Suppose you observe a binary system containing a main-sequence star and a brown dwarf. The orbital period of the system is 1 year, and the average separation of the system is 1 AUAU . You then measure the Doppler shifts of the spectral lines from the main-sequence star and the brown dwarf, finding that the orbital speed of the brown dwarf in the system is 22 times greater than that of the main-sequence star.

Complete Question:

How massive is the brown dwarf

Mass of the brown dwarf, M₂ = 4.132 * 10²⁷ kg

Explanation:

Let M₁ = Mass of the main-sequence star

M₂ = Mass of the brown dwarf

v₁ = speed of the main-sequence star

v₂ = speed of the brown dwarf

The average separation of the system r₁ = r₂ = 1 AU

Orbital speed of the brown dwarf is 22 times that of the main - sequence star

That is, v₂ = 22 v₁

Since the sun is a main sequence star

Mass of the main - sequence star = 2 * 10³⁰ kg

Centripetal force, F = Mv²/r

M₁v₁²/r₁² = M₂v₂²/r²

M₁v₁² = M₂v₂²

M₂/M₁ = v₁²/v₂²

M₂/M₁ = (1/22)²

M₂/M₁ = 0.002066

M₂ = 0.002066M₁

M₂ = 0.002066 * 2 * 10³⁰

M₂ = 4.132 * 10²⁷ kg

frequency or wavelenth of em radiation in a vacuum

force is based upon mass and acceleration.the statement is not always true.

it can be explained from newton's second law of motion.

as per newton's second law of motion the rate of change of momentum is directly proportional to the applied force and takes place along the direction of force.

mathematically it can be written as -

                                              ⇒      

here   k is the proportionality constant whose value is 1.

                                                            [a is the acceleration=dv/dt]

hence we see that force not only depends on mass and acceleration but also on velocity and rate of change of mass. force f will depend on mass and acceleration only when mass is constant.

hence the statement that force is based upon mass and acceleration is false.