Jan Dangerfield, Stuart Haring and, Julian Gilbey Solutions for Exercise 8: EXERCISE 3E

Three coplanar forces act on a particle, as shown in the following diagram. $X$ has components $0 \mathrm{N}$ in the $x$ -direction and $20 \mathrm{N}$ in the $y$ -direction. $Y$ has components $25 \mathrm{N}$ in the $x$ -direction and $-10 \mathrm{N}$ in the $y$ -direction. $Z$ has components $-10 \mathrm{N}$ in the $x$ -direction and $-15 \mathrm{N}$ in the $y$ -direction. Find the magnitude and direction of the resultant of the three forces.

Four coplanar forces act on a particle, as shown in the following diagram. Show that the $x$ component and $y$ component of the resultant are equal. Hence, determine the direction of the resultant force.

A boat of mass $1000 \mathrm{kg}$ is pulled by three tugboats. One pulls due north with force $500 \mathrm{N},$ one pulls due east with force $350 \mathrm{N}$ and one pulls on a bearing of $040°$ with a force of $250 \mathrm{N}.$ Find the bearing and acceleration of the resultant motion.

In a competition of strength, four people pull a mass with ropes at different angles. The direction in which the mass moves determines the winner. Arjun wants the mass to go north, Bob wants it to go east, Chen wants it to go south and David wants it to go west. The men pull with the forces in the directions shown in the diagram. Find the direction of the resultant motion and determine who wins.

A rowing boat of mass $120 \mathrm{kg}$ is being pulled from rest by three boats. One pulls north with a force of $100 \mathrm{N},$ the second pulls on a bearing of $020°$ with a force of $80 \mathrm{N}$ and the third pulls on a bearing of $045°$ with a force of $90 \mathrm{N}.$ There is resistance from the water of $200 \mathrm{N}$ directly against the motion. Find the bearing and acceleration of the resultant motion.

A hovercraft has an engine providing a force of $150 \mathrm{N}$ on a bearing of $340°.$ The wind blows on a bearing of $310°,$ which means the hovercraft accelerates from rest on a bearing of $320°.$ Find the force of the wind on the hovercraft.

The wind is blowing a boat with force $F.$ The motor of the boat can exert a driving force of $D \mathrm{N},$ where $D<F.$ Show with a diagram that, whatever direction the wind is taking the boat with the motor switched off, the motor is capable of deflecting the direction by a maximum of ${\sin }^{-1}\frac{D}{F}.$

A building is unstable after a natural disaster. A car is stuck under the building and needs to be dragged out as quickly as possible, although the exact direction is less important. Three people can pull ropes, one due north, one at a bearing of $010°$ and one at a bearing of $030°.$ Akhil can pull with a force of $300 \mathrm{N},$ Ben can pull with a force of $240 \mathrm{N}$ and Khadijah can pull with a force of $210 \mathrm{N}.$ Find who should pull each rope to maximise the acceleration and what the net force will be.