Two ropes are connected to a steel cable that supports a hanging weight (?Fig. P5.61?). (a) Draw a free-body diagram showing all of the forces acting at the knot that connects the two ropes to the steel cable. Based on your diagram, which of the two ropes will have the greater tension? (b) If the maximum tension either rope can sustain without breaking is 5000 N, determine the maximum value of the hanging weight that these ropes can safely support. Ignore the weight of the ropes and of the steel cable.
Solution 57P Step 1 : Let us consider the figure given In this question, we need find which cable has more tension What is maximum value of weights the cable can sustain, if the maximum tension either of the cable can handle is 5000 N Step 2 : Let us consider the data given Angle for cable 1 = 60 0 1 0 Angle for cable 2 =240 Let us find which cable has maximum tension Let us denote the tension along cable 1 as T and1tension along cable 2 as T 2 Case (1) Tension along horizontal length is given by For cable 1 T cos 1 1 Similarly for cable 2 it is given as T cos 2 2 Since there is no acceleration along horizontal direction We can write as T 1os =1T cos2 ----2-----------(1) Case ( 2) Tension along vertical direction It is obtained using For cable 1 T sin 1 1 Similarly for cable 2 it is given as T si2 2 Since we have mass M pulling down the cable along vertical direction we have T 1in +1T sin 2 = M -2--------------(2) Step 3 : We need to find the value of mass M Let us consider equations ( 1) T 1os = 1 cos 2 2 Rearrange this equation we get T2cos 2 T 1 cos 1 Consider equation (2) T 1in +1T sin 2 = M 2 Substitute the value of T in1equation (2) we get T2cos 2 cos 1 sin 1 T sin2 = M 2 We can write as T 2os ×2tan + T1sin 2= M 2 T 2cos 2 tan + s1n ) = M2 Substitute the value for = 60 and = 40 0 1 2 T 2cos 40 × tan 60 + sin 40) = M T (0.7660 × 1.7320 + 0.6427) = M 2 T 21.9696) = M It is given that the cable will break when the tension reaches 5000N Thus we can write as 5000 N × (1.9696) = M M = 6428 Hence we have the maximum weight the cables can sustain as 6428 N