Introduction to Mechanical Engineering Homework 2

HOMEWORK 2 EQUILIBRIUM - CONCURRENT SYSTEM OF FORCES SYSTEM ELEMENTS

Background

Rock climbing is one of the world’s favorite adventure sports. In the Black Hills of South Dakota this recreational activity is very popular. Rock climbing involves scaling the face of a rock wall vertically. Generally, two climbers work together and use a system of ropes and anchors to catch accidental falls in case they occur. There are different styles and types of rock climbing, but the most common ones are top roping and lead climbing. In this example, we will focus on top roping.

Top roping is when a climber is attached to a rope that goes through an anchor at the top, while a person, called the belayer, stands on the ground at the bottom and constantly removes the slack in the rope as the climber ascends. This is shown in Figure 1. A picture of lead climbing is presented in Figure 2.

Source: https://climbersrock.net/climbing-types Figure 1. Top roping [1].

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Source: https://www.ukclimbing.com/news/2014/05/melissa_le_neve_climbs_wallstreet_8c-68923 Figure 2. Lead climbing [2].

The most extreme form of rock climbing is called free soloing or free climbing, shown in Figure 3. Here, climbers use no safety equipment. This form of climbing is only performed by the most advanced and professional rock climbers due its extreme danger.

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Source: https://www.nationalgeographic.com/magazine/2019/02/alex-honnold-made-ultimate-climb-el-capitan- without-rope/ Figure 3. Alex Honnold making his free solo up El Capitan [3].

Video

Watch the video given below to get a closer look at rock climbing and some of the techniques that are involved in climbing up a vertical slab of stone.

Rock Climbing Technique (duration: 4:34) [4] URL: https://www.youtube.com/watch?v=GdvmNTZzYZ4

Systems Perspective

For rock climbing to be conducted safely, various pieces of high-quality gear are required. Since a system is a combination of interacting elements organized to achieve one or more stated purposes, the rock-climber and his/her gear can be viewed as a system. Also, since a system element is a discrete part of a system that can be implemented to fulfill specified requirements, each piece of equipment used by the rock-climber can be considered a system element. Table 1 lists the gear frequently used for top roping and the function of each element. Figure 4 shows some of the gear that is commonly used for top roping, lead climbing, or traditional climbing.

The operating environment or context of a system is the collection of elements that specifically do not belong to the system but that interact with the system. In the case of rock climbing, the operating environment of the system includes rock faces and the weather.

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Table 1. Gear frequently used for top roping and their functions [5]. Element Function A belay device uses friction to control the descent of a climber, or to 1. Belay Device stop a fall. 2. Carabiners Lightweight connection devices. A harness is used to connect the rope to a climber. Other equipment 3. Harness such as a powder bag can also be attached to it. 4. Helmet A device to protect the climber’s head. Quickdraws are used to connect ropes to anchors in the rock face. 5. Quickdraws They allow the rope to move freely through the loop without too much friction. Rope is attached to the harness and anchors. Two types are used in climbing, static and elastic. Elastic ropes are usually used for 6. Rope belaying. They stretch if a climber falls and absorbs energy, reducing the impact felt by a climber. Static ropes are used in anchoring systems. Climbing shoes protect the climber’s feet and provide traction on the 7. Climbing Shoes rock face. Nuts and Hexes are wedged in cracks in the rock face and act as 8&9. Nuts and Hexes anchor points. They are used to limit the distance of a fall. 10. “Figure 8” A “Figure 8” is commonly used for rappelling down the rock face. 11. Chalk Bag Holds chalk powder that is used to give the fingers more traction.

Sources: https://weighmyrack.com https://www.rothco.com/product/rothco-figure-8-climbing-ring https://www.backcountry.com/the-north-face-chalk-bag-pro Figure 4. Rock climbing equipment [6-8].

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Problem 1

Watch the following video on basic scuba diving gear and make a table like Table 1 listing the different elements and their function. Make sure that you list every element mentioned in the video.

Video URL: https://www.youtube.com/watch?v=O9yJMQNqoRM [9]

Problem 2

Given: Consider the case of a rock-climber scaling a rock face as shown in Figure 5. The climber has a mass 75 kg and uses just one leg to keep himself off the wall. In the configuration shown, the climber is in static equilibrium. Also, as shown in Figure 6, the normal force between the shoe of the climber and the rock makes an angle of 30° with the horizontal, and the rope makes an angle of 10° with the vertical.

Find: The magnitude of the tension in the rope and the magnitude of the normal reaction force acting on the shoe. Express your answers to three significant figures. Use the SI system of units.

Assumptions: To simplify the system, we will consider a concurrent system of forces.

Source: https://www.99boulders.com/climbing- gear-for-beginners Figure 6. FBD of the Rock Climber, Coordinate Figure 5. Rock climbing scenario [10]. Axes.

Problem 3

Given: With the help of a friend, a rock-climber is attempting to scale a rock face in Yosemite National Park. The rock-climber and his friend are in static equilibrium in the situation shown in Figure 7. Assume that the rope goes directly downwards as shown in the FBD. The weight of the rock-climber is 160 lb and the person on the ground weighs more than him. The tension in the continuous rope is uniform at all points.

Find: The tension in the quickdraws.

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Source - caricature: https://rockandice.com/climb-safe/climb-safe-top-roping-is-not-so-safe/ Source - anchor: http://www.climbri.org/AMC/Instruction/Sport-Climbing/draws-on-anchor2.jpg Figure 7. Top roping scenario [11,12].

References 1. Top roping: https://climbersrock.net/climbing-types 2. Lead climbing: https://www.ukclimbing.com/news/2014/05/melissa_le_neve_climbs_wallstreet_8c-68923 3. Free solo: https://www.nationalgeographic.com/magazine/2019/02/alex-honnold-made- ultimate-climb-el-capitan-without-rope/ 4. Rock climbing technique: https://www.youtube.com/watch?v=GdvmNTZzYZ4 5. Rock climbing equipment: https://en.wikipedia.org/wiki/Rock-climbing_equipment 6. Rock climbing gear: https://weighmyrack.com 7. “Figure 8”: https://www.rothco.com/product/rothco-figure-8-climbing-ring 8. Chalk bag: https://www.backcountry.com/the-north-face-chalk-bag-pro 9. Scuba diving gear video: https://www.youtube.com/watch?v=O9yJMQNqoRM 10. Outdoor scene: https://www.99boulders.com/climbing-gear-for-beginners (Visited on 4/16/2019) 11. Rock climber: https://rockandice.com/climb-safe/climb-safe-top-roping-is-not-so-safe/ 12. Anchor: http://www.climbri.org/AMC/Instruction/Sport-Climbing/draws-on-anchor2.jpg

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