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GRAVITATION 185

Table 8.1 Data from measurement of as the planet goes around. Hence, ∆ A /∆t is a
planetary motions given below constant according to the last equation. This is
confirm Kepler’s Law of Periods the law of areas. Gravitation is a central force
10
(a ≡ Semi-major axis in units of 10 m. and hence the law of areas follows.
T ≡ Time period of revolution of the planet
in years(y).
2
3
Q ≡ The quotient ( T /a ) in units of t Example 8.1 Let the speed of the planet
10 y m .) at the perihelion P in Fig. 8.1(a) be v P and
-3
-34
2
the Sun-planet distance SP be r P . Relate
Planet a T Q {r P , v P } to the corresponding quantities at
the aphelion {r A, v A }. Will the planet take
Mercury 5.79 0.24 2.95 equal times to traverse BAC and CPB ?
Venus 10.8 0.615 3.00
Earth 15.0 1 2.96 Answer The magnitude of the angular
Mars 22.8 1.88 2.98 momentum at P is L p = m p r p v p , since inspection
Jupiter 77.8 11.9 3.01
tells us that r p and v p are mutually
Saturn 143 29.5 2.98 = m p r A v A . From
Uranus 287 84 2.98 perpendicular. Similarly, L A
Neptune 450 165 2.99 angular momentum conservation
Pluto* 590 248 2.99 m p r p v p = m p r A v A
The law of areas can be understood as a v r
p A
consequence of conservation of angular or = t
momentum whch is valid for any central force . v A r p
A central force is such that the force on the Since r A > r p , v p > v A .
planet is along the vector joining the Sun and The area SBAC bounded by the ellipse and
the planet. Let the Sun be at the origin and let the radius vectors SB and SC is larger than SBPC
the position and momentum of the planet be in Fig. 8.1. From Kepler’s second law, equal areas
denoted by r and p respectively. Then the area are swept in equal times. Hence the planet will
swept out by the planet of mass m in time take a longer time to traverse BAC than CPB.
interval ∆t is (Fig. 8.2) ∆A given by
8.3 UNIVERSAL LAW OF GRAVITATION
∆A = ½ (r × v∆t) (8.1)
Hence Legend has it that observing an apple falling
from a tree, Newton was inspired to arrive at an
∆A /∆t =½ (r × p)/m, (since v = p/m) universal law of gravitation that led to an
= L / (2 m) (8.2) explanation of terrestrial gravitation as well as
where v is the velocity, L is the angular of Kepler’s laws. Newton’s reasoning was that
momentum equal to ( r × p). For a central the moon revolving in an orbit of radius R was
force, which is directed along r, L is a constant m
subject to a centripetal acceleration due to
earth’s gravity of magnitude
Johannes Kepler
(1571–1630) was a 2 2
V 4π R m
scientist of German a m = = 2 (8.3)
origin. He formulated R m T
the three laws of where V is the speed of the moon related to the
planetary motion based
π
on the painstaking time period T by the relation V = 2 R m /T. The
observations of Tycho time period T is about 27.3 days and R m was
Brahe and coworkers. Kepler himself was an already known then to be about 3.84 × 10 m. If
8
assistant to Brahe and it took him sixteen long we substitute these numbers in Eq. (8.3), we
years to arrive at the three planetary laws. He get a value of a much smaller than the value of
is also known as the founder of geometrical m
acceleration due to gravity g on the surface of
optics, being the first to describe what happens
to light after it enters a telescope. the earth, arising also due to earth’s gravitational
attraction.
* Refer to information given in the Box on Page 182

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