Determining mass of a weight on a three spring scale setup

[merricksdad]

Homework Statement

Given three spring scale readings, positioned at unknown angles, find the mass of the weight hanging from all three scales without using trig, and without measuring the angles. You have only a yard stick. This is a static equilibrium problem.

Homework Equations

Not allowed to use trig. Can only use a^2+b^2=c^2.

The Attempt at a Solution

I assume I have to measure vertical and horizontal components by hand, as I am not allowed to take an angular reading. What I don't understand is how do I take a reading without some known level plane, other than the floor, and without a level also attached to my yardstick? If I had a level plane, at or around the top of the weight, I could measure <x,y,z> displacement.

The experiment gives hints that we're supposed to somehow use Similar Triangles, and that the vector components make right triangles. Obviously vector components would make right triangles if I could measure angles and use trig to get those components, but I'm told not to use trig.

One image shows the apparatus pushed up against a white board. If I can't use trig, how does that help me in a 3 spring scale setup? I don't get it at all. I want to just remove the weight and place it on a scale...

I know I must calculate the opposing force using the vectors by summing their component parts. But how do I accurately get the component parts with only knowing the readings on the scales?

 

[haruspex]

Homework Statement

Given three spring scale readings, positioned at unknown angles, find the mass of the weight hanging from all three scales without using trig, and without measuring the angles. You have only a yard stick. This is a static equilibrium problem.

Homework Equations

Not allowed to use trig. Can only use a^2+b^2=c^2.

The Attempt at a Solution

I assume I have to measure vertical and horizontal components by hand, as I am not allowed to take an angular reading. What I don't understand is how do I take a reading without some known level plane, other than the floor, and without a level also attached to my yardstick? If I had a level plane, at or around the top of the weight, I could measure <x,y,z> displacement.

The experiment gives hints that we're supposed to somehow use Similar Triangles, and that the vector components make right triangles. Obviously vector components would make right triangles if I could measure angles and use trig to get those components, but I'm told not to use trig.

One image shows the apparatus pushed up against a white board. If I can't use trig, how does that help me in a 3 spring scale setup? I don't get it at all. I want to just remove the weight and place it on a scale...

I know I must calculate the opposing force using the vectors by summing their component parts. But how do I accurately get the component parts with only knowing the readings on the scales?

I'm not getting the picture. If you cannot post an image, please try to describe the set-up in much greater detail.

 

[merricksdad]

Sorry, I don't have a picture. It appears to be a simple physics project, with three poles, to which are attached a spring scale (reading Newtons), and from each scale is a cord which is fastened to the same weight ball, in the center, suspended above the table by the cords. The weight is not marked. We have only the setup, and a yardstick. Find the mass of the weight. The distance between the poles appears arbitrary. Angle measured from the weight to the pole appears to be similar, if not same, for each pole, but the angle cannot be measured for the experiment, it must be derived from vector components.

So we had this problem in lab today, but we never got to it. The TA ended up saying that if we actually did it, we'd get extra credit instead. No chance. From what I gather, the best thing we could possibly do is to measure the vector z-component by assuming the poles are completely vertical, or by testing that they were at least nearly vertical by measuring their distances from each other at both top and bottom of the poles. We could get the z-components from the measurement of the pole length taken up by the string and the scale, to the ball. And the xy plane component from measuring the distance from the top of the weight to the pole. My brain is dying right now, but somewhere in there we could also separate the x and y components from the xy plane measurement.

I certainly hope this one doesn't come back in another lab, unless the TA is going to explain it to us. Unfortunately, more than half the lab was the requirement of trying to figure it out without first being shown how to do it. If you've done this before, I'd definitely like to hear it.

 

[haruspex]

Are the poles the same heights? What are you allowed to measure... the heights, the distances between the poles, the distances between the tops of the poles, the lengths of the strings..?

 

[merricksdad]

The poles are about the same heights. The position of the spring scale on the pole is about the same, but they are not at the top of the pole. We could have measured anything we wanted to with the ruler.

 

[gneill]

When you say that you aren't allowed to use trig, does that mean no use of a calculator with sin, cos, tan, etc., functions, or no triangle based geometry at all other than Pythagoras' theorem? What about simple similar triangles to take advantage of related ratios?

 

[haruspex]

The poles are about the same heights. The position of the spring scale on the pole is about the same, but they are not at the top of the pole. We could have measured anything we wanted to with the ruler.

