Find the Cartesian representation of the tension at A pulling toward B Find the

Question

Find the Cartesian representation of the tension at A pulling toward B Find the moment of the applied tension load of 9600 N about the origin. Observe that the effective total load at the origin is equal to the load resultant (the find that force quiz 3) you have calculated plus the moment at the origin from your cross product The stress analyst will use this information and formulas from Strength of Materials to calculate combined stresses The design group can now specify the material and shape of the parts. Cross product M ___________ = r Times F = _________ i [()() - ()()] = ___________ i j [()() - ()()] = ___________ j k [()() - ()()] = ___________ k M ____________ = [() i + () j + () k]

Explanation / Answer

Given that the tension in the cable AB is T = 9600 N.

Coordinates of various points are O(0,0,0), A(1.6,0,2) and B(2.4,1.5,0).

Tension in the cable at point A acts from A to B. Thus the unit vector along the direction AB from A to B is

nAB = {AB / AB} = [{OB - OA} / AB]

===> nAB = {(2.4 i + 1.5 j) - (1.6 i + 2 k)} / AB

===> nAB = {0.8 i + 1.5 j - 2 k}/{0.82+1.52+(-2)2}

===> nAB = {0.8 i + 1.5 j - 2 k}/2.625

Now tension in the cable AB at point A can be expressed as

TAB = T * nAB = (9600/2.625) * {0.8 i + 1.5 j - 2 k} N

===> TAB = 2925.847 i + 5485.963 j - 7314.618 k N

This is the tension in the cable AB acting at A towards B in Cartesian form.

Position vector from origin O to the point of application of the force, A is

r = OA = 1.6 i + 2 k

Moment of the applied tension load of TABat the origin O is given by

M = r x TAB= (1.6 i + 2 k) x (2925.847 i + 5485.963 j - 7314.618 k) N-m

Note the following vector cross products

i x i = j x j = k x k = 0.

i x j = - j x i = k, j x k = - k x j = i and k x i = - i x k = j.

===> M = {1.6 * 5485.963 i x j - 1.6 * 7314.618 i x k + 2 * 2925.847 k x i + 2 * 5485.963 k x j}

===> M = - 10971.926 i +17555.082 j + 8777.541 k N-m

Magnitude of the moment of the applied tension load at A is

M = {(-10971.926)2 + 17555.0822 + 8777.5412} N-m.

===> M = 22485.758 N-m.

This is the moment of the applied tension load at A about the origin at O.

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