Torque applied by a continuous mass instead of point particle

[crudux_cruo]

I came across this 'problem' when I was trying to think about how a torsion spring would apply torque in something like a miniature catapult.

I understand that in the context of something like turning a wrench, we can find the net torque on the wrench by treating the hand applying the force as a single vector and calculating the cross product of the distance from the bolt being turned and that force vector.

I don't have an artistic bone in my body so I'll try my best to describe it with words, but imagine you have a seesaw and somebody lays across one of the sides of it. Their head is a certain distance from the fulcrum and their feet just barely reach the edge of the seesaw.

How would you go about finding the net torque the person applies to the seesaw? Would it involve integrating the force applied by each point of the persons body with respect to the distance of the fulcrum of each point (or vice versa)? Can you approximate the net torque by applying the persons total gravitational force to a single point where their feet would be? I'm probably overcomplicating it, but I'm having trouble piecing it together by myself.

 

[Baluncore]

I'm probably overcomplicating it, but I'm having trouble piecing it together by myself.

Since you are considering a static balance, you would find the persons centre of mass, then apply all their weight at that point. To find the persons centre of mass, get them to lie balanced across the see-saw fulcrum. Mark them where they cross the fulcrum.

 

[anuttarasammyak]

Would it involve integrating the force applied by each point of the persons body with respect to the distance of the fulcrum of each point (or vice versa)? Can you approximate the net torque by applying the persons total gravitational force to a single point where their feet would be?

You may easily find some pictures of Giant Prayer Wheels in Tibetan in web. Many people around the wheel contribute torque to rotate the wheel. Total torque comes from summing up their contribution. If you have more micro view, you can distinguish right hand contribution and left hand contribution of a man for summing up, fingers and go further to infinitesimal parts of touching skins for integration.

 

[crudux_cruo]

Since you are considering a static balance, you would find the persons centre of mass, then apply all their weight at that point. To find the persons centre of mass, get them to lie balanced across the see-saw fulcrum. Mark them where they cross the fulcrum.

I haven't gotten to the next chapter on equilibrium, so maybe the intuition will click into place after I read through that. Would the same principle apply when working out the net torque applied by a torsion spring along a length of wood (ie the mini catapult thing I was thinking of earlier)?

 

[Baluncore]

Would the same principle apply when working out the net torque applied by a torsion spring along a length of wood (ie the mini catapult thing I was thinking of earlier)?

I guess it would depend on how you coupled the torsion spring to the length of wood.

If instead of laying down, a person stood on the see-saw, their centre of mass would be in a known position along the plank. So long as they do not lie down, then bend their knees, put their arms over their head, or touch their toes, their centre of mass will be in a consistent position.

If they were dynamically changing geometry then you must find the mass and centre of mass position for each limb or body part. You can then sum the torques = ( position * weight ) due to all body parts.

 

[jbriggs444]

The "center of mass" is, if you chase it all the way down, itself an integral. One integrates the position of each mass element multiplied by the mass of that mass element. Then one divides by total mass. The result is a sort of weighted average position.

This is useful because one can take this pre-computed integral and multiply by a uniform force field (such as gravity). By a trick of algebra, the result is the same as the weighted average point of application of the resulting gravitational force.

I do not have a clear picture of the sort of catapult you have in mind. Are you talking about a torsion spring such as the Romans used (twisted horse hair as I recall). I think they used ballistae rather than onagers like this one.

 

[crudux_cruo]

I do not have a clear picture of the sort of catapult you have in mind. Are you talking about a torsion spring such as the Romans used (twisted horse hair as I recall). I think they used ballistae rather than onagers like this one.

A few weeks ago when I was working through the chapter (in Fundamentals of Physics), I had a difficult time getting a handle on how to actually use integration to calculate the CoM of rigid objects. I feel like the book brushes past it very quickly, and I have had trouble finding a source that explains it in detail without losing me. Honestly it probably comes down to a lack of mathematical maturity and weak integration skills.

