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Exercise S-1.1.5

Practice Exercise in Technical Mechanics 1, Rigid Body Statics.

Topic: Central Force Systems - First Basic Task: Reduction to a Single Force

Practice Exercise S-1.1.5

Central Force Systems - First Basic Task: Reduction to a Single Force

Problem Statement

The traction rope of a conveyor system runs at an angle of \(\gamma = 40°\) to the vertical from the pulley. Due to the mass of the conveyor basket, a rope force F1 = 50 kN results.

Central Force System
Fig. 1: Problem statement
  1. What is the magnitude of the resultant of the two rope forces that act as bearing loads in pulley bearings A?
  2. At what direction angle \(\alpha_R\) (measured from the positive x-axis) does the resultant act?
Before You Embark on This Quest: A Pre-Mission Briefing

Before you dive into this adventure, let's do a quick check to see if you're ready for the challenge.

What's the Big Deal?
What topic are we dealing with here? Can you explain why?

All the lines of action of the attacking forces intersect at one point. So, we're dealing with a central force system.

Need a Little Refresher?

If you're not quite familiar with central force systems, no worries! Here are some refresher courses to get you up to speed:

Alright, You're Ready for the Task!

Good luck!

Calculating the Resultant Force

First, visualize the central force system for this problem. Draw a Cartesian coordinate system with the origin (0,0) at the intersection of the lines of action (w1,1 and w1,2) of the two rope forces.

Then, draw the two forces F1 (F1,1 and F1,2) shifted to this intersection point. You can do this according to the Principle of Transmissibility of Forces without changing the force system.

Consider the direction angles \(\alpha_{1,1}\) and \(\alpha_{1,2}\) of the forces, and plot them in your sketch. The direction angle of a force is measured counterclockwise from the positive x-axis.

Your sketch should look something like this:

Central Force System
Fig. 2: Sketch

The values for F1,1, F1,2, \(\alpha_{1,1}\) and \(\alpha_{1,2}\) are:

  • \(F_{1,1}\) = 50 kN (according to the problem statement)
  • \(F_{1,2}\) = 50 kN (according to the problem statement)
  • \(\alpha_{1,1}\) = 270° (according to the problem statement, sketch)
  • \(\alpha_{1,2} = \alpha_{1,1} + \gamma = 270° + 40° = 310°\)
  1. What is the magnitude of the resultant of the two rope forces that act as bearing loads in pulley bearings A?

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