An earth's satellite near the surface of the earth takes about 90 min per revolution. A satellite orbiting the moon also takes about 90 min per revolution. Then which of the following is true? (\(\rho\)m is the density of the moon and ρe is the density of the earth).
- \(\rho_m<\rho_e\)
- \(\rho_m>\rho_e\)
- \(\rho_m=\rho_e\)
- No conclusion can be made about the densities
The Correct Option is C
Solution and Explanation
This is because the time period of a satellite in orbit depends on the mass of the celestial body it's orbiting and the radius of its orbit. In this case, both satellites have the same time period of 90 minutes per revolution, indicating that the ratio of the cube of the semi-major axis of their orbits (a3) to the sum of their masses (M) is the same.
For Earth's satellite: ae3=constant
For Moon's satellite: am3=constant
Since both satellites have the same time period and the same constants of proportionality, we can equate the two equations: Meae3=Mmam3
Given that the mass of the moon (Mm) is much smaller than the mass of the Earth (Me), the only way for this equation to hold true is if the densities of the moon (ρm) and Earth (ρe) are equal, i.e., ρm=ρe.
The correct option is(C): \(\rho_m=\rho_e\)
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Concepts Used:
Gravitational Potential Energy
The work which a body needs to do, against the force of gravity, in order to bring that body into a particular space is called Gravitational potential energy. The stored is the result of the gravitational attraction of the Earth for the object. The GPE of the massive ball of a demolition machine depends on two variables - the mass of the ball and the height to which it is raised. There is a direct relation between GPE and the mass of an object. More massive objects have greater GPE. Also, there is a direct relation between GPE and the height of an object. The higher that an object is elevated, the greater the GPE. The relationship is expressed in the following manner:
PEgrav = mass x g x height
PEgrav = m x g x h
Where,
m is the mass of the object,
h is the height of the object
g is the gravitational field strength (9.8 N/kg on Earth) - sometimes referred to as the acceleration of gravity.