Jan Dangerfield, Stuart Haring and, Julian Gilbey Solutions for Exercise 5: EXERCISE 3C

A mass of $5 \mathrm{kg}$ is held in equilibrium by two ropes with tensions of $30 \mathrm{N}$ and $40 \mathrm{N}.$ Find the angles that the ropes make with the vertical.

A mass of $7 \mathrm{kg}$ is held in equilibrium by two ropes. One has tension $20\mathrm{N}$ and acts at $40°$ to the upwards vertical. Find the tension in the other rope and the angle that it makes with the upwards vertical.

A ship is held in place by two ropes with forces $40 \mathrm{N}$ and $35 \mathrm{N},$ as shown in the diagram, which prevent the wind blowing it away. The wind has force $F$ and acts at an angle $\theta$ to the $35 \mathrm{N}$ force, as shown. Find the sizes of $\mathrm{\theta }$ and $F.$

Three ropes pull a boat, which remains in equilibrium. The ropes act due north and on bearings of $100°$ and $210°.$ The one acting north has tension $25 \mathrm{N}.$ Find the tensions in the other ropes.

A box has two ropes holding it in place. It is pushed by a force of $10 \mathrm{N}.$ The angles between the force and the ropes are $120°$ and $150°.$ Find the tensions in the ropes.

An $8 \mathrm{N}$ force, a $9 \mathrm{N}$ force and a $10 \mathrm{N}$ force on an object result in no net force. Find the angle between the $8 \mathrm{N}$ and the $9 \mathrm{N}$ forces.

A particle is held in equilibrium by three forces. Two of the forces have the same size, $F \mathrm{N}.$ Prove that the third force acts along the line of the angle bisector of the lines of action of the other two forces.

Four forces on an object, $A, B, C$ and $D,$ result in no net force. If the angle between forces $A$ and $B$ is $\alpha$ and the angle between forces $C$ and $D$ is $\gamma ,$ show that $A^2+B^2+2\mathrm{ABcos\alpha }=C^2+D^2+2CD\cos \gamma .$