Forces and Equilibrium

Three coplanar forces of magnitudes $40\text{N}$, $30\text{N}$ and $X\text{N}$ act at a point in the directions shown in the diagram.

Given that the forces are in equilibrium, find the values of $\theta$ and $X$.

A block of mass $12\text{ kg}$ is being pulled by a rope up a rough plane. The plane is inclined at an angle of $20^\circ$ above the horizontal. The rope pulling the block is parallel to a line of greatest slope of the plane. The coefficient of friction between the block and the plane is $0.4$. The acceleration of the block is $2\text{ m s}^{-2}$.

Find the tension in the rope.

Three coplanar forces of magnitudes $P\text{ N}$, $5\text{ N}$ and $10\text{ N}$ act at a point $O$, as shown in the diagram. The resultant of the three forces has magnitude $Q\text{ N}$ and acts in a direction perpendicular to the force of magnitude $P\text{ N}$.

Find the value of $P$ and the value of $Q$.

A particle $P$ of mass $m\text{kg}$ is attached to one end of a light inextensible string of length $2.6\text{m}$. The other end of the string is attached to a fixed point on a horizontal ceiling, and the string is taut. The particle is held in equilibrium by a force of magnitude $35\text{N}$, acting in a vertical plane which is perpendicular to the ceiling and contains the string. The force acts in a direction perpendicular to the string (see diagram). The tension in the string is $T\text{N}$ and the vertical distance of $P$ from the ceiling is $2.4\text{m}$.

Find, in either order, the value of $m$ and the value of $T$.

A particle $P$ of mass $3\text{kg}$ is projected with a speed of $8\text{ms}^{-1}$ up a line of greatest slope of a rough plane inclined at $30^\circ$ to the horizontal. $P$ is projected from a point $A$ on the plane and comes to instantaneous rest at a point $B$ on the plane. $P$ then slides back down the plane. The coefficient of friction between $P$ and the plane is $\frac{1}{12}\sqrt{3}$.

Using an energy method throughout, find the speed of $P$ at the instant it returns to $A$.

Three blocks $P$, $Q$ and $R$, of masses $25\text{ kg}$, $20\text{ kg}$ and $m\text{ kg}$ respectively, are held in equilibrium by three light inextensible strings $OP$, $OQ$ and $OR$. The strings $OP$ and $OR$ both pass over small fixed smooth pulleys $A$ and $B$ respectively, with $P$ and $R$ hanging vertically below the pulleys. The block $Q$ hangs vertically below the point $O$. The angle between $OA$ and the vertical is $30^\circ$ and the angle $BOQ = \alpha^\circ$ (see diagram).

Find the value of $m$ and the value of $\alpha$.

Coplanar forces of magnitudes $45\text{ N}$, $28\text{ N}$, $72\text{ N}$ and $35\text{ N}$ act at a point in the directions shown in the diagram.

Find, in either order, the magnitude and direction of the resultant force.

A particle $P$ of mass $m\text{ kg}$ is in equilibrium on a rough plane inclined at an angle $\theta^\circ$ to the horizontal. The equilibrium of $P$ is maintained by a force of magnitude $8mg\text{ N}$ making an angle $\theta^\circ$ with a line of greatest slope (see diagram). The coefficient of friction between $P$ and the plane is $0.5$ and $P$ is on the point of slipping down the plane.

Find the value of $\theta$.