Question

A student weighing 160 pounds hangs for dear life from a cable tied to two other cables fastened to a support as shown above. The upper cables make angles of 39º and 55º with the horizontal. These upper cables are not as strong as the vertical cable and will break if the tension in them exceeds 500 N.

Answer 1

The intention is to determine whether the cables will resist the tension or will break.

There are three tensions

Applyng Newton's Second Law to the student, the tension of the only cable that holds the student has to equal his weight,

T = weight = m*g = 160 lbs / 2.2046 lbs/kg * 9.8 m/s=711 N

Now apply Newton's Second Law to the joint of the cables

There you have that the equilibrium of forces leads to that the sum of the up-components of the other two cables = the tension T just found, i.e. 711 N.

Now find the up-components of the tensions of other two cables:

sin 39 = T_1up / T_1 => T_1up = T_1*sin(39)

sin 55 = T_2up / Ts => T_2up = T_2*sin(55)

Total up tension = T_1*sin(39) + T_2*sin(55)

Newton's second law => total up tension = tension of the cable that holds the student

T_1*sin (39) + T_2*sin(55) = 711 N  [equation 1]


Now find the equation from the horizontal equilibrium.

Horizontal-components fo the tension of the other two cables

cos 39 = T_1 left / T_1 => T_1 left = T_1*cos(39)

cos 55 = T_2 right / T_2=> T_2 right = T_2*cos(55)

Second Newton's Law and non movement => left-component = right component.

T_1 * cos(39) = T_2 cos(55)    [equation 2]

Equation 1 and equation 2 form a systems of two equations with two variables (T_1 and T_2).

When you solve it you find:

T1 = 711 / [sin(39) + tan(55)*cos(39)] = 711 / 1.739 = 408.9 N

T_2 = cos (39)*408.9 / cos (55) = 553. 9 N

Therefore this cable will break because the tension calculated exceeds 500 N.

Answer 2

T1 = tension in rope at 39 deg 
T2 = tension in rope at 55 deg 

Then for the vertical components: 
T1vertical = T1*sin(39) 
T2vertical = T2*sin(55) 

T1vertical + T2vertical = 160 

Now for the horizontal: 
T1horizontal = T1cos(39) 
T2horizontal = T2cos(55) 

T1horizontal = T2horizontal 
T1cos(39) = T1cos(55)

T1 = 711 / sin(39) + tan(55) x cos(39)
     = 711 / 1.739 = 408.9 N

T2 = cos (39) x 408.9 / cos (55)
     = 553. 9 N