Exercise from Theory and Problems of Physics for Engineering and Science.

In the crane here the boom is $3.2 \text{m}$ long and weighs $1200 \text{N}$. The cable can support a tension of $10000 \text{N}$. The weight is attached $0.5 \text{m}$ from the end of the boom. What maximum weight can be lifted?

I would add a image, but Imgur is failing right now. Here's my attempt to explain in the meantime...

Let there be a base. Towards the left is one end of the cable. Towards the right is one end of the boom. The weight is attached to the other end of the cable. The boom is up $45^\circ$ from the base. The cable is $30^\circ$ from the base (by hanging over the other end of the boom).

       _/ / |
cable_/  /  |
   _/   /   O weight
 _/    /boom
/ 30  / 45


  • weight's weight: $W_w$
  • boom's weight: $W_b$
  • cable's tension: $T$
  • boom's length: $l$

Let's take the torque relative to the base of the boom.

$ \tau = \tau_{W_w} + \tau_{W_b} - \tau_T = \sin{45} W_w l + \sin{45} W_b \frac{1}{2}l - \sin{15} T l = 0 $

$W_w = \frac{\sin{15} T - \frac{1}{2} \sin{45} W_b}{\sin{45}} \approx 3060 $

Note that $l$ is lost in the final equation. I find this dubious because if $l$ is larger there should be a larger $\tau_{W_b}$ because the CM is higher.

max weight: $3060 \text{N}$

Is my solution correct? If not, where and why?


closed as off-topic by Floris, sammy gerbil, Yashas, John Rennie, Jon Custer Mar 1 '17 at 13:57

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  • $\begingroup$ Looks like the right approach - but note that "check my work" questions are considered off topic on this site. $\endgroup$ – Floris Feb 28 '17 at 19:21
  • $\begingroup$ @Floris -- Oh, I apologize. I followed everything in the meta on HW questions, so I thought it was alright. $\endgroup$ – Fine Man Feb 28 '17 at 19:25
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    $\begingroup$ It might have been hard to find, but per our recommendation on asking homework questions, "It's not enough to just show your work and ask where you went wrong. If you just need someone to check your work, you can always seek out a friend, classmate, or teacher. As a rule of thumb, a good conceptual question should be useful even to someone who isn't looking at the problem you happen to be working on. $\endgroup$ – Floris Feb 28 '17 at 19:33
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    $\begingroup$ @Floris -- As a homeschooler, I have no friends (or, none that know more than me) or classmates, and I'm self-taught (with the help of books, of course), so this is the only "helper" resource for me. What if I asked this in one of the chat rooms? $\endgroup$ – Fine Man Feb 28 '17 at 19:35
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    $\begingroup$ @SirJony If you rephrase the question to just be about why conceptually your answer doesn't depend on $l$, I think that would be a good question. $\endgroup$ – Brian Moths Feb 28 '17 at 19:37

Your approach is correct; your doubt ("why do I lose $\ell$ from the equation?") comes about from the fact that you think "if the boom is longer, the torque is bigger". Which is true.

However, the arm of the tension of the cable gets longer by the same factor - and this is why things cancel.

Your intuition is not happy about this, because you know that a longer boom is normally heavier. However, this is only true if the weight of the boom is a function of length - and in this problem, that was not the case.

I hope this helps.

  • $\begingroup$ OK, I see. BTW, I notice you say "approach is correct". Are you suggesting that I made some algebraic error in the process? $\endgroup$ – Fine Man Feb 28 '17 at 21:06
  • $\begingroup$ I am saying I did not try to replicate your calculation. $\endgroup$ – Floris Feb 28 '17 at 21:08
  • $\begingroup$ Oh, OK. :) I'm pretty sure I didn't make any stupid errors (but I'll verify just in case). $\endgroup$ – Fine Man Feb 28 '17 at 21:09

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