Equilibrium and Statics involving a mass attached to three strings

In summary, the problem of finding the forces of the three strings in the given image is statically indeterminate as there are only two equations of statics for three supports. Additional equations must be introduced to solve the system. If the strings are completely inextensible, there is no solution. However, if the strings have some extensibility, a solution can be found by considering small extensions and using proportions to calculate the tensions in the strings. This method involves some differential calculus and may be beyond the grade 11 syllabus.
  • #1
alingy1
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Homework Statement



Find the forces of the three strings of this image.

http://www.flickr.com/photos/79276401@N05/8395815988/in/photostream

Homework Equations



Using gravitational constant to find force of the mass.
Using algebra to find the forces of the three strings.

The Attempt at a Solution


I tried to create two equations, one for the x-axis and another for the y-axis. However, in the x-axis, the two strings of 25° cancel themselves. Do you think there is a missing data? This blocks me from using substitution to find the other forces. How would you proceed?
 
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  • #2
You have three supports but only two equations of statics. Therefore, the problem is statically indeterminate. In order to solve this system, additional equations must be introduced.
 
  • #3
If the strings are completely inextensible then there is no way to solve this. You can obtain consistent solutions by setting e.g. the tension in the centre string to 0, or by setting the other two to 0.
In the real world, all strings are at least a little extensible. If you consider a small extension to the centre string, you can calculate the extensions to the other two. Taking the tensions to be in proportion to these, a solution can be found.
 
  • #4
Actually, I have never seen string extensibility in grade 11. Your reply interests me. Could you explain how we could proceed to find the three forces? How do we find how much they extend? Very bizarrely, the teacher said this question was in last year's exam...
 
  • #5
It requires a little differential calculus, so I'm guessing that also puts it beyond your syllabus.
Let the two angled strings be attached w from the centre. If the centre string has length x, the other two have length √(x2+w2). If the centre string is stretched by amount δx the other two are each stretched √((x+δx)2+w2) - √(x2+w2) ≈ xδx/√(x2+w2) = δx sec(θ). So the tension in the side strings is sec(θ) times that in the centre string. That's enough extra info to solve it.
 

FAQ: Equilibrium and Statics involving a mass attached to three strings

1. What is the definition of equilibrium in a system involving a mass attached to three strings?

Equilibrium in this system refers to a state in which the forces acting on the mass are balanced, resulting in no net acceleration.

2. How is the tension in each string calculated in this type of equilibrium problem?

The tension in each string is calculated by using the principles of statics and Newton's laws of motion. The sum of the forces in the x and y directions must equal zero, and the sum of the torques must also equal zero.

3. What is the significance of the angles of the strings in this type of equilibrium problem?

The angles of the strings are crucial in determining the magnitude and direction of the tension forces. They also affect the overall stability and balance of the system.

4. How does the addition of a mass to one of the strings impact the equilibrium of the system?

Adding a mass to one of the strings changes the distribution of forces and can result in a shift in the angles of the strings. This can affect the overall equilibrium of the system and may require recalculating the tension in each string.

5. Can the equilibrium of a system with a mass attached to three strings be affected by external forces?

Yes, external forces such as wind or vibrations can disrupt the equilibrium of the system by adding additional forces and changing the angles of the strings. This must be taken into account when solving equilibrium problems in real-world scenarios.

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