Answered step by step
Verified Expert Solution
Link Copied!

Question

1 Approved Answer

The particle M in Figure 1 is suspended by using the two (soft) ropes MA and MB, which are made of the same material and

The particle M in Figure 1 is suspended by using the two (soft) ropes MA and MB, which are made of the same material and have equal diameters. The x and y coordinates of points A and B, measured in metres, are (-1,2) and (1,2), respectively. In the design, point M is required to be located at a grid point (blue circle) on the grid that is uniform and defined over the region 1.2 m x 1.2 m, 0.5 m y 1.5 m with the elements being squares of side lengthx=y=0.1m . Let F1andF2be the internal force vector in MA and MB, respectively. Your tasks are to find possible grid points for particle M to be suspended, and which grid points the ropes can have a minimum diameter. It is noted that one can adapt the lengths of the ropes to new locations of point M, the ropes are soft (i.e. subjected to only tensile forces), and the lower the internal force magnitude the smaller the diameter of the rope will be.

For each location of point M, the conditions for point M to be in equilibrium (Figure 2) are

  • the total force vector acting on point M is zero

F1+F2+W=0 equation (1)

whereW =(0,-W) is the weight (W is the magnitude of W).

  • F1=(F1x,F1y)is a tensile force lying on MA.
  • F2=(F2x,F2y)is a tensile force lying on MB

Solve the directions ofF1 andF2 are chosen from M to A and from M to B, respectively (i.e. tensile forces), and the vector equation (1) is converted into the following algebraic equation set:

F1x+F2x=0 equation (2)

F1y+F2yW=0 equation (3)

or

MAxAxMF1+MBxBxMF2=0 equation (4)

MAyAyMF1+yBMByMF2=W equation (5)

where

MA=(xAxM)2+(yAyM)2&MB=(xBxM)2+(yByM)2

are the lengths of the ropes MA and MB, respectively. The unknowns to be found in equations (4) & (5) are the two force magnitudesF1 andF2 (2 equations and 2 unknowns). If the negative force magnitude is obtained, the corresponding rope is not in tension and thus the equilibrium conditions are not satisfied.

W= 5

Perform hand calculations to solve:

1. Establish the algebraic equation system based on (4)-(5) for each following case: M(1, 0.5) m and M(0, 0.5) m.

2. Express the two systems of algebraic equations in Requirement 1 in matrix form.

3. Find the internal forces for both cases in Requirement 1. Report the answers rounded to three significant figures.

Step by Step Solution

There are 3 Steps involved in it

Step: 1

blur-text-image

Get Instant Access to Expert-Tailored Solutions

See step-by-step solutions with expert insights and AI powered tools for academic success

Step: 2

blur-text-image

Step: 3

blur-text-image

Ace Your Homework with AI

Get the answers you need in no time with our AI-driven, step-by-step assistance

Get Started

Recommended Textbook for

Applied Physics

Authors: Dale ewen, Neill schurter, P. erik gundersen

10th Edition

978-0136116332, 136116337, 9780132830096, 978-0132109277

More Books

Students also viewed these Physics questions