Wire tension that might involve tensors

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Discussion Overview

The discussion revolves around the tension in a wire connecting two masses, ##m_1## and ##m_2##, as one mass is pulled along a track at a constant acceleration. The problem involves mechanical engineering dynamics, potentially incorporating tensor analysis, and examines how the tension varies with the angle ##\theta## of the wire relative to the x-axis. Participants explore the dynamics of the system, including the forces acting on each mass and the equations of motion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes a scenario where a known force ##F## is applied to mass ##m_1##, which is connected to mass ##m_2## via a wire at a constant angle ##\theta##, seeking to determine the tension in the wire.
  • Another participant suggests that a Free Body Diagram (FBD) for both masses is necessary to analyze the forces and solve for tension.
  • Some participants express skepticism about the need for tensor analysis, arguing that the problem is straightforward and can be solved with basic dynamics.
  • A participant proposes an intuitive guess for the tension components based on trigonometric relationships but is challenged to derive the equations of motion instead.
  • Concerns are raised about the relationship between the applied force ##F## and the tension ##T##, with some participants insisting that they cannot be collinear, while others question the implications of this assertion.
  • There is a suggestion to break down the forces acting on each mass into components parallel and perpendicular to the guide to clarify the dynamics.
  • One participant notes that when attempting to set up the equations of motion, trigonometric terms cancel out, leading to confusion about the tension's dependence on the angle ##\theta##.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the necessity of tensor analysis or the correct approach to setting up the equations of motion. There are competing views on the relationship between the applied force and the tension in the wire, and the discussion remains unresolved regarding the correct formulation of the problem.

Contextual Notes

Participants mention the need for multiple Free Body Diagrams and the introduction of the mass of the Earth, ##m_e##, which adds complexity to the analysis. There are unresolved issues regarding the kinematic relationships between the two masses and the implications of the applied force's direction.

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I need help with this problem. This is not a homework assignment, so please don’t send it over to the homework forum.

It involves mechanical engineering dynamics that probably are more subtle and advanced then first year mechanical engineering dynamics. It might involve tensor analysis. I don’t know.

We let a mass, designated as ##m_2##, slide along a straight track with one degree of freedom with no friction. The track is rigidly attached to the earth. There is a wire attached to ##m_2## so that it can be pulled at a constant acceleration. Attached on the other end of the wire is a second mass, designated as ##m_1##. A known force of magnitude F is applied to ##m_1##, such as a magnetic force, accelerating ##m_1## and ##m_2## along the track. We define the motion of ##m_1##-##m_2## to be in the positive y-direction. The wire makes an angle ##\theta## with respect to the x-axis, and for this problem we keep the angle ##\theta## constant. Thus, the givens are: ##m_1##, ##m_2##, F, and the angle ##\theta##. I want to know the tension in the wire for any constant angle ##\theta## between 0 and 90 degrees with the above givens.

On the surface this might seem like a trivial problem, but when I do a deeper analysis, it seems to be more complicated than I can handle. For example, when ##\theta## is zero degrees, intuitively, the tension in the wire would be at a maximum, and there would be no acceleration of ##m_1##-##m_2## in the positive y-direction, and when ##\theta## is 90 degrees, the tension would be a minimum, and there would be a maximum acceleration of m1-m2. But what would the tension in the wire be at any angle between zero and 90 degrees? I can’t get my head around this. Could someone please help me with this?
 
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This does not sound too difficult, but I urge you to post a figure showing the situation and identifying the variables for clarity. If you do, I think you will have the help you need in short order.
 
Thank you, Berkeman.

CWatters is definitely correct about the FBD, except that you need two FBDs, not just one. Draw them, write the equations of motion for each body, and solve for the tension. If you get stuck, but sure to post the work you have done.

I cannot imagine why you think tensors might be required in this simple problem. It really is not that hard at all.
 
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Dr.D said:
Thank you, Berkeman.

CWatters is definitely correct about the FBD, except that you need two FBDs, not just one. Draw them, write the equations of motion for each body, and solve for the tension. If you get stuck, but sure to post the work you have done.

I cannot imagine why you think tensors might be required in this simple problem. It really is not that hard at all.
So, I need to draw a FBD for ##m_1## and another FBD for ##m_2##, and then solve a system of equations from the two diagrams? I will try.
 
The diagram link opens to a blank page for me ?
 
As an intuitive guess, I think the final expression for T would be as follows:

The tension force with respect to the x-axis would be:

##T_x = cos\theta F##

The tension force with respect to the y-axis would be:

##T_y = sin\theta \frac{m_2}{m_1 + m_2} F##

Thus, T would be:

##T = \sqrt{T_x^2 + T_y^2}##

Is this correct?
 
  • #10
Rather than guess, why not write the equations of motion for the particles and work it out mathematically?
 
  • #11
Nidum said:
The diagram link opens to a blank page for me ?

It opened OK for me using Libre Office.
 
  • #12
Dr.D said:
Rather than guess, why not write the equations of motion for the particles and work it out mathematically?
When I try to set up the equations of motion, all of the trigonometric expressions cancel out. I don't know how to set up the equations of motion correctly.
 
  • #13
Write the sum of forces on each body, once in the direction parallel to the guide and again in the direction perpendicular to the guide.

Be sure to post your work if you need more help.
 
  • #14
Dr.D said:
Write the sum of forces on each body, once in the direction parallel to the guide and again in the direction perpendicular to the guide.

Be sure to post your work if you need more help.
I think this is what I am suppose to do:

Forces on ##m_1##:

$$a_y = \frac{sin\theta F – sin\theta T} {m_1}$$

$$a_x = \frac{cos\theta F – cos\theta T} {m_1}$$Forces on ##m_2##. (I have to introduce the constant ##m_e##, the mass of the earth).

$$a_y = \frac{sin\theta T} {m_2}$$

$$a_x = \frac{cos\theta T} {m_e + m_2}$$But I am not sure what to do next, for if I equate the terms for ##a_y## and ##a_x##, all the trig expressions drop out, which leads to an answer that doesn't make sense. (The tension is constant for all angles of ## \theta##).
 
  • #15
There is an error in your diagram that is probably leading you astray. The applied force F cannot be colinear with the wire.

In addition, you need to think through the kinematic relations between the motions of the two particles.
 
  • #16
Dr.D said:
There is an error in your diagram that is probably leading you astray. The applied force F cannot be colinear with the wire.

In addition, you need to think through the kinematic relations between the motions of the two particles.
Why is the applied F not colinear with the wire? I do not understand that. The tension force must be colinear with the wire, so why not F? I have defined the force F to be colinear with the wire, but if this is not physically possible, can you please explain why this is not possible? In addition, if F is not colinear, then this would cause ##m_1## to rotate and this is a complication I did not define to happen.
 
Last edited:
  • #17
Both m1 and m2 must accelerate parallel to the guide. If F and T are colinear, there is no component of the sum of forces on m1 available to accelerate m1.

Rather than drawing F, draw simply two components, Fx, and Fy, and don't worry about exactly what direction they are in. This should clarify what happens considerably.

This is not a hard problem; do not make it one. But do not impose misunderstanding up on by insisting the F and T be colinear.
 

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