Question

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.

Answer 1

According to the Newton’s second law, the equation for the net force is,

Here, m is the total mass of the system and a is the acceleration of the system.

Draw the free body diagram for the given system as follows,

(a)

Find the tension which is greater.

The free body diagram shows all the forces acting at the knot connecting the two ropes to the steel cable.

The net horizontal force acting is,

Here, is the horizontal component of the tension , and is the horizontal component of the tension .

Rewrite the above equation as follows,

Substitute for , and for in above equation.

From the above relation between the tension and tension , it is cleared that the tension is 0.65 times greater than the tension , that is,

Therefore, the rope which making an angle has greater tension.

(b)

Calculate the maximum value of the hanging weight.

The net vertical force acting on the system is,

Here, is the vertical component of the tension , is the vertical component of the tension , and is the hanging weight force.

Rearrange the above equation for w.

Substitute 5000 N for , for , and for in above equation.

Therefore, the hanging weight in the rope is .