The radius of the Earth's orbit around the Sun is 1.50 × 10¹¹ m. The Earth takes 3.15 × 10⁷ s to complete the orbit. What is the mass of the Sun? [G = 6.67 × 10⁻¹¹ N m² kg⁻²]
Answer (1)
To find the mass of the Sun, we can use Newton's law of universal gravitation and the centripetal force equation. The gravitational force between the Earth and the Sun provides the necessary centripetal force to keep the Earth in orbit.
The gravitational force is given by: F = r 2 G ⋅ M ⋅ m
Where:
F is the gravitational force,
G = 6.67 × 1 0 − 11 N m 2 kg − 2 is the gravitational constant,
M is the mass of the Sun,
m is the mass of the Earth, and
r = 1.50 × 1 0 11 m is the radius of the Earth's orbit.
The centripetal force needed to keep the Earth moving in a circle is: F = r m ⋅ v 2
Where v is the orbital speed of the Earth.
We know v can also be expressed as: v = T 2 π r
Where T = 3.15 × 1 0 7 s is the time taken for one complete orbit (1 year).
Substituting v = T 2 π r into the centripetal force equation, we equate the gravitational and centripetal forces: r 2 G ⋅ M ⋅ m = r m ⋅ ( T 2 π r ) 2
Simplifying gives: G ⋅ M = T 2 4 π 2 ⋅ r 3
Solving for M , the mass of the Sun, we get: M = G ⋅ T 2 4 π 2 ⋅ r 3
Substituting the known values: M = 6.67 × 1 0 − 11 × ( 3.15 × 1 0 7 ) 2 4 × π 2 × ( 1.50 × 1 0 11 ) 3
Calculating this expression will provide the mass of the Sun. Be sure to consistently check units and carry out calculations carefully to find M .
