Problem: A light cable is used to support three loads of 35 kN, 40 kN and 50 kN respectively. The cable is tied at its end to two pegs at the same level and 60m apart. The loads divide the distance between the pegs into four equal parts. If the length of the cable is 8.5m, find the shape of the cable and tension in its different segments.

Assumptions: The cable is assumed to be massless and inextensible. The loads are assumed to be point loads and the weight of the cable is neglected.

Solution:

Step 1: Draw the free body diagram of the cable and loads.

Step 2: Determine the reaction forces at the pegs. Since the cable is massless and inextensible, the horizontal reaction forces at the pegs must be equal and opposite. The vertical reaction force at one of the pegs can be determined using the equilibrium equation in the vertical direction.

Step 3: Divide the distance between the pegs into four equal parts. Label the points where the loads are applied as A, B and C. Label the points where the cable changes direction as D and E.

Step 4: Assume a parabolic shape for the cable between each pair of points. The equation of the parabola can be written as y=ax^2+bx+c where x is the distance from the left peg, y is the height of the cable at that point, and a, b, and c are constants to be determined.

Step 5: Apply the boundary conditions at each point to determine the constants a, b, and c. At points A and C, the height of the cable is zero. At point B, the height of the cable is the sum of the loads divided by the length of the segment. At points D and E, the slope of the cable is zero.

Step 6: Use the equations of the parabolas to determine the tension in each segment of the cable. The tension in each segment can be determined using the equation T=ρg√(1+(dy/dx)^2)Δx where ρ is the mass per unit length of the cable, g is the acceleration due to gravity, dy/dx is the slope of the cable at that point, and Δx is the length of the segment.

Step 7: Check the solution for consistency and accuracy.

Conclusion: The shape of the cable is a series of parabolic curves. The tension in each segment of the cable can be determined using the equations of the parabolas and the equation for tension.