The distance of the centres of Moon and the Earth is $D$. The mass of the Earth is 81 times the mass of the Moon. At what distance from the centre of the Earth, the gravitational force on a particle will be zero?

The gravitational field in a region is given by $$\mathbf{E}=5 \mathrm{~N} / \mathrm{kg} \hat{\mathbf{i}}+12 \mathrm{~N} / \mathrm{kg} \hat{\mathbf{j}}$$. The change in the gravitational potential energy of a particle of mass $$1 \mathrm{~kg}$$ when it is taken from the origin to a point ($$5 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}$$) is

A geostationary satellite revolves around the Earth in a circular orbit of radius $$4 R$$. Here, $R$ is the radius of the Earth. Then, the time period of another satellite moving in a circular orbit of radius $$2 R$$ is:

A skylab or mass $$m \mathrm{~kg}$$ is first launched from the surface of the earth in a circular orbit of radius $$2 R$$ (from the centre of the earth) and then it is shifted from this circular orbit to another circular orbit of radius $$3 R$$. The minimum energy required to place the lab in the first orbit and to shift the lab from first orbit to the second orbit are