Lessons · Engineering · equilibrium of a particle
Equilibrium of a particle: the forces add to nothing
When a small body is at rest, the forces on it sum to zero in every direction: ΣF_x = 0 and ΣF_y = 0.
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 sign hangs from two cables and the question is whether the cables are big enough. Two equations, written from the picture, give the tension in each, and nothing else does.
How to think about it
Draw the point and every force on it. Split each angled force into x and y parts. Write the sum in x equal to zero, the sum in y equal to zero, and solve. Two equations, two unknowns.
Worked example
A 30 kg lamp hangs from two cords, each at 45° to the horizontal: W = 30 × 9.81 = 294.3 NThe weight first, in newtons.
ΣF_x = 0: T_1 cos 45° − T_2 cos 45° = 0, so T_1 = T_2 = TSymmetry, read straight off the horizontal equation.
ΣF_y = 0: 2 T sin 45° − 294.3 = 0Both cords pull up; the lamp pulls down.
T = 294.3 / (2 × 0.7071) = 208.1 NEach cord carries more than half the weight, because part of its pull is wasted sideways.
Your turn
The same lamp on cords at 30° to the horizontal. Write the vertical equation.
2 T sin − 294.3 = 0
Solve one, graded on the server
The trap
Expecting the tension to fall as the cords get flatter. It rises: at 10° each cord carries nearly three times the lamp's weight, because almost all of its pull is sideways and only the sliver that is vertical holds the lamp up.