Statics
Statics
Statics deals with bodies at rest. This chapter develops equilibrium of a particle under concurrent forces, resolution in two perpendicular directions, Lami's theorem, friction on rough planes and inclined surfaces, and triangular force diagrams when three non-parallel forces act at a point.
Conditions for equilibrium of a particle
A particle is in equilibrium when the vector sum of all forces is zero, so the sums of components in two perpendicular directions are each zero. Choose axes to reduce the number of unknowns: along the plane and perpendicular to it on an incline, or horizontal and vertical with hanging strings. A triangle of forces closes when three forces hold a particle in equilibrium, and Lami's theorem gives each force proportional to the sine of the angle between the other two.
Friction and rough surfaces
For a body on a rough incline with only its weight, normal reaction and friction acting, impending downward slip gives mg sin θ = μ_s mg cos θ, or tan θ = μ_s. Additional applied forces change this balance. On a horizontal surface with a pulling force applied at angle φ above the horizontal, R = mg − P sin φ.
Worked example
A body of weight 50 N is held in equilibrium by two light strings: one horizontal and one making 40° with the horizontal. Find the tensions. Let the slant tension be T and the horizontal tension H. Vertical: T sin 40° = 50, so T = 50/0.6428 ≈ 77.8 N. Horizontal: H = T cos 40° ≈ 77.8 × 0.766 ≈ 59.6 N. Both answers are physically reasonable: the slant string carries the weight and must be larger, while the horizontal string only balances the horizontal component.
Traps that cost marks
- Resolving with sin and cos swapped for the chosen angle.
- Forgetting the normal reaction when a force pulls upward at an angle, which changes R.
- Applying Lami's theorem to more than three forces or to forces not meeting at one point.
- Using μR where the body is not at limiting equilibrium.
- Sign errors from taking both tensions as acting in the same direction.
Moments and Equilibrium
Moments and Equilibrium
Moments and equilibrium concerns rigid bodies rather than particles: turning effects of forces, the principle of moments, couples, the centre of mass of rods and simple composite bodies, and problems where bodies rest on supports, hinges or rough ground. This is where the paper most often combines geometry with force balance.
Moment of a force
The moment about a point is force × perpendicular distance from the point to the line of action, measured in N m, and its sense is clockwise or anticlockwise. When the distance is measured along the surface rather than perpendicular, use the perpendicular component of the force instead: for a force P at angle φ to a rod, the moment about the end is (P sin φ) × length. A couple is two equal, opposite, parallel forces; its moment is Fd and is the same about every point.
Equilibrium of a rigid body
Two conditions must both hold: the resultant force is zero (resolved in two perpendicular directions) and the total moment about any point is zero. The principle of moments states that for equilibrium the sum of clockwise moments about a point equals the sum of anticlockwise moments. Taking moments about a point where an unknown force acts eliminates that unknown, which usually makes the problem one step.
Centre of mass
A uniform rod has its weight at its midpoint. For a light rod carrying point loads, the weight total acts at the centre of mass found by taking moments of the individual weights about one end. Toppling occurs when the vertical through the centre of mass falls outside the base.
Worked example
A uniform plank AB of length 5 m and weight 200 N rests on supports at A and B. A load of 300 N is placed 2 m from A. Find the reactions. Take moments about A: R_B × 5 = 200 × 2.5 + 300 × 2 = 500 + 600 = 1100, so R_B = 220 N. Resolving vertically: R_A + R_B = 500, so R_A = 280 N. Check by taking moments about B: R_A × 5 = 200 × 2.5 + 300 × 3 = 1400, giving R_A = 280 N as required.
Traps that cost marks
- Using the slant distance instead of the perpendicular distance.
- Mixing clockwise and anticlockwise terms in one sum without signs.
- Putting the weight of a uniform rod at one end instead of the midpoint.
- Forgetting the load in the vertical resolution after taking moments.
- Failing to check the answer by taking moments about the second support.