Lessons · Engineering · equilibrium of a rigid body
Equilibrium of a rigid body: forces balance and turning balances
A body that is not moving has ΣF_x = 0, ΣF_y = 0 and ΣM = 0 about any point. Three equations, so three unknowns can be found.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
What it is for
A plank across two trestles, a person standing near one end. Which trestle carries more, and how much? The site foreman wants the number before anyone walks out, and moments give it in three lines.
How to think about it
Free-body diagram first. Then take moments about a point that one unknown passes through, so it vanishes from that equation and the other unknown comes out alone. Finish with the force sum, and check with a moment about a different point.
Worked example
Three equations: ΣF_x = 0, ΣF_y = 0, ΣM = 0The whole of planar statics for a rigid body.
A 3 m plank of 150 N on trestles at its ends; a 450 N person 1 m from AThe plank's weight acts at its middle, 1.5 m from A.
ΣM_A = 0: R_B × 3 − 150 × 1.5 − 450 × 1 = 0 → R_B = 675 / 3 = 225 NAbout A, R_A has no arm and drops out. R_B comes out alone.
ΣF_y = 0: R_A + 225 − 150 − 450 = 0 → R_A = 375 NUp equals down. A carries more, because the person is nearer A.
Check ΣM_B: 375 × 3 − 150 × 1.5 − 450 × 2 = 1125 − 225 − 900 = 0 OKAn equation you did not use to solve, used to check. It costs one line and catches almost everything.
Your turn
A 2 m beam, 100 N load at 0.5 m from A, R_B at 2 m. Write the moment equation about A.
R_B × 2 − 100 × = 0
Solve one, graded on the server
The trap
Taking moments about a point the unknown you want passes through. That makes it vanish. Choose the pivot so the reaction you do NOT want yet is the one that drops out.