Engineering statics equilibrium problem

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Discussion Overview

The discussion revolves around solving an engineering statics equilibrium problem involving the forces acting on a bead suspended by an elastic cord. Participants are working through the geometry and force calculations related to the positions of points A, B, C, and D, as well as the forces acting on the bead in relation to the bar AC.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents the coordinates of points A, B, C, and D and attempts to find the coordinates of B based on the midpoint between A and C.
  • Another participant questions the method used to find the x-coordinate of B, suggesting it should also be the midpoint of A and C's x-coordinates.
  • There is a discussion about calculating the tension in the cord BD using the spring constant and unstretched length, leading to a negative value for tension, which is noted as unusual.
  • Participants explore the conversion of tension into a force vector and the use of dot products to find the magnitude of P, with one participant expressing uncertainty about the negative result of the dot product.
  • There is a query about finding the reaction force between bead B and rod AC, with a suggestion to consider the cross product, although it is acknowledged that there are infinitely many vectors perpendicular to AC.
  • One participant expresses urgency in needing to resolve the problem quickly, indicating the time-sensitive nature of the discussion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct approach to finding the coordinates of point B or the implications of the negative tension value. Multiple competing views and methods are presented, and the discussion remains unresolved regarding the specifics of the calculations and the physical interpretations.

Contextual Notes

There are limitations in the assumptions made about the coordinates and the calculations of forces, particularly regarding the treatment of tension as negative and the implications for the direction of forces. The discussion also reflects uncertainty in the application of vector operations and their physical meanings.

Alison A.
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Homework Statement


Here is the prompt/picture
FBD.png


The hint given is a FBD of the bead is recommended to being this problem. Find the coordinates of B so that both the magnitude and orientation of the elastic cord force can be properly represented. Also, two mutually orthogonal normal force directions (to bar AC) need to be included to permit a general representation of normal forces acting on the bead. A shortcut to solving for the elastic cord force can be obtained by writing the vector equilibrium equation, then taking the dot product of that equation with a unit vector pointing in the direction of bar AC. Since the normal forces are by definition perpendicular to the bar, their contribution is zero, and a single scalar equations remains for the for P.

Homework Equations


ΣFx=0
ΣFy=0
ΣFz=0
Basic trigonometry

The Attempt at a Solution


Alright so I started off by finding the coordinates of all the points
A (124, 0, 0)mm
B (?, 31, 21)mm
C (0, 62, 42)mm
D (62, 0, 62)mm

To find B I made a right triangle with A and C to find the magnitude. Since B is in the middle of AC as stated, I divided the magnitude I found and got 65.5. So B (65.5, 31, 21)mm.

I found the unit vector of AC like the hint suggested and found that to be (-0.856, 0.428, 0.290).

I am stuck here, I don't know how to go about finding the vector equilibrium.

Thank you for any help!
 
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Alison A. said:

The Attempt at a Solution


Alright so I started off by finding the coordinates of all the points
A (124, 0, 0)mm
B (?, 31, 21)mm
C (0, 62, 42)mm
D (62, 0, 62)mm

To find B I made a right triangle with A and C to find the magnitude. Since B is in the middle of AC as stated, I divided the magnitude I found and got 65.5. So B (65.5, 31, 21)mm.

Why did you do this? If the y coordinate and z coordinate of point B are both halfway between the y and z coordinates of points A and C, it stands to reason that the x coordinate of B will also be halfway between the x coordinates for these two points.
I found the unit vector of AC like the hint suggested and found that to be (-0.856, 0.428, 0.290).

I am stuck here, I don't know how to go about finding the vector equilibrium.

Thank you for any help!

Find the correct location for point B.

Once you know this, you can then find the tension force in the cord BD, given the spring constant and its unstretched length.
 
SteamKing said:
Why did you do this? If the y coordinate and z coordinate of point B are both halfway between the y and z coordinates of points A and C, it stands to reason that the x coordinate of B will also be halfway between the x coordinates for these two points.Find the correct location for point B.

Once you know this, you can then find the tension force in the cord BD, given the spring constant and its unstretched length.

A (124, 0, 0)mm
B (62, 31, 21)mm
C (0, 62, 42)mm
D (62, 0, 62)mm

TBD=k(l-l0) = 4(20-51.4) = -125.6
 
Alison A. said:
A (124, 0, 0)mm
B (62, 31, 21)mm
C (0, 62, 42)mm
D (62, 0, 62)mm

TBD=k(l-l0) = 4(20-51.4) = -125.6
Tensions are usually taken to be positive. The stretch in the spring would be calculated as stretched length - unstretched length = 51.4 - 20 = 31.4 mm
The tension would therefore be 4 N/mm * 31.4 mm = 125.6 N.
 
So I add this value with the unit vector in the direction of AC?
 
Alison A. said:
So I add this value with the unit vector in the direction of AC?
The tension in the cord is pulling the bead at B against the bar AC. You want to convert the magnitude of this tension, 125.6 N, into a force vector FBD parallel to BD.

Then, according to the directions in the problem statement, you want to take the dot product of FBD and the unit vector uAC to find out what magnitude P must be.
 
So the unit vector of BD is (0, -0.6031, 0.7977), then multiplied by the tension is (0, -75.75, 100.2)

The dot product of that and uAC is -3.367 which doesn't seem right...
 
Alison A. said:
So the unit vector of BD is (0, -0.6031, 0.7977), then multiplied by the tension is (0, -75.75, 100.2)

This looks OK.

The dot product of that and uAC is -3.367 which doesn't seem right...

I think it's correct. I think the fact that FBD ⋅ uAC is negative means that P is in the direction of point C.
 
The answer was positive, hm, anyway I got it right. So for finding the reaction between bead B and rod AC, it would be just the force that is perpendicular to P right? I know the hint says nothing about it but wouldn't the cross product between them?
 
  • #10
I know there are infinitely number of vectors perpendicular to bar AC, but I don't know how to find it specifically where bead B is. Thank you so much for your help, I need to figure out these problems within the next hour. :cry:
 

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