How Do You Calculate the Tension and Elastic Properties of Suspended Wires?

[BadlyAddicted]

Homework Statement

Two wires each of 2.50m long, one of steel of diameter 1.0mm and the other of copper of diameter 0.56mm are suspended from two points in the same horizontal plane 200mm apart. The lower ends of the wires are fixed to the two ends of a uniform bar 200mm long and of weight 20.0N

A mass of 6.0kg is suspended from the center of the bar. The bar tilts at 0.8degrees to the horizontal. The young modulus of steel is greater than that of copper. Assume the wires remain vertical.

i) What is the tension in each wire?
ii) Calculate the extension of steel
iii) Determine the young modulus of the copper wire
v) Calculate the strain energy stored in the two wires.

The acceleration of free fall is 9.8ms^-2 and the Young modulus of steel is 2.0 * 10^11 Pa.

Homework Equations

The equations to be used are probably these (from my knowledge):
strain = extension/length
stress = tension/area
young modulus = stress/strain = FL/Ae

The Attempt at a Solution

i) [ 20 + (6 * 9.8) ] / 2= 39.4

If someone could work out ii and iii for me showing all the working i'd appreciate. i believe i can proceed from there

 

[Nate810]

with the information given.ii) Extension of steel = [ ( 20 + (6 * 9.8) ) / 2 ] * 0.01m / (2.50m)= 0.00796miii) Young Modulus of Copper = FL/Ae = [ ( 20 + (6 * 9.8) ) / 2 ] * 0.01m / (0.56mm * 0.00796m) = 2.37 * 10^11 Pa

 

[Moetasim]

ii) To calculate the extension of steel, we need to use the formula for Young's modulus: E = stress/strain. We know the stress (tension) in the steel wire is equal to the weight of the bar (20N) plus the weight of the mass (6kg * 9.8m/s^2 = 58.8N), divided by the cross-sectional area of the wire (pi * (1.0mm/2)^2 = 0.785mm^2). Therefore, the stress is (20N + 58.8N) / 0.785mm^2 = 99.2N/mm^2.
To find the strain, we can use the formula strain = extension/length. We know the length of the wire is 2.50m, so we just need to find the extension. The angle of tilt of the bar (0.8 degrees) is very small, so we can assume that the extension of the wire is equal to the length of the wire that is displaced by the angle of tilt. This can be found using trigonometry: sin(0.8 degrees) = extension/2.50m. Solving for extension, we get 0.034m. Therefore, the strain is 0.034m/2.50m = 0.0136.
Finally, we can calculate the extension of steel using the formula E = stress/strain. Plugging in our values, we get extension = (99.2N/mm^2) / (0.0136) = 7294.1mm = 7.29m.

iii) To determine the Young modulus of the copper wire, we can use the same formula as in part ii: E = stress/strain. We know the stress (tension) in the copper wire is equal to the weight of the bar (20N) plus the weight of the mass (6kg * 9.8m/s^2 = 58.8N), divided by the cross-sectional area of the wire (pi * (0.56mm/2)^2 = 0.246mm^2). Therefore, the stress is (20N + 58.8N) / 0.246mm^2 = 351.2N/mm^2. We also know the strain, which we calculated in part ii (0.0136).