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I am trying to solve the following problem, but got lost in the process. enter image description here

I am trying to find the minimum/maximum angle ( θ ) needed to have the system in equilibrium without breaking the ropes which max tension can only support up to 700 lbs. Angle AB is fixed. The equations are as follows:

Eq1: ∑ Fx = TAB cos (25) - TAC cos (θ) = 0

Eq2: ∑ Fy = TAB sin (25) + TAC sin (θ) = 1000

Solving Eq1 for TAC, we have:

Eq3: TAC = TAB cos (25) / cos (θ)

You can conclude that at all times TAC is always greater than TAB. Therefore you can set TAC to 700 lbs in either Eq 1 or Eq2 and try to find θ But that still left me with the unknown TAB. How can I find θ ?

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  • $\begingroup$ Angle AB can't be fixed, as the weight will cause both ropes to elongate. You might want to double-check the statement of the question. Another note is that the equation for rotational equilibrium is missing. $\endgroup$
    – r13
    Jun 4 at 22:52
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TL;DR: There is something wrong with the problem definition. The minimum force on the $T_{AC}$ is approximately 906 lbs at $\theta=65\deg$ (in that case $T_{AB}= 422[lbs]$)


NOTE: From (Eq.3) what you really get is that:

$$ \frac{T_{AC}}{ T_{AB}} =\frac{ cos (25)} {cos (θ)} $$

which actually can be interpreted, as:

  • $T_{AC}>T_{AB}$ when $\theta > 25\deg$
  • $T_{AC}<T_{AB}$ when $\theta < 25\deg$

So basically you need to check first what happens at 25$\deg$.


at 25$\deg$

At 25$\deg$, there is symmetry so:

$$T_{AC} =T_{AB}$$

So from the second equation you can obtain that:

$$ ∑ Fy = T_{AB}= T_{AC}= \frac{1000}{sin (25)}=2366$$

which means that the force is already exceeding 1000.

So you would need to have an angle $\theta >25$.

Because then $T_{AB}<T_{AC}$, and if $T_{AC}= 700 [lbs]$ then $T_{AB}$ will also be $T_{AB}<700$.


Solving for $\theta$.

Since $\theta >25\deg$, $T_{AC}>T_{AB}$ and therefore you can substitute in Eq.2, the following form of eq.3 $ T_{AB} =T_{AC}\frac{cos (θ)}{ cos (25)} $ :

$$T_{AB} sin (25) + T_{AC} sin (θ) = 1000$$

$$T_{AC}\frac{\cos (θ)}{ \cos (25)} sin (25) + T_{AC} \sin (θ) = 1000$$

$$T_{AC}\left(\frac{\cos (θ)}{ \cos (25)} sin (25) + \sin (θ)\right) = 1000$$

$$T_{AC}\left(\cos (θ)\tan(25) + \sin (θ)\right) = 1000$$

$$T_{AC}\left(\cos (θ)\tan(25) + \sin (θ)\right) = 1000$$

From this equation we can solve for $T_{AC} = 1000\frac{1}{\cos (θ)\tan(25) + \sin (θ)}$, and we can plot this equation.

enter image description here

The problem is that the minimum value of $T_{AC}$ is at approx 900[lbs].

The reason is that the maximum value for the expression $\cos (θ)\tan(25) + \sin (θ)$ is for $\theta= 65\deg$. So for 65 deg, you can get the minimum $T_{AC}\approx 906$. (In that case, $T_{AB} = 422 [lbs]$).

So, you can't really protect the AC rope from breaking.

if you only needed $T_{AB}$ to be under 700[lbs], then you can achieve that for angles $\theta \ge 45\deg$. see graph below

enter image description here

#%%
import numpy as np
import matplotlib.pyplot as plt

# %%
xs = np.linspace(25,85,100)
ts = xs*np.pi/180
Ts = 1000/(np.cos(ts)*np.tan(25*np.pi/180)+ np.sin(ts))
TABs = Ts*np.cos(ts)/np.cos(25*np.pi/180)
# %%
plt.figure(figsize=(10,8))
plt.plot(xs,Ts, label='$T_{AC}$')
plt.plot(xs,TABs, label='$T_{AB}$')
plt.grid()
plt.legend(fontsize=15)
plt.xlabel('Angle[deg]', fontsize=15)
plt.ylabel('Forces [lbs]', fontsize=15)
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