So assume the points of attachment to the poles are all the same height, and that you know the distances from the weight to these three points. Suppose, for now, that you also know how far below those points is the weight.
What balance of forces equation can you write?

 

[merricksdad]

When you say that you aren't allowed to use trig, does that mean no use of a calculator with sin, cos, tan, etc., functions, or no triangle based geometry at all other than Pythagoras' theorem? What about simple similar triangles to take advantage of related ratios?

No trig functions on the calculator. No component calculation using same trig. Yup, just the Pythagorean theorem.

 

[merricksdad]

So assume the points of attachment to the poles are all the same height, and that you know the distances from the weight to these three points. Suppose, for now, that you also know how far below those points is the weight.
What balance of forces equation can you write?

If I make these assumptions, yes I can make general vector component assumptions if I define one of the poles' strings as equaling one axis on the x-y plane, correct? But I'm making assumptions, right? And then would I not even need the Pythagorean theorem for anything except final calculation?

 

[Stephen Tashi]

Some random thoughts:
Taking a coordinate system where the the force of the weight is all in the z-direction, one could write 6 equations involving the x,y and z components of the strings. There are 3 equations that come from the equilibrium of the forces in each of the x,y,z directions and 3 equations that relate the components of a force on a cable to the total force on it.

One could write 3 more equations stating that the x and y components of the weight are zero and that the z component is the total weight, but its simpler to omit the x and y components of the weight from the list of unknowns. So there are 10 unknowns - the unknown weight plus the 9 unknown component forces.

Now, thinking like a physicist, what is the mathematical meaning of the under-determined system of equations? If we rotated the positions of the cables about the z-axis we would transform one solution of the problem into a different solution. There also might be ways to reflect the position of the cables about a vertical plane and transform one solution to another solution.

So, can we turn this problem into an example of how group theory is applied to physics? It's probably not what the lab intended, but it would be a relief to find an example of applying group theory to physics that was easy to understand!

 

[haruspex]

if I define one of the poles' strings as equaling one axis on the x-y plane,

I did not say assume the strings are all in a horizontal plane, if that is what you mean The weight is some distance below the points of attachment to the poles. Finding that distance is the hard part. You cannot measure it directly. How can you deduce it?

Edit: to clarify, I am suggesting assuming the attachments to the poles are all at the same height just to see if we can solve that case. Then we look at whether that assumption can be dropped.

 

[J Hann]

Draw a vertical from W to the supporting beam.
This gives you 3 right triangles with Tn for the hypotenuse of each triangle (the scale readings).
Looking at 1 triangle, let T1 be the scale reading, and h the distance to the beam and maybe x1 the distance along the beam.
There is a "closed" force triangle here with sides T1, W1 and Fx1.
This is similar to the triangle LT, h, and x1 (your linear measurements for this triangle)
Now just relate W1 to T1 using your measurements from the similar triangle.
(To prove this note that W1 = T1 sin theta = T1 * x1 / LT.

 

[Stephen Tashi]

One image shows the apparatus pushed up against a white board.

I can't visualize what that means.

 

[merricksdad]

Draw a vertical from W to the supporting beam.
This gives you 3 right triangles with Tn for the hypotenuse of each triangle (the scale readings).
Looking at 1 triangle, let T1 be the scale reading, and h the distance to the beam and maybe x1 the distance along the beam.
There is a "closed" force triangle here with sides T1, W1 and Fx1.
This is similar to the triangle LT, h, and x1 (your linear measurements for this triangle)
Now just relate W1 to T1 using your measurements from the similar triangle.
(To prove this note that W1 = T1 sin theta = T1 * x1 / LT.

This is what we were shown a week later in the class which goes along with the lab. The class professor said he's not in control of the lab layout or requirements. Now I understand, and see why nobody got to that part of the lab. Thank you guys for helping out. It sounds like the professor was not happy with requirement to not use trig.

 

[haruspex]

This is what we were shown a week later in the class which goes along with the lab. The class professor said he's not in control of the lab layout or requirements. Now I understand, and see why nobody got to that part of the lab. Thank you guys for helping out. It sounds like the professor was not happy with requirement to not use trig.

No, you do not need trig, not even Pythagoras.
If we assume the floor is level, we can measure the distance from floor to attachment point on each pole and from floor to attachment point on the weight. The differences give the vertical extents of the three strings, h_i, i=1, 2, 3.
We can measure the distances between upper and lower attachment points on each string, i.e. the string lengths, L_i.
If the tensions are T_i, you can write the vertical component of each as a simple function of its h, L and T.