A marble rolls from the top of a big sphere, as shown in the diagram.

$\mathrm{b}.$ Deduce that at that instant, the normal reaction force on the marble from the sphere is given by $N=m g (3 \cos \theta -2)$.

A marble rolls from the top of a big sphere, as shown in the diagram.

Hence, determine the angle $\theta$ at which the marble loses contact with the sphere.

Consider two spherical bodies of mass $16 M$ and $M$ as in the diagram.

There is a point $\mathrm{P}$ somewhere on the line joining the masses where the gravitational field strength is zero.

$\mathrm{a}.$ Determine the distance of point $\mathrm{P}$ from the centre of the bigger mass in terms of $d$, the centre-to-centre distance separating the two bodies.

Consider two spherical bodies of mass $16 M$ and $M$ as in the diagram.

There is a point $\mathrm{P}$ somewhere on the line joining the masses where the gravitational field strength is zero.

$\mathrm{b}.$ Draw a graph to show the variation of the gravitational field strength $g$ due to the two masses with the distance $x$ from the centre of the larger mass.

Consider two spherical bodies of mass $16 M$ and $M$ as in the diagram.

There is a point $\mathrm{P}$ somewhere on the line joining the masses where the gravitational field strength is zero.

$\mathrm{c}.$ A small point mass $m$ is placed at $\mathrm{P}$.

$\mathrm{i}.$ State the force on $m$.

Consider two spherical bodies of mass $16 M$ and $M$ as in the diagram.

There is a point $\mathrm{P}$ somewhere on the line joining the masses where the gravitational field strength is zero.

$\mathrm{c}.$ A small point mass $m$ is placed at $\mathrm{P}$.

$\mathrm{ii}.$ The small mass $m$ is slightly displaced to the left of $\mathrm{P}$. State and explain whether the net force on the point mass will be directed to the left or to the right.

Consider two spherical bodies of mass $16 M$ and $M$ as in the diagram.

There is a point $\mathrm{P}$ somewhere on the line joining the masses where the gravitational field strength is zero.

Describe qualitatively the motion of the point mass after it has been displaced to the left of $\mathrm{P}$.

A satellite is in a circular orbit around a planet of mass $M$, as shown in the diagram.

$\mathrm{a}.\mathrm{i}.$ On a copy of the diagram draw arrows to represent the velocity and acceleration of the satellite.

A satellite is in a circular orbit around a planet of mass $M$, as shown in the diagram.

$\mathrm{a}.\mathrm{ii}.$ Explain why the satellite has acceleration even though its speed is constant.