Two forces, each of size $8 \mathrm{N},$ have a resultant of $13 \mathrm{N}.$
The two given forces of magnitude $8 \mathrm{N}$ act on a particle of mass $m \mathrm{kg},$ which remains at rest on a horizontal surface with no friction. The normal contact force between the surface and the particle has magnitude $7 \mathrm{N}.$ Find $m$ and the acute angle that one of the $8 \mathrm{N}$ forces makes with the surface.

A particle $P$ is in equilibrium on a smooth horizontal table under the action of four horizontal forces of magnitudes $6 \mathrm{N}, 5 \mathrm{N}, F \mathrm{N}$ and $F \mathrm{N}$ acting in the directions shown. Find the values of $\alpha$ and $F.$

A particle $P$ of weight $21 \mathrm{N}$ is attached to one end of each of two light inextensible strings, $S_1$ and $S_2,$ of lengths $0.52 \mathrm{m}$ and $0.25 \mathrm{m}$ respectively. The other end of $S_1$ is attached to a fixed point $A,$ and the other end of $S_2$ is attached to a fixed point $B$ at the same horizontal level as $A.$ The particle $P$ hangs in equilibrium at a point $0.2 \mathrm{m}$ below the level of $AB$ with both strings taut (see diagram). Find the tension in $S_1$ and the tension in $S_2.$