Question

A wagon weighing 35KN is attached to a wire rope and moving down an incline plane at speed of 3.6Km/hr when the rope jams and the wagon is suddenly brought to rest. If the length of the rope is 60 meters at the time of sudden stoppage, calculate the maximum instantaneous stress and maximum instantaneous elongation produced. Diameter of rope = 30mm. E = 200GN/m^2.

Answer 1

Weight of the wagon W = 35KN

Speed of the wagon, v = 3.6 km/hr = 1m/sec

Diameter of the rope, d = 30mm = 0.03m

Length of the rope at the time of sudden stoppage, L = 60m

Maximum instantaneous stress σ_i ;

The kinetic energy of the wagon = 1/2.mv² = Strain energy

= 1/2. (35 × 1000 /9.81) × 12 = 1783 Nm or J ...(i)

This energy is to be absorbed by the rope at a stress σ_i

Now, strain energy stored = σ_i ².A.L/2E

= {σ_i² × π/4 × (0.032) × 60}/(2 × 200 × 10⁹) = 0.0106 σ_i²/1011 ...(ii)

 Equating equation (i) and (ii)

σ_i = 129.69 × 106 N/m² = 129.69 MN/m²

Maximum instantaneous elongation of the rope, 

δL Using the relation,

δL = σi .L/E = ( 129.69 × 106 × 60)/(200 × 10⁹)

δL = 389.07 × 10^–4 m = 38.9 mm