Question

Assume the universe has a constant universal mass density, ρ = ρ U uniform throughout the universe. How much longer or shorter would a year be in this case rather than it currently is with zero density? Ignore friction. Give your answer as a function of density

Answer 1

Let the uniform mass density of the universe be .

The earth revolves around the sun due to a centripetal force acting on it which is nothing but the gravitational force of attraction between the sun and earth.

i.e where m is the mass of the earth, M_s is the mass of the sun,R is the distance between the earth and the sun and G is the gravitational constant.

Since where T₀ is the period of the revolution which is equal to one year, we obtain on substituting the above value in the above equation,

.

If the universe has a uniform mass density , the equivalent mass of the sun .Thus, the equation of motion of the earth around the sun becomes,

. Thus the new time period of the earth's revolution around the sun is

. This is found to be shorter than 1 year.