Christian is making a Tyrolean traverse as shown in Fig....

Christian is making a Tyrolean traverse as shown in Fig. $4-35 .$ That is, he traverses a chasm by stringing a rope between a tree on one side of the chasm and a tree on the opposite side, $25 \mathrm{~m}$ away. The rope must sag sufficiently so it won't break. Assume the rope can provide a tension force of up to $29 \mathrm{kN}$ before breaking, and use a "safety factor" of 10 (that is, the rope should only be required to undergo a tension force of $2.9 \mathrm{kN}$ ) at the center of the Tyrolean traverse. ( $a$ ) Determine the distance $x$ that the rope must sag if it is to be within its recommended safety range and Christian's mass is $72.0 \mathrm{~kg} .(b)$ If the Tyrolean traverse is incorrectly set up so that the rope sags by only one-fourth the distance found in $(a),$ determine the tension force in the rope. Will the rope break?

Key Concepts

Safety Factors in Engineering

A safety factor is an additional design criterion used to ensure that a structure can support loads beyond those expected under normal conditions, thereby providing a margin of safety. By designing systems such that the expected forces are only a fraction of the maximum capacity, engineers reduce the risk of failure due to unforeseen overloads or imperfections in the material or construction.

Cable Sag and its Approximation

When a cable is suspended between two supports and subjected to loads, it forms a curve typically approximated by a catenary or a parabolic shape under certain simplifying conditions. The sag, or vertical displacement at the midpoint relative to the supports, is a key parameter that influences the tension in the cable. Approximations based on the sag help in estimating the forces in the cable for safe design and analysis.

Tension in a Cable

Tension is the force transmitted through a cable or rope when it is subjected to a load. It is not uniformly distributed along the length and typically reaches its maximum near the supports. Understanding how tension varies with distance, sag, and applied loads is crucial for ensuring that the cable or rope does not experience forces exceeding its safe limits.

Static Equilibrium Analysis

This concept involves setting the sum of all forces and moments in a structure equal to zero, ensuring that the system is at rest. In the context of cables and ropes under load, static equilibrium allows us to derive relationships between applied loads, sag, and tension forces, making it a fundamental tool in structural analysis and design.

Transcript

00:01 Christian is making an entirely interverse as shown in figure 435.

00:04 That is, he traverses a chasm by stringing a rope between a tree and one side of the chasm in a tree on the opposite side, 25 meters away.

00:14 The rope must act sufficiently so won't break.

00:16 Assume the rope can provide attention force up to 29kn before breaking and use the safety factor of 10.

00:33 That is, the rope should only be required to undergo a tension force of 2 .9kn.

00:37 At the center of the tireline traverse, this should happen.

00:41 Okay, so for a, we want to determine the distance x at the rope and a sag if it is to be within its recommended safety range in christian's mass is 72 kilo kilograms.

00:52 So here's what we know.

00:54 We know the distance is 25 meters.

00:57 We know that we can undergo up to 29kn.

01:11 We know that the tension force is 2 .9 and the mass is 72 kilograms.

01:30 So we're looking for distance x at the rope must sag.

01:36 We want a quick visual here.

01:40 Then we've got our tension force vectors and we've got theta, the angle.

02:00 We're going to have our gravity.

02:04 Gravity and we've got our distance of 25 and we want to find x.

02:18 One way to find x is by using theta.

02:24 We could say that the tangent of theta is equal to opposite over adjacent and so the opposite could be the value of x over the adjacent which would be 25 over 2.

02:52 We just need to find theta.

02:59 There are two forces acting on this man...

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