Caternary Problem: Solve for Tension & Angles

  • Thread starter Radarithm
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In summary: I completely forgot that catenaries are chains! In summary, the tension at the lowest point is zero because the angle \theta is equal to the horizontal.
  • #1
Radarithm
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Homework Statement


In most problems in this book, the ropes, cords, or cables have so little mass compared to other objects in the problem that you can safely ignore their mass. But if the rope is the only object in the problem, then clearly you cannot ignore its mass. For example, suppose we have a clothesline attached to two poles (Fig 5.61). The clothesline has a mass [itex]M[/itex] and each end makes an angle [itex]\theta[/itex] with the horizontal. What are (a) the tension at the ends of the clothesline and (b) the tension at the lowest point? (c) Why can't we have [itex]\theta=0[/itex]? (d) blahblahblah [For a more advanced treatment of this curve, see K.R. Symon, Mechanics, 3rd Ed)

IMG_0741.jpg


Homework Equations



Newton's Laws

The Attempt at a Solution


Unfortunately, I have absolutely no experience with problems such as this. My work is here (look at the second attempt, the first one was garbage): http://s15.postimg.org/xoou3zhh7/IMG_0743.jpg , and I am pretty sure I have the wrong answer; if I feel like it's too easy I'm definitely doing it wrong.
 
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  • #2
Radarithm said:

Homework Statement


In most problems in this book, the ropes, cords, or cables have so little mass compared to other objects in the problem that you can safely ignore their mass. But if the rope is the only object in the problem, then clearly you cannot ignore its mass. For example, suppose we have a clothesline attached to two poles (Fig 5.61). The clothesline has a mass [itex]M[/itex] and each end makes an angle [itex]\theta[/itex] with the horizontal. What are (a) the tension at the ends of the clothesline and (b) the tension at the lowest point? (c) Why can't we have [itex]\theta=0[/itex]? (d) blahblahblah [For a more advanced treatment of this curve, see K.R. Symon, Mechanics, 3rd Ed)

IMG_0741.jpg


Homework Equations



Newton's Laws

The Attempt at a Solution


Unfortunately, I have absolutely no experience with problems such as this. My work is here (look at the second attempt, the first one was garbage): http://s15.postimg.org/xoou3zhh7/IMG_0743.jpg , and I am pretty sure I have the wrong answer; if I feel like it's too easy I'm definitely doing it wrong.

Hi Radarithm!

The tension acts along the rope, not horizontally. You only need to show mg at the centre of rope. Can you proceed now?
 
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  • #3
Doesn't the tension (on one end) need to support only half of the rope?
 
  • #4
Radarithm said:
Doesn't the tension (on one end) need to support only half of the rope?

Nope but its component would do. ;)
 
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  • #5
Got the answer I was looking for. Thanks for the help!
 
  • #6
nice :smile:

btw, it's catenary :wink:

(from the latin "catena" meaning "chain", cf. concatenation)
 
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  • #7
tiny-tim said:
nice :smile:

btw, it's catenary :wink:

(from the latin "catena" meaning "chain", cf. concatenation)

*facepalm* :redface: Thanks for correcting me.
 

Related to Caternary Problem: Solve for Tension & Angles

1. What is the catenary problem?

The catenary problem is a mathematical and engineering problem that involves finding the shape and tension of a hanging cable or chain under its own weight.

2. How is the tension in a catenary determined?

The tension in a catenary is determined by balancing the force of gravity with the tension in the cable. This can be mathematically expressed as T = λg, where T is the tension, λ is the weight per unit length of the cable, and g is the acceleration due to gravity.

3. What are the key factors that affect the shape and tension of a catenary?

The key factors that affect the shape and tension of a catenary are the weight per unit length of the cable, the distance between the supports, and the angle at which the cable is hung.

4. How do you solve for the tension and angles in a catenary problem?

To solve for the tension and angles in a catenary problem, you can use mathematical equations and principles such as the catenary equation, the law of cosines, and the law of sines. You can also use computer simulations and models to find the most accurate solutions.

5. What are some practical applications of solving catenary problems?

Solving catenary problems is important in engineering and construction, as it helps determine the appropriate tension and shape of cables and chains used in structures such as bridges, power lines, and suspension systems. It is also useful in understanding natural phenomena such as the shape of stalactites and the curve of a hanging rope or chain.

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