G RAVITATION · Part 3

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. . (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 and F is a constant. F , F are called the focii. Join the points F and F and extend the line to intersect the ellipse at points P and A as shown in Fig. . (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. . Law of areas : The line that joins any planet to the sun sweeps equal areas in equal intervals of time (Fig. . ). This law comes from the observations that planets appear to move slower when they are farther from the sun than when they are nearer. Fig. . The planet P moves around the sun in an elliptical orbit. The shaded area is the area ∆ A swept out in a small interval of time ∆ t. . Law of periods : The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse traced out by the planet. Table . gives the approximate time periods of revolution of eight * planets around the sun along with values of their semi-major axes. Table . Data from measurement of planetary motions given below confirm Kepler’s Law of Periods (a ≡ Semi-major axis in units of m. ≡ Time period of revolution of the planet in years(y). Q ≡ The quotient ( T /a ) in units of - y m - .) Planet Q Mercury . . . Venus . . . Earth . . Mars . . . Jupiter . . .