State the Kelper's law which is represented by the relation $r^{3} \propto T^{2}$

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Solution

Third law of Kepler states that square of time period of revolution of a planet is directly proportional to the cube of semi-major axis of the elliptical path on which it is revolving. In case of circular path, square of time period of revolution of a planet is directly proportional to the cube of radius of revolution.

$T^{2} \propto r^{3}$

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State the Kepler's law which is represented by the relation $r^{3} \propto T^{2}$ .

Q. Imagine a universe in which the law of gravitation has the inverse - first - power form

\to F = \frac{G M m}{r}

(a) Show that the speed of a planet in a circular orbit is independent of the orbit radius. (b) Show that in such a universe Kelper's third law would take the form

\frac{T_{1}}{T_{2}} = \frac{r_{1}}{r_{2}} .

Q. A partical of mass m moves along a straight line with constant speed in the x - direction , a distance b from the x - axis (Fig. 4.17). Show that Kelper's second law is satisfied by demonstrating that the two shaded triangles in the figure have the same area when

t_{4} - t_{3} = t_{2} - t_{1} .

Q. Can Kelper's laws be applied to atomic structure as electrons are also bound to nucleus like planets to the sun ?

According to one of the Kepler's laws of planetary motion :
(a) $r^{2} \propto T^{3}$
(b) $r \propto T^{2}$
(c) $r^{3} \propto T^{2}$
(d) $r^{3} \propto \frac{1}{T^{2}}$

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State the Kelper's law which is represented by the relation r3∝T2

State the Kelper's law which is represented by the relation $r^{3} \propto T^{2}$