Pulled Aside — the Lamp, the Cable and the Cord
A lamp hangs from a ceiling cable and is pulled sideways by a horizontal cord. Now there are three forces and one of them is horizontal, which is exactly the case where resolving into two perpendicular axes pays for itself: each axis gives one equation, and neither needs the other. A spring balance on the wall reads the cord's tension, so the answer can be checked rather than taken on trust.
Como usar esta simulação
- Set the angle the cable makes with the vertical, from hanging free to 55°
- Change the mass of the lamp from 1 kg to 8 kg
- Press Let go of the cord to remove the horizontal force and watch the lamp swing
- Use the camera bar to read the spring balance close up
O que observar
- A 3 kg lamp on a cable 25° from the vertical gives T = 32.4 N and P = 13.7 N
- The spring balance reads exactly the P the equations give — it is the same number
- The cable tension is always larger than the weight, and grows as the angle grows
- ΣF_x and ΣF_y are separately zero at every setting, and each has only two terms
- With the cord slack the cable alone carries the weight and P falls to zero
- Released, the lamp is no longer in equilibrium: it accelerates along its arc
A física por trás
At the point where the cable, the cord and the lamp meet there are three forces: the cable tension T along the cable, the cord tension P horizontally, and the weight W = mg straight down. Resolving along the horizontal gives T sin θ = P; along the vertical, T cos θ = W. From those, T = W / cos θ and P = W tan θ. Because cos θ is never greater than one, the cable always carries more than the weight of the lamp, and the further aside it is pulled the more it carries. Let the cord go and the horizontal force disappears; the remaining two no longer balance, so the lamp is not in equilibrium and swings until friction has taken the energy away.