Example: Statics

A 75 kg gymnast hangs vertically from a pair of parallel rings.

Question:

  1. If the ropes supporting the rings are attached to the ceiling directly above, what is the tension in the ropes?
  2. If the ropes are supported so that they make an angle of 45o with the ceiling, what is the tension on the ropes?
  3. Analyze the tension on the ropes as the angle $\theta$ between the ceiling and the ropes becomes smaller and smaller.

Answer:

  1. In this part, there are no forces in the x direction. In the y direction we have
    $\sum$ Fy = T1 + T2 - mg = 0

    Since the tension has to be the same in both ropes, T1 = T2 = T, and we get:

    T = mg = (75 kg $\cdot$ 9.81 m/s2) = 370 N

     

  2. In this part, there are forces in the x and y directions. We will work this part for a general angle $\theta$ and then plug in $\theta$=45o in the end:
    $\sum$ Fx = T1 cos$\theta$ - T2 cos$\theta$ = 0
    $\sum$ Fy = T1 sin$\theta$ + T2 sin$\theta$ - mg = 0

    From the upper equation we again get T1 = T2 = T, and from the lower equation we then get:

    T = mg/2sin$\theta$ = 75 (9.81)/2sin(45) = 520 N

     

  3. The answer to part 3 is that, as one can see from the last formula, as $\theta$ goes to 0, T becomes infinitely big.

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