Solution Found!
A cylindrical wire of radius R elongates when subjected to
Chapter 3, Problem 15E(choose chapter or problem)
A cylindrical wire of radius elongates when subjected to a tensile force
. Let \(L_{0}\) represent the initial length of the wire and let \(L_{1}\) represent the final length. Young's modulus for the material is given by
\(Y=\frac{F L_{0}}{\pi R^{2}\left(L_{1}-L_{0}\right)}\)
Assume that \(F=800 \pm 1 \mathrm {~N}\), \(R=0.75 \pm 0.1 \mathrm{~mm}\), \(L_{0}=25.0 \pm 0.1 \mathrm{~mm}\), and \(L_{1}=30.0 \pm 0.1 \mathrm{~mm}\).
a. Estimate , and find the uncertainty in the estimate.
b. Of the uncertainties in F, R, \(L_{0}\), and \(L_{1}\), only one has a non-negligible effect on the uncertainty in . Which one is it?
Equation Transcription:
Text Transcription:
L_0
L_1
Y={FL_0}over{{pi}R^2(L_1-L_0)}
F=800{+/-}1 N
R=0.75{+/-}0.1 mm
L0=25.0{+/-}0.1 mm
L1=30.0{+/-}0.1 mm
L_0
L_1
Questions & Answers
QUESTION:
A cylindrical wire of radius elongates when subjected to a tensile force
. Let \(L_{0}\) represent the initial length of the wire and let \(L_{1}\) represent the final length. Young's modulus for the material is given by
\(Y=\frac{F L_{0}}{\pi R^{2}\left(L_{1}-L_{0}\right)}\)
Assume that \(F=800 \pm 1 \mathrm {~N}\), \(R=0.75 \pm 0.1 \mathrm{~mm}\), \(L_{0}=25.0 \pm 0.1 \mathrm{~mm}\), and \(L_{1}=30.0 \pm 0.1 \mathrm{~mm}\).
a. Estimate , and find the uncertainty in the estimate.
b. Of the uncertainties in F, R, \(L_{0}\), and \(L_{1}\), only one has a non-negligible effect on the uncertainty in . Which one is it?
Equation Transcription:
Text Transcription:
L_0
L_1
Y={FL_0}over{{pi}R^2(L_1-L_0)}
F=800{+/-}1 N
R=0.75{+/-}0.1 mm
L0=25.0{+/-}0.1 mm
L1=30.0{+/-}0.1 mm
L_0
L_1
ANSWER:
Answer:
Step 1 of 3
In this question we are asked to estimate the Young’s modulus of cylindrical wire of radius
and the uncertainty in the estimate.
Young’s modulus for the material is given by
…………….(1)
Given parameters:
Tensile force = 800
N
Radius = 0.75
0.1 mm
Initial length mm
Final length 0.1 mm
The quantity which we are estimating has several measurements. In this question we are measuring the radius , tensile force
, Initial length
and Final length
. These quantity mentioned here is uncertain and independent measurements.
So here we can use the following basic formula.
……………...(2)
Where ,
,........,
are independent measurements and uncertainties are
,
,.........,
Equation 2 is called the multivariate propagation of error formula.