A-Level CAIE Mathematics AS 4.1 Forces and equilibrium Question Bank

A uniform solid cone has weight 5 N and base radius $0.1 \mathrm{~m} . A B$ is a diameter of the base of the cone. The cone is held in equilibrium, with A in contact with a rough horizontal surface and A B vertical, by a force applied at B. This force has magnitude 3 N and acts parallel to the axis of the cone (see diagram). Calculate the height of the cone.

A uniform rectangular block has a square base A B C D with $A B=B C=0.4 \mathrm{~m}$. The height of the block is $h \mathrm{~m}$. The block is placed with its base on a rough plane inclined at $30^{\circ}$ to the horizontal. The block does not slide. It is given that the block is on the point of toppling when the diagonal A C lies along a line of greatest slope. Calculate h.

A crate of mass 500 kg is being pulled along rough horizontal ground by a horizontal rope attached to a winch. The winch produces a constant pulling force of 2500 N and the crate is moving at constant speed. Find the coefficient of friction between the crate and the ground.

Coplanar forces of magnitudes $40 \mathrm{~N}, 32 \mathrm{~N}, P \mathrm{~N}$ and 17 N act at a point in the directions shown in the diagram. The system is in equilibrium. Find the values of P and $\theta$.

A small smooth ring R of mass 0.2 kg is threaded onto a light inextensible string A R B. The two ends of the string are attached to points A and B on a sloping roof inclined at $45^{\circ}$ to the horizontal. A horizontal force of magnitude $P \mathrm{~N}$, acting in the plane A R B, is applied to the ring. The section B R of the string is perpendicular to the roof and the section A R of the string is inclined at $70^{\circ}$ to the horizontal (see diagram). The system is in equilibrium. Find the tension in the string and the value of P.

A smooth ring R of mass $m \mathrm{~kg}$ is threaded on a light inextensible string A R B. The ends of the string are attached to fixed points A and B with A vertically above B. The string is taut and angle $A R B=90^{\circ}$. The angle between the part A R of the string and the vertical is $45^{\circ}$. The ring is held in equilibrium in this position by a force of magnitude 2.5 N , acting on the ring in the direction B R (see diagram). Calculate the tension in the string and the mass of the ring.

Three coplanar forces of magnitudes $F \mathrm{~N}, 20 \mathrm{~N}$ and 30 N act at a point P, as shown in the diagram. The resultant of the three forces acts in a direction perpendicular to the force of magnitude $F \mathrm{~N}$. Find the value of F.

A particle of mass 0.2 kg is resting in equilibrium on a rough plane inclined at $20^{\circ}$ to the horizontal.

A particle of mass 2 kg is initially at rest on a rough horizontal plane. A force of magnitude 10 N is applied to the particle at $15^{\circ}$ above the horizontal. It is given that 10 s after the force is applied, the particle has a speed of $3.5 \mathrm{~m} \mathrm{~s}^{-1}$.

Coplanar forces of magnitudes $7 \mathrm{~N}, 6 \mathrm{~N}$ and 8 N act at a point in the directions shown in the diagram. Given that $\sin \alpha=\frac{3}{5}$, find the magnitude and direction of the resultant of the three forces.