## Gravitation - Kepler's Laws Of Planetary Motion

Numerical Problems with Solutions

**Question 1: The planet Neptune travels around the Sun with a period of 165 years. Show that radius of orbit is approximately 30 times that of the Earth's orbit, considering their orbits circular.**

Answer:

Let T₁ = T(earth) = 1 year

T₂ = T(Neptune) = 165 years

Let R₁ and R₂ be the radii of the circular orbits the Earth and the Neptune respectively.

∴

T₁² R₁³ --- = ---- T₂² R₂³ R₁³ × T₂² ∴ R₂³ = ---------- T₁² R₁³ × 165² R₂³ = ---------- = 165²R₁³ 1² R₂ ≈ 30R₁

**Question 2: A planet of mass 'm' moves around the sun of mass M in an elliptical orbit. The maximum and minimum distance of the planet from the sun are r₁ and r₂ respectively. The time period of the planet is proportional to**

(a) r₁³⸍²

(b) r₂³⸍²

(c) (r₁ + r₂)³⸍²

(d) (r₁ - r₂)³⸍²

Answer: (c) (r₁ + r₂)³⸍²

**Question 3: If the distance between the earth and the sun were half its present value, the number of days in a year would have been**

(a) 64.5

(b) 129

(c) 182.5

(d) 730

Answer: (b) 129

**Question 4: The time period of the satellite of the earth is 5 hr. If the separation between earth and satellite is increased to 4 times the previous value, then what will be the new time period of satellite.**

Answer:

T₁² R₁³ --- = ---- T₂² R₂³ ∴ T₂² = 64 × 25 ∴ T₂ = 40 Hours

☛See also:

Laws Of Motion (Revision Test)

Gravitation - Understanding Mass and Weight

Gravitation - Kepler's Laws

Centre Of Mass and Rotational Motion (Assignment)

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