Question

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Answer 1

Answer:

a) TAB = 4905[N]; b) TPB = TRB = 4275.8[N]; c) TAB = 43600 [N]; d) N = 8 girders; f) α = 19.54[deg]; g) TAC = 211868.8[N]; h)

Explanation:

This problem is related to the third law of Newton, and we must use the static for the solution of this problem.

a)

For the tension in cable AB, we can make a free body diagram making a cut in the cable and analyzing its internal force. (The free body diagram can be found in the attached image01). Since the velocity is constant there is not acceleration and the sum of forces can be equal to zero.

$T_{AB} = 4905 [N]$

b)

To find the tension in the cable PR, we need to make a free body diagram of the girder with the cables. (The free body diagram can be found in the attached image01 ), First, we need to make a sum of forces in the x-axis and we can realize that the tension in cables PB and RB are equals, then making a sum of forces in the y-axis we can determine the force in each of the cables.

$T_{PB}=4275.8[N] and T_{RB}=4275.8[N]$

c)

The force that will topple the crane, will be the force of the load or enough amount of girders that makes the reaction on the support G equal to zero. Therefore we need to draw a free body diagram with the forces acting in each of the supports (in the attached image02 we can find this analysis).

In this analysis we must have in count all the external forces.

And for the solution we can make a sum of moments around the point H, keeping in mind that the force in the support G is zero.

$T_{AB} = 43600 [N]$

d)

This analysis is simply because we have the force = 43600 [N], now we divide this value by the gravity and we will have the mass.

m = 43600/9.81

m = 4444.44[kg]

Then dividing this amount between the mass of one girder we can obtain the number of the girders.

N = 4444.44 / 500

N = 8.8 (girders) or with 8 girders will be enough load to avoid the topple of the crane.

e)

For this solution we can dran the jib with all the forces acting over it, the free body diagram can be seen in the attached image03.

f)

This angle can be found using a trigonometric analysis. In the attached image04 we can see two right triangles with blue and red color. So we need to find each of the angles on those two triangles.

For the red triangle we know all the dimensions of its sides, therefore the angle α1, formed between the jib and the clable AB will be equal to:

$tan(\alpha _{1} )=\frac{9}{13}\\\alpha _{1} = 34.7[deg]$

Now for the blue triangle the horizontal distance will be x = 9 + (4 - 0.5); x = 12.5 [m].

The vertical distance is y = 9 [m]

For the blue triangle we know all the dimensions of its sides, therefore the angle α2

$tan(\alpha _{2} )=\frac{12.5}{9}\\\alpha _{2} = 54.24[deg]$

Now with those two angles, we can find the angle between the cable AC and the Jib.

α = α2 - α1 = 54.24 - 34.7 = 19.54 [deg]

g)

We need to multiply the 500 kg by 5 and we will have the tension in the cable AB

TAB = 500*9.81*5 = 24525 [N]

With this force we can make the free body diagram on the jib and find the tension in the cable AC, this free body diagram can be found in the attached image05. The angle between the cable AC and cable AB is 54.24.

Then we make a sum of moments in point D to find the tension in the cable AC.

TAC = 211868.8[N]

h)

Because now the movement includes some acceleration and the sum of forces won't be equal to zero, this time all the forces will be equal to the product of the mass by the acceleration, this product will increase the tension in the cables and the stability of the system.