What is the magnitude of the force exerted by the bolt on the O-ring as the gymnast hangs from the bottom end of the rope?(Images have background info used to solve the problem) **Example 5.1: One-Dimensional Equilibrium****Problem:**A gymnast hangs stationary from a rope, which is bolted to the ceiling. The weights of the gymnast, the rope, and the O-ring are 500 N, 100 N, and 50 N, respectively. What are the magnitudes of the tensions at both ends of the rope?**Solution:****Step 1:**First, we sketch the situation (Figure 5.1a). Then we draw three free-body diagrams: for the gymnast, the rope, and the O-ring (Figures 5.1b–d). The forces acting on the gymnast are 500 N down and the upward force exerted on her by the rope. We don’t include the downward force she exerts on the rope because it doesn’t affect the situation. **Figure 5.1:**- **Diagram (a):** Sketch of the situation showing the gymnast hanging from the rope attached to an O-ring.- **Diagram (b):** Free-body diagram for the gymnast, showing weight (W = 500 N) downward and tension (T1) upward.- **Diagram (c):** Free-body diagram for the rope, showing tension T1 at the top and T2 at the bottom with weight (Wrope = 100 N).- **Diagram (d):** Free-body diagram for the O-ring, showing weight (W = 50 N) downward with a tension (Tcell) at the top.The diagrams illustrate action-reaction pairs, indicating the tensile forces and weights acting on each object.For a detailed solution, refer to the **Video Tutor Solution** linked via the QR code. **Applications of Newton’s Laws**---**Equilibrium of Forces on a Rope**Consider the gymnast, whose motionless state implies equilibrium. From Equation 5.1, we calculate:Sum of forces, \( \Sigma F_y = 0 \); thus, \( T_1 + (-500 \, \text{N}) = 0 \).This leads to:\[ T_1 = 500 \, \text{N} \]This ensures the forces balance, confirming that the gymnast's weight is supported.The forces on the gymnast do not form an action-reaction pair, but tension plays a significant role. Analyzing the rope:\[ \Sigma F_y = T_2 + (-100 \, \text{N}) + (-500 \, \text{N}) = 0 \]From this, we derive:\[ T_2 = 600 \, \text{N} \]In equilibrium, this tension is the force needed at the top of the rope, where it is anchored, to support the gymnast's weight. Observing the forces, \( T_2 \) must support both the gymnast and any additional weight or force at the top. **Practice Problem:**Calculate the force required at the bottom of the rope, where the gymnast is hanging: \[ \text{Answer: } 650 \, \text{N} \]---**TABLE 5.1: Approximate Breaking Strengths**| Material | Breaking Strength ||---------------------------|---------------------|| Thin wire (sine) | 50 N || Nylon clothesline rope | 400 N || Small aluminum hanger | \(3 \times 10^2 \, \text{N}\) || Steel cable | \(3 \times 10^4 \, \text{N}\) |Understanding the breaking strength of ropes is crucial, as it determines the maximum tension a rope can sustain before failure. These approximate values help in choosing suitable materials under various conditions.

Answered: Image /qna-images/answer/1c990c4d-5670-4f0d-8b73-bb2262037da5.jpg

in the future. EXAMPLE 5.1 One-dimensional equilibri23 22 Now let's apply the problem-solving strategy outlined above to a simple one-dimensional equilibrium prob- lem. A gymnast hangs from the lower end of a rope connected to an O-ring that is bolted to the ceiling. The weights of the gymnast, the rope, and the O-ring are 500 N, 100 N, and 50 N, respectively. What are the magnitudes of the tensions at both ends of the rope? SOLUTION SET UP First we sketch the situation (Figure 5.1a). Then we draw three free-body diagrams, for the gymnast, the rope, and the O-ring (Figure 5.1b, c, and d). The forces acting on the gymnast are her weight (magnitude 500 N) and the upward force (magnitude T₁) exerted on her by the rope. We don't include the downward force she exerts on the rope because it isn't a force that acts on her. We take the y axis to be directed vertically upward, the x axis horizontally. There are no x com- ponents of force; that's why we call this a one-dimensional problem. ►FIGURE 5.1 Our diagrams for this problem. We sketch the situation (a), plus free-body diagrams for the gymnast (b), rope (c), and O-ring (d). //////// O-ring Rope (a) Gymnast 7₂ H Gymnast Rope -X →→→X WG-500 N †₁ (b) Video Tutor Solution These are action-reaction pairs. (c) 4. We-100 N ceil O-ring -X W-50 N *L (d) CONTINUED

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...