A steel (Young\'s modulus 2.0 x 10 11 N/m 2 ) wire is strung between two support
ID: 2259673 • Letter: A
Question
A steel (Young's modulus 2.0 x 1011 N/m2) wire is strung between two supports attached to a ceiling. Initially, there is no tension in the wire when it is horizontal. A 99-N picture is then hung from the center of the wire, as the drawing illustrates, so the ends of the wire make angles of 26
A steel (Young's modulus 2.0 x 1011 N/m2) wire is strung between two supports attached to a ceiling. Initially, there is no tension in the wire when it is horizontal. A 99-N picture is then hung from the center of the wire, as the drawing illustrates, so the ends of the wire make angles of 26 degree with respect to the horizontal. What is the radius of the wire?Explanation / Answer
It almost looks like there is not enough information, but with a little imagination, we can supply the numbers we need.
Formula
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Y = F * L/(dL * Area)
Suppose that the wire is initially 2L units long. That means that the wire will stretch to a length of 2 times the hypotenuse of the triangle - 2L.
Find the length of the hypotenuse
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cos(26) = L / hypotenuse
hypotenuse = L/cos(26) = 1.1126 L
Next
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Find the change in length
1.1126 L - L = 0.1126L
The total change in length is twice this amount because the hypotenuse is 1/2 the total length of the wire.
L/dL = 2L/2*0.1126L = 8.881
Sin(26)*T + Sin(26)*T = 86N This is the vertical component of the tension. The horizontal components cancel out. 2*sin(26)*t = 86
T = 98.09 N
F = 98.09N
A = ???
Y = 2*10^11 N/m^2
L/dL = 8.881
A = 98.09*8.881/2*10^11
A = 4.3557 * 10^-9
pi*r^2 = 4.357 * 10^-9
r^2 = 1.386 * 10^-9
r = 3.7325 * 10^-5 m
r = 3.7325*10^-2 mm which is not very much.
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