Pulley dynmamics involving acceleration and time-varying tension.

[Super Kirei]

I came up with this question while thinking about pulleys. Most textbook pulley problems involve the force on the load being equal to the load's weight so that there's no acceleration. But what if we had the following setup. There's an anchor on the ceiling and there's a heavy crate on the floor attached to a moveable pulley and then a fixed pulley on the ceiling near the anchor and they're all connected by a steel cable that we'll assume is strong enough to handle all tensions without breaking or stretching and light enough to be considered mass less. Rather than just pull down on the cable and enjoy a mechanical advantage of two, I build some scaffolding and put a heavy crate on top of the scaffolding and connect the steel cable to that. Then I push the crate off of the scaffolding but the steel cable has some slack so the crate is free falling before the cable becomes taut and starts to exert a force on the other crate. Let's assume that the second crate is heavy enough or falls far enough that the initial force exercted on the first crate is enough to get it moving rather than the second crate just bouncing on the end of the cable. Just eyeballing it I think they're will be acceleration and time-varying tension on the cable but I don't really know how to write the differential equation.

 

[Nugatory]

a steel cable that we'll assume is strong enough to handle all tensions without breaking or stretching

The "no-stretch" assumption makes this problem effectively insoluble. At the moment that the slack in the cable is taken up one end of the cable will be moving and the other end won't. Either the cable stretches a bit or the stationary end changes its speed instantaneously, implying a momentarily infinite acceleration and force.

Try assuming that the cable is a stiff spring, such that the tension in it is a linear function of the small change in its length. Then you can just do F=ma for each mass, where the force is tension in the cable, and set up the differential equations.

 

[Super Kirei]

Why would the stationary end of cable need to change it's speed instantaneously? The end that's attached to the crate has been moving all along, it's just that the cable wasn't taut, there were some loops of cable coiled up on top of the crate. And I'm not sure how to set up the equations using F=ma. Maybe I can use the fact the the total displacement of the crate attached to the moving pulley over any given interval of time must be half that of the crate attached to the moving end of the cable.

 

[Nugatory]

Why would the stationary end of cable need to change it's speed instantaneously? The end that's attached to the crate has been moving all along, it's just that the cable wasn't taut, there were some loops of cable coiled up on top of the crate. And I'm not sure how to set up the equations using F=ma. Maybe I can use the fact the the total displacement of the crate attached to the moving pulley over any given interval of time must be half that of the crate attached to the moving end of the cable.

I'm sorry, I wasn't clear. By "the stationary end" I meant the end that is attached to the crate on the floor, so is not moving at the moment that the last of those loops of slack are taken up. Once the cable goes taut, there are only two possibilities: either both ends of the cable are moving at the same speed, or the cable is stretching. Because the stationary end wasn't moving at the instant that the cable went taut, it can't be moving when the cable is taut without an instantaneous change in velocity.

 

[PJC01]

I find this question very interesting and it brings up some important concepts in pulley dynamics. Firstly, it is important to note that in a pulley system, the tension in the cable is always equal throughout the entire cable. This means that the tension in the cable connecting the two crates will be the same as the tension in the cable connecting the scaffolding to the anchor on the ceiling.

In this scenario, we have two crates connected by a cable, with one crate being heavier than the other. When the heavier crate is pushed off the scaffolding, it will start to fall and the cable will become taut, exerting a force on the lighter crate. This will cause the lighter crate to accelerate in the same direction as the heavier crate.

The acceleration of the system will depend on the mass of the crates and the force exerted by the falling crate on the cable. As the lighter crate starts to move, the tension in the cable will increase, causing the acceleration to decrease. This results in a time-varying tension in the cable.

To write the differential equation for this scenario, we would need to consider the forces acting on each crate, including the weight of the crate, the tension in the cable, and the acceleration of the crates. This can be a complex equation to solve, but it is possible to model the system using Newton's laws of motion and the concept of conservation of energy.

In conclusion, pulley dynamics involving acceleration and time-varying tension can be a challenging but interesting problem to solve. It requires a thorough understanding of the principles of pulley dynamics and the ability to apply mathematical concepts to model the system.