Tangent aside, I was hoping to build a tiny little mangonel catapult in the next month or so as a way to apply what I have learned so far. The first thing that came to mind was to get a torsion spring, and store energy by compressing one end with the swinging arm. I figured that would involve applying the spring force over a length (as in the spring is literally in contact with several centimeters of the arm), as opposed to just a single point.

Frankly, I have rewritten this post several times and I still feel like I am being incredibly clumsy with my wording here. Let me know what needs further elaboration and I'll try my best to explain it better.

 

[jbriggs444]

figured that would involve applying the spring force over a length (as in the spring is literally in contact with several centimeters of the arm), as opposed to just a single point.

Frankly, I have rewritten this post several times and I still feel like I am being incredibly clumsy with my wording here. Let me know what needs further elaboration and I'll try my best to explain it better.

If I understand this right, the end of the spring is rounded and fits into a rounded slot in the arm. So the "point of contact" is spread out over several centimeters.

You have a figure for the force applied by the spring in mind. You do not know how much torque this amounts to. You want to multiply by the length of the moment arm to figure it out. But you don't know where within that rounded slot to count as the end of the moment arm.

If it were me, I'd probably make force measurements at the far end of the lever and judge torque from that. And maybe make some control measurements. How much variation in force required before the hinge budges either way. How much support force required if the spring is removed? Are there any places where anything is binding? Maybe some high speed video to measure acceleration.

But I may not be clear on what you think you know and what you are trying to calculate from that.

 

[crudux_cruo]

I made a very crude drawing, but I think that we are roughly thinking along the same lines.

and for reference the spring I am going for (which I'm sure you are already aware of, but I need to give context to the terrible thing I just drew)

I had figured that I would use rotational work to find the KE it would have at whatever optimal angle, and use an approximation of rotational inertia to find angular velocity, and then the projectiles tangential velocity.

Like you suggested, the plan was to use my phone's 240fps video mode and record a launch in front of some reference point. Then I'd compare my predictions with the footage.

The springs come with a measure of their max torque, and I'm assuming the torque is a measure of the spring force at the very end of one of the uncoiled segments. I'm not too sure how I'd convert that to a linear(?) force that would be applied to the arm.

How does one go about taking force measurements, in this instance? Would I just get one of those force sensors and estimate things through trial and error?

 

[anuttarasammyak]

It is called a helical torsion spring. In your design I wonder one arm of the helical torsion spring cannot keep contacting not on a point but full contact along the bar to the arm of catapult during the motion. If you allow the contact is on a point of the end of the spring arm, your difficulties would be lessened.

 

[crudux_cruo]

It is called a helical torsion spring.

Thank you! I noticed that the torsion spring I wanted to use wasn't the only kind so I appreciate knowing the proper name for it.

If you allow the contact is on a point of the end of the spring arm, your difficulties would be lessened.

My only concern is that I am not sure how I'd accomplish that without bending and possibly fatiguing the spring arm. Maybe with the actual materials in front of me I'd have a better idea. It seems like it would be simplify the torque calculation a fair bit, though.

 

[jrmichler]

Some experimenting with a torsion spring catapult will help answer your questions. And you can get a torsion spring complete with catapult arm, release mechanism, and mounting base for a few dollars at your local hardware store:

 

[jbriggs444]

Some experimenting with a torsion spring catapult will help answer your questions. And you can get a torsion spring complete with catapult arm, release mechanism, and mounting base for a few dollars at your local hardware store:

The mousetrap design is especially easy to analyze since the point of application of the spring to the trap does not matter. The axis for the spring coincides with the axis of the trap.

 

[crudux_cruo]

After doing a bit of reading, I realize that I am jumping the gun and there are concepts I haven't learned yet that I'll cover soon. It seems one of those concepts (Equilibrium) will address the confusion I am having right now, so I probably could have avoided the trouble of making this thread by just being patient for another week.

I appreciate the responses though! I think it gives me more context for things moving forward.

 

[hutchphd]

But it makes your learning this week much better.
If you are not confused, you are not learning