Group Problem: You are planning to build a log cabin in Northern Minnesota on a remote hill with a beautiful view of the setting sun. You will drag the logs up a long sometimes rocky hill to the building site by means of a rope attached to a winch. You will need a rope for this job so you aim to know how much weight the rope would safely support. You are operating on a tight budget so matching the rope strength would be a cost saver. You know that the logs are heavy, and estimate the heaviest as 1,000 lbs. From maps you verify the hill is steeped at an angle of 8 = 70° with respect to the vertical, and you estimate a coefficient of kinetic friction between a log and the earthen hill as 0.5. When pulling a log you will ensure that the uphill acceleration is never more than 3.0 ft/s². The maximum recommended load is th of the nominal strength for the ropes considered as stated on the product labels, you have three ropes in mind: 12 kN, 18 kN, & 24 kN. Which one of these three rope strengths is best?
Group Problem: You are planning to build a log cabin in Northern Minnesota on a remote hill with a beautiful view of the setting sun. You will drag the logs up a long sometimes rocky hill to the building site by means of a rope attached to a winch. You will need a rope for this job so you aim to know how much weight the rope would safely support. You are operating on a tight budget so matching the rope strength would be a cost saver. You know that the logs are heavy, and estimate the heaviest as 1,000 lbs. From maps you verify the hill is steeped at an angle of 8 = 70° with respect to the vertical, and you estimate a coefficient of kinetic friction between a log and the earthen hill as 0.5. When pulling a log you will ensure that the uphill acceleration is never more than 3.0 ft/s². The maximum recommended load is th of the nominal strength for the ropes considered as stated on the product labels, you have three ropes in mind: 12 kN, 18 kN, & 24 kN. Which one of these three rope strengths is best?
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Transcribed Image Text:**Group Problem:** You are planning to build a log cabin in Northern Minnesota on a remote hill with a beautiful view of the setting sun. You will drag the logs up a long sometimes rocky hill to the building site by means of a rope attached to a winch. You will need a rope for this job so you aim to know how much weight the rope would safely support. You are operating on a tight budget so matching the rope strength would be a cost saver. You know that the logs are heavy, and estimate the heaviest as 1,000 lbs. From maps you verify the hill is steeped at an angle of θ = 70° with respect to the vertical, and you estimate a coefficient of kinetic friction between a log and the earthen hill as 0.5. When pulling a log you will ensure that the uphill acceleration is never more than 3.0 ft/s². The maximum recommended load is \(\frac{1}{10}\)th of the nominal strength for the ropes considered as stated on the product labels, you have three ropes in mind: 12 kN, 18 kN, & 24 kN. Which one of these three rope strengths is best?
**SOLUTION:** Using the step-wise strategy that we have practiced throughout this course, for forces that includes the drawing of Freebody and Force Diagrams, and an inventory of known quantities: max. log weight: W = 1,000 lbs, hill incline: θ = 20° (w.r.t. horizontal), coefficient of kinetic friction: μ = 0.5, and maximum acceleration for the log: a_max = 3 ft/s². Express the maximum log weight as a mass: m_log = ______ kg.
**STEP A:** Identify the forces on the Scenario, Freebody, and Force Diagrams below:
- **Scenario Diagram:** Displays a log on a hill inclined at 20° with a rope attached to a winch.
- **Freebody Diagram:** Shows a log with two vectors: one pointing up along the incline and another perpendicular to the incline.
- **Force Diagram:** Depicts a point with vectors indicating forces in both +x and +y directions.
These diagrams illustrate the forces acting on the log as it is pulled up the hill.
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