Maths › Mechanics › Statics of a particle
Statics of a particle
Nothing moves, so everything cancels: the resultant force on a particle in equilibrium is zero in every direction you care to resolve. Two directions, two equations, and the unknowns fall out.
Builds on Friction and inclined planes.
IN THIS TOPIC
- Resolve a set of coplanar forces in two perpendicular directions and set each sum to zero.
- Solve string-and-weight equilibrium problems for unknown tensions.
- Handle equilibrium on rough surfaces, where friction supplies the balancing force.
WHAT YOU PROBABLY THINK
If a particle is in equilibrium, there are no forces acting on it.
Zero, twice
A particle in equilibrium has zero resultant force. In the plane, that is two scalar statements: the components sum to zero horizontally and vertically, or along and perpendicular to a slope, whichever pair of directions makes the forces simplest. Choose the axes before writing anything: the best pair usually lines up with the most awkward force.
WORKED EXAMPLE
A weight on two strings
A 10 N weight hangs from two strings making 30° and 60° with the horizontal. Find both tensions.
Horizontally: T₁ cos 30° = T₂ cos 60°, so T₂ = √3 T₁.
Vertically: T₁ sin 30° + T₂ sin 60° = 10, so 0.5T₁ + 1.5T₁ = 10.
T₁ = 5 N and T₂ = 5√3 ≈ 8.7 N.
The check: the steeper string carries more of the weight, as it should.
Equilibrium with friction
On rough surfaces, friction joins the balance sheet as the force that makes equilibrium possible, sized by whatever the other forces demand, capped at μR. A block resting on a slope is held by friction mg sin θ up the slope; the question “what is the largest force before it slips?” is limiting equilibrium, F = μR, plus the same two resolutions as ever.
The wording deserves care: show equilibrium is possible by showing the needed friction is within the ceiling; find the friction force by resolving, never by quoting μR.
THE EXAM BIT
- Two resolutions, each set to zero, each labelled with its direction: that layout earns the method marks by itself.
- Pick axes to simplify: along and perpendicular to a slope beats horizontal and vertical whenever a slope exists.
- In “on the point of” questions, add F = μR to the two equilibrium equations and solve the trio.
- Tension pulls away from the particle along the string; getting one arrow backwards poisons both equations.
CHECK YOURSELF
A particle of weight 20 N on a smooth slope inclined at 30° is held by a string parallel to the slope. Find the tension and the normal reaction.
Show a hint
Resolve along the slope, then perpendicular to it.
Show the answer
Along: T = 20 sin 30° = 10 N.
Perpendicular: R = 20 cos 30° ≈ 17.3 N.
Equilibrium: components sum to zero in two chosen directions.
Choose axes that flatten the geometry, and let friction fill the gap up to μR.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Resolve a set of coplanar forces in two perpendicular directions and set each sum to zero.
- Solve string-and-weight equilibrium problems for unknown tensions.
- Handle equilibrium on rough surfaces, where friction supplies the balancing force.
No animated video for this topic yet; these notes stand alone.