A solid sphere of mass ‘\(M\)’ and radius ‘\(a\)’ is surrounded by a uniform concentric spherical shell of thickness \(2a\) and mass \(2M\). The gravitational field at distance ‘\(3a\)’ from the centre will be:
1. \(\frac{GM}{9a^2}\)
2. \(\frac{2GM}{9a^2}\)
3. \(\frac{2GM}{3a^2}\)
4. \(\frac{GM}{3a^2}\)

A test particle is moving in a circular orbit in the gravitational field produced by a mass density \(\rho_{(r)}=\frac{K}{r^2}\). Identify the correct relation between the radius \(R\) of the particle's orbit and its period \(T\):

The mass density of a planet of radius \(R\) varies with the distance \(r\) from its centre as \(\rho(r)=\rho_0\left(1-\frac{r^2}{R^2}\right) \) Then the gravitational field is maximum at : 
1. \( r=\sqrt{\frac{3}{4}} R \)
2. \( r=\sqrt{\frac{5}{9}} R \)
3. \( r=R \)
4. \( r=\frac{1}{\sqrt{3}} R \)

The acceleration due to gravity on the earth's surface at the poles is \(g\) and the angular velocity of the earth about the axis passing through the pole is \(\omega\). An object is weighed at the equator and at a height \(h\) above the poles by using a spring balance. If the weights are found to be same, then \(h\) is: (\(h<<R\), where \(R\) is the radius of the earth)
1. \( \frac{R^2 \omega^2}{8 g} \)
2. \(\frac{R^2 w^2}{4 g} \)
3. \(\frac{R^2 w^2}{g} \)
4. \(\frac{R^2 \omega^2}{2 g}\)