Would this pulley system only halve the force?

TLDR: Is this only a 2:1 pulley system?

Hi there, I’m trying to build a Murphy bed for my roommate. Long story short we’re using an electric hoist + pulley system to raise and lower the bed.

The setup is that the motor is behind the base of the bed on the left. It has a 440 lb pulling capacity. I was planning on running it up the top of the frame, down diagonally where two pulleys would be on each corner of the bed, the wire running from one side of the bed to the other, back up diagonally, up to another pulley on the other side of the top of the frame, then all the way down and anchored to the wall/ground.

The bed is ~330 lbs (~1468 N), so I was doing an analysis of the system to try to see what the pulley ratio would be.

My first assumption is that the two pulleys at the bottom of the bed act as one pulley in practice, as they don’t seem to provide any mechanical advantage. So I simplified my model into a 3 pulley system, but it seems like the weight of the bed is only split on that middle pulley. The other two pulleys don’t seem to provide any mechanical advantage either, they just seem to redirect the force.

Is this analysis correct? Is this only a 2:1 pulley system?

• Yes. ............... Feb 27, 2022 at 4:02

Yes, your calculations are correct and the motor will lift the bed.

Basically in a pulley system, one can count the number of ropes supporting the load, N.

Then the pulling force needed to lift the load is, $$\ T=\frac{1}{N}$$

Intuitively you can assume the system is at balance and stationary, it will be easier to see. If we need to pull very fast then we have to allow for acceleration then

$$T=\frac{mg+ma}{N}$$

Of course, we need to pull the rope N times the amount of lift.

Note that the top right pulley does not move so it is doing nothing and may be removed. The right hand rope would then be attached to the wall at the top.

However, I think your drawing shows a support under the end of the bed. This is required because if someone sits on one corner the pulley arrangement alone allows the bed to twist. An improvement would be to keep the two top pulleys and use two ropes each tied to the base of the bed and both wound up by the motor. Alternately, legs that drop out and support the end would solve that problem.

Figure 1. Force vectors.

The tension in the diagonal ropes in your second analysis is $$\frac {734}{cos(A)}$$ where A is the angle from the vertical. The vertical component will be 734 N on each side and there will be opposing horizontal forces so the rope tension will be higher than 734 N. You'll need to take this into account when specifying the rope, pulleys and anchoring to the wall.

As an aside note that the tension in the rope is proportional to \$\frac 1 {cos(A)}. As A approaches 90° the tension goes to infinity. A tightrope requires huge tension compared to a slack rope.