Two steel wires are connected together, end to end, and attached to a wall as sh

Question

Two steel wires are connected together, end to end, and attached to a wall as shown below. The two wires have the same length and elastic modulus, but the ratio of the radius of the first wire to the radius of the second wire is 5 : 4. As the wires are initially the same length, the midpoint of the combination coincides with the connection point. An applied force then stretches the combination by 2.300 mm while the two wires stay connected together. After the wires are stretched, what is 5 the distance from the midpoint to the connection point?

Explanation / Answer

Hi,

In this case we must know how to calculate the stress of the combination as well as the deformation of said combination.

A key data is the relation of the radii : r1/r2 = 5/4

The relation of the areas (considering that said areas are circles) is: A1/A2 = 25/16

The relation of the stresses is: S1/S2 = 16/25 (considering that S = F/A)

The relation of the changes in the length is: L1/L2 = 16/25 (1) (considering that L/L = S/Y)

Note: Y is the Young's module, which is equal for each wire, according to what the problem says.

Knowing that the sum of both variations (L1 and L2) is equal to the total variation (L) we have that:

L = 2.300 mm = L1 + L2 (2)

Using (1) and (2) we can find the value of each variation in the length:

L1 = (16/25)L2 :::::::::: L2 = (25/41) L = (25/41) 2.300 mm ::::::::: L2 = 1.402 mm ::::::: L1 = 0.898 mm

The distance L + L1 is the distance from the wall to the connection point.

The distance L + L/2 is the distance from the wall to the midpoint.

So the value of would be:

= (L + L/2) - (L + L1) ::::::: = L/2 - L1 = [ 2.3/2 - 0.898 ] mm = 0.252 mm

I hope it helps.

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