Spring scale tension

[ymnoklan]

Homework Statement

What does the scale read?

Relevant Equations

Sigma F = 0, T = W

What does the scale read? At first I thought it might be zero as the two weights would cancel out. However, when thinking a bit further I wonder what the scale really reads. Is it the tension?

 

[jbriggs444]

I wonder what the scale really reads. Is it the tension?

We are not allowed to answer the question for you.

If you eliminated the right hand weight and tied the right hand cord to a table leg instead, would the scale reading change?

 

[ymnoklan]

We are not allowed to answer the question for you.

If you eliminated the right hand weight and tied the right hand cord to a table leg instead, would the scale reading change?

No? Because when the right-hand cord is tied to the table, the table exerts an equal and opposite force (tension) to balance the 100 N force from the weight.

 

[jbriggs444]

No? Because when the right-hand cord is tied to the table, the table exerts an equal and opposite force (tension) to balance the 100 N force from the weight.

What then would the scale reading be with the right hand cord tied off?

 

[kuruman]

Say the reading on the scale is F. Suppose I connect two additional spring scales on the vertical pieces of string as shown in the figure on the right.
Please answer the following four questions.

1. How is the reading of the left scale related to the reading of the right scale? Why?
2. How is the reading of each vertical scale related to the reading F of the middle scale? Why?
3. Would the reading on either one of the side scales change if their top ends were attached to a ceiling instead? Why?
4. What is F ?

 

[ymnoklan]

View attachment 355480Say the reading on the scale is F. Suppose I connect two additional spring scales on the vertical pieces of string as shown in the figure on the right.
Please answer the following four questions.

1. How is the reading of the left scale related to the reading of the right scale? Why?
2. How is the reading of each vertical scale related to the reading F of the middle scale? Why?
3. Would the reading on either one of the side scales change if their top ends were attached to a ceiling instead? Why?
4. What is F ?

1. The readings of the left and right scales are equal because the system is symmetrical with equal weights 100 N on both sides.
2. Each vertical scale will read half of the value of F because the middle scale measures the total tension in the horizontal rope. This tension is equally divided by the two vertical scales because the forces from the hanging weights are transmitted through the vertical strings.
3. No, because the tension in each vertical string depends only on the weight of the object hanging from it (100 N in the case).
4. The reading on the middle scale is equal 200 N because the middle scale measures the total tension in the horizontal segment of the rope. Since there is a 100 N weight pulling on each end of the horizontal rope, the total force transmitted through the middle scale is F = 100 N + 100 N = 200 N.

so... would 200 N be correct then?

 

[Orodruin]

Your reasoning for 2 is incorrect. Try not thinking about the scales - imagine they are not there - what is the tension in each part of the rope and why?

 

[kuruman]

so... would 200 N be correct then?

To answer that, look at this assertion

The reading on the middle scale is equal 200 N because the middle scale measures the total tension in the horizontal segment of the rope.

You are proposing the rule that A spring scale reading is the sum of the tensions at each end. Let's assume that this rule is correct. You will agree that your rule applies regardless of whether the scale is vertical or horizontal.

Please apply this rule to one of the vertical scales. What do you know and what does your rule force you to conclude?

 

[ymnoklan]

Your reasoning for 2 is incorrect. Try not thinking about the scales - imagine they are not there - what is the tension in each part of the rope and why?

Tension in the vertical sections of the rope:
Each 100 N weight is supported by the vertical sections of the rope. For equilibrium, the tension in each vertical section of the rope must equal the weight it supports: 100 N.

Tension in the horizontal section of the rope:
The middle (horizontal) section of the rope connect the two vertical sections. At the junctions where the horizontal and vertical sections meet, the forces must balance; each vertical section of the rope exerts a horizontal tension force equal to the tension in the horizontal section. Since the two weights are equal (100 N), the tension in the horizontal section of the rope is the same throughout. This tension is also equal to the tension in the vertical sections.
The middle scale measures the tension in the horizontal section of the rope: 100 N.

