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Why does every cable have 2 different tensions?

  1. Sep 21, 2012 #1
    1. The problem statement, all variables and given/known data
    I was recently introduced to the concept of non-massless cables. I am used to solving problems where the cable is massless, and therefore tension on both ends of the cable are the same.

    However, my TA mentioned that this is only true in 3 instances.
    1. If the cable is massless.
    2. If there is no acceleration on the cables (that is Fnet=0)
    3. (Actually a special case of 2) When the cable is in free-fall.

    The reason for this is never explained however. When I was doing a problem, with a cable in between two blocks, I initially tried to make tensions on both ends of the cable to be the same. Yet, I was told this was incorrect, because the cable has mass.

    Can someone clarify the reason why cables always have two different tensions, even massless ones (where t1=t2)?
     
  2. jcsd
  3. Sep 21, 2012 #2
    A cable with mass doesn't just have two different tensions. The tension in the cable varies with location along the entire cable, in order for each small section of the cable to be accelerated. Draw a free body diagram on a small section of cable and use Newton's second law to predict how the tension changes with position if the mass per unit length of the cable is ρ.
     
  4. Sep 21, 2012 #3

    PhanthomJay

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    If a cable with mass is strung 'horizontally' between 2 blocks spaced a distance L apart, it assumes an equilibrium position in the shape of the catenary curve, which is very closely approximated by a parabolic curve when the sag is less than the a few percent of the horizontal span, where the amount of sag depends upon the total length of the cable between the blocks. Tension forces in cables always act along the length of the cable, and vary along its length. At the low point of the curve, where the tangent to the curve is horizontal, the tension force in the cable must be horizontal, and this is known as the horizontal tension in the cable. From equilibrium considerations in the horizontal direction, the horizontal tension is the same at any point in the curve, but there will also be a vertical component of the tension due to the cable's weight. This implies that at any point in the curve away from the low point, the tension in the cable is greater than the tension at the low point, becoming a maximum at the support where the angle that the tangent of the curve makes with the horizontal is greatest (and a minimum at the low point).
    I think you may have not understood the TA's explanation, or maybe he or she stated it incorrectly. A cable with mass does not have equal tension in it throughout its length, even if it is at rest, whether strung vertically or horizontally. You might want to clarify what you mean by a 'cable between 2 blocks'.
     
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