Q4. Use NCERT’s Table 7.1 to check the third law for Saturn: a = 143 × 10 m and T = 29.5 years. Compare T/a with the Earth’s (a = 15.0 × 10 m, T = 1 year).
Saturn: 29.5 = 870 and 143 ≈ 2.92 × 10, so T/a ≈ 2.98 × 10 in these units, that is 2.98 × 10 y m. Earth: 1/15.0 ≈ 2.96 × 10, that is 2.96 × 10 y m. They agree to within 1%.
Q5. Two satellites orbit the same planet in circles of radii r and 9r. Find the ratio of their periods.
T ∝ r3/2, so the outer one has 93/2 = 27 times the period of the inner one.
Q6. A planet’s nearest and farthest distances from its star are 2 AU and 6 AU. Find (a) the semi-major axis, (b) the ratio of its speeds at the nearest and farthest points, and (c) its period, if the star has the same mass as the Sun.
(a) a = (2 + 6)/2 = 4 AU. (b) vP/vA = rA/rP = 3. (c) T = 43/2 = 8 years, because T = a (years, AU) holds for any star of the Sun’s mass.