 

[kuruman]

I think you got it.

 

[Orodruin]

The middle scale measures the tension in the horizontal section of the rope: 100 N.

Indeed.

 

[ymnoklan]

To answer that, look at this assertion

You are proposing the rule that A spring scale reading is the sum of the tensions at each end. Let's assume that this rule is correct. You will agree that your rule applies regardless of whether the scale is vertical or horizontal.

Please apply this rule to one of the vertical scales. What do you know and what does your rule force you to conclude?

I think I got it at last. But I am curious; does this have anything to do with stepping on a scale? I mean, if you step on a body scale, your weight will push down on the scale with say 600 N. At the same time, as the scale is not moving, so the normal force will push back up on the scale from the ground with 600 N, so the sum of the forces is zero - yet the scale will show 600 N, not zero. Or is this a completely different phenomenon?

 

[haruspex]

I think I got it at last. But I am curious; does this have anything to do with stepping on a scale? I mean, if you step on a body scale, your weight will push down on the scale with say 600 N. At the same time, as the scale is not moving, so the normal force will push back up on the scale from the ground with 600 N, so the sum of the forces is zero - yet the scale will show 600 N, not zero. Or is this a completely different phenomenon?

It is related. What do you think is inside the scale?

 

[ymnoklan]

It is related. What do you think is inside the scale?

What do you mean by 'inside' the scale? I would guess there is some kind of force meter and a converter that divides the force by the acceleration to display the mass.

 

[jbriggs444]

What do you mean by 'inside' the scale? I would guess there is some kind of force meter and a converter that divides the force by the acceleration to display the mass.

Many designs are possible. One of the older and simpler bathroom scale designs is an arrangement of levers connected to a spring. The spring is further connected with gears to a dial with readings painted on it.

The spring is the "force meter".
The painted markings in conjunction with the levers and gears are the "converter"^*.
There is normally a little thumb wheel that can be rotated to adjust the zero point.

Other scale designs include one with levers and gears and a an internal counter-weight. "Honest weight, no springs".

Modern designs use electronic load cells and a digital readout.

(*) In the case of a spring scale, conversion from a measured force to an inferred mass amounts to dividing the force by the local acceleration of gravity and further multiplying by some constant for unit conversion. Any such sequence of multiplications and divisions by known constants can be achieved by painting the markings on the scale in the right places.

In the case of a balance scale like the Toledo there is no conversion from force to mass. The test mass is compared to a reference mass using a force proxy under the assumption that the local gravitational acceleration is constant throughout the volume occupied by the scale. The result is a mass measurement.

Norman Rockwell seems to have had a passion for noticing scales.

 

[ymnoklan]

Many designs are possible. One of the older and simpler bathroom scale designs is an arrangement of levers connected to a spring. The spring is further connected with gears to a dial with readings painted on it.

The spring is the "force meter".
The painted markings in conjunction with the levers and gears are the "converter"^*.
There is normally a little thumb wheel that can be rotated to adjust the zero point.

Other scale designs include one with levers and gears and a an internal counter-weight. "Honest weight, no springs".

View attachment 355515

Modern designs use electronic load cells and a digital readout.

(*) In the case of a spring scale, conversion from a measured force to an inferred mass amounts to dividing the force by the local acceleration of gravity and further multiplying by some constant for unit conversion. Any such sequence of multiplications and divisions by known constants can be achieved by painting the markings on the scale in the right places.

In the case of a balance scale like the Toledo there is no conversion from force to mass. The test mass is compared to a reference mass using a force proxy under the assumption that the local gravitational acceleration is constant throughout the volume occupied by the scale. The result is a mass measurement.

Norman Rockwell seems to have had a passion for noticing scales.

View attachment 355517View attachment 355518View attachment 355519

Thank you so much for sharing this text! I really enjoyed learning about the evolution of scales and how they’ve developed from simple mechanical designs to modern digital systems. The details about levers, springs, and balance scales were especially fascinating, and I loved the connection to Rockwell as well. I appreciate you taking the time to share something so interesting!