Mathematics · Chapter 8
Study notes aligned to the official NEB syllabus.
Mechanics applies mathematics to forces and motion. Its two halves are statics, the study of forces in equilibrium, here focused on the resultant of parallel forces, and dynamics, the study of motion under force, here covering Newton's laws of motion and projectile motion. Forces are vectors, so the vector methods of earlier chapters carry over directly, and the calculus of the previous unit describes how velocity and displacement evolve in time.
Two forces are parallel when they act along parallel lines. Their combined effect is a single resultant force, and finding its magnitude and line of action is the basic problem of this section.
Two like parallel forces $P$ and $Q$ (acting in the same direction) have a resultant
$$R = P + Q,$$
acting in the same direction and along a line that divides the join of their points of application internally in the inverse ratio of the forces. If the forces act at points $A$ and $B$ and the resultant acts at $C$, then
$$P \cdot AC = Q \cdot BC,$$
so $C$ lies nearer the larger force.
Two unlike parallel forces $P$ and $Q$ (acting in opposite directions, with $P > Q$) have a resultant
$$R = P - Q,$$
acting in the direction of the larger force, along a line dividing the join externally in the inverse ratio of the forces, again with $P \cdot AC = Q \cdot BC$ but $C$ now outside the segment on the side of the larger force. When $P = Q$ the two unlike forces form a couple, whose resultant force is zero but which produces a turning moment.
Worked example. Like parallel forces of $6,\text{N}$ and $4,\text{N}$ act at points $A$ and $B$ that are $0.5,\text{m}$ apart. Their resultant is $R = 10,\text{N}$, acting between them at the point $C$ where $6\cdot AC = 4\cdot BC$. With $AC + BC = 0.5$, this gives $AC = 0.2,\text{m}$ and $BC = 0.3,\text{m}$, so the resultant acts $0.2,\text{m}$ from the larger force.
Mechanics applies mathematics to forces and motion. Its two halves are statics, the study of forces in equilibrium, here focused on the resultant of parallel forces, and dynamics, the study of motion under force, here covering Newton's laws of motion and projectile motion. Forces are vectors, so the vector methods of earlier chapters carry over directly, and the calculus of the previous unit describes how velocity and displacement evolve in time.
Two forces are parallel when they act along parallel lines. Their combined effect is a single resultant force, and finding its magnitude and line of action is the basic problem of this section.
Two like parallel forces and (acting in the same direction) have a resultant
acting in the same direction and along a line that divides the join of their points of application internally in the inverse ratio of the forces. If the forces act at points and and the resultant acts at , then
so lies nearer the larger force.
Two unlike parallel forces and (acting in opposite directions, with ) have a resultant
acting in the direction of the larger force, along a line dividing the join externally in the inverse ratio of the forces, again with but now outside the segment on the side of the larger force. When the two unlike forces form a couple, whose resultant force is zero but which produces a turning moment.
Worked example. Like parallel forces of and act at points and that are apart. Their resultant is , acting between them at the point where . With , this gives and , so the resultant acts from the larger force.