\vec{L}$ is the angular momentum equal to $(\vec{r}\times \vec{p})$. For a central force, which is directed along $\vec{r}$, $\vec{L}$ is a constant

as the planet goes around. Hence, $\Delta \vec{A}/\Delta t$ is a constant according to the last equation. This is

the law of areas. Gravitation is a central force and hence the law of areas follows.

Example 7.1 Let the speed of the planet at the perihelion P in Fig. 7.1(a) be $v_P$ and the Sun-planet distance SP be $r_P$. Relate $\{r_P,v_P\}$ to the corresponding quantities at the aphelion $\{r_A,v_A\}$. Will the planet take equal times to traverse BAC and CPB ?

Answer The magnitude of the angular momentum at P is $L_P=m_p{r}_p{v}_p$, since inspection tells us that ${\vec{r}}_p$ and ${\vec{v}}_p$ are mutually perpendicular. Similarly, $L_A=m_p{r}_A{v}_A$. From angular momentum conservation $$m_p{r}_p{v}_p=m_p{r}_A{v}_A$$

or