Show that the critical velocity of body revolving in cercular orbit very close to the surface of a planet of radiuis R and meandencity.
To show the critical velocity of a body revolving in a circular orbit very close to the surface of a planet, we can make use of the gravitational force and centripetal force acting on the body.
1. Consider a body of mass m revolving in a circular orbit very close to the surface of a planet of radius R and mean density ρ.
2. According to Newton's law of universal gravitation, the gravitational force between the body and the planet is given by:
F_gravity = G * (m * M) / r^2
where G is the gravitational constant, M is the mass of the planet, and r is the distance between the body and the center of the planet.
3. At the critical velocity, the centripetal force acting on the body is equal to the gravitational force. The centripetal force is given by:
F_centripetal = m * (v_critical)^2 / r
where v_critical is the critical velocity of the body.
4. Equating the gravitational force and the centripetal force:
G * (m * M) / r^2 = m * (v_critical)^2 / r
5. Simplify by canceling out the mass m:
G * M / r = (v_critical)^2
6. Rearrange the equation to solve for the critical velocity:
v_critical = √(G * M / r)
So, the critical velocity of a body revolving in a circular orbit very close to the surface of a planet is given by the square root of the product of the gravitational constant, the mass of the planet, and the reciprocal of the distance between the body and the center of the planet.