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Modulus Of Resilience Of A Molded Phenolic Plastic

Data from a tension test to determine the elastic properties of a molded phenolic (synthetic resin) plastic are as follows:

Specimen diameter, d = 0.400 __+__ 0.001 in.

Gage length, l = 1 __+__ 0.01 in.

Load at the proportional limit, P = 500 __+__ 20 lb

Elongation due to P, δ = 0.0030 __+__ 0.0001 in.

(a) Determine the modulus of resilience, u_{r}.

(b) Calculate the probable maximum relative error in u_{r}.

**The strings**:
S_{7}P_{3}A_{31} (Force - Pull).
**The math**:

Pj Problem of Interest is of type *force* (pull). Modulus of resilience is strain energy absorbed per unit volume. Energy and Work problems are in general force problems. In this case the force is tension force (pull).

Formulas of interest:

Modulus of resilience u_{r}, is strain energy absorbed per unit volume.

u_{r} = Pδ/2V = σ_{pl}ε_{pl}/2 ---------(1)

Where P is load at the proportional limit. Proportional limit is the maximum stress a material can sustain without deviating from the law of stress-strain proportionality.

δ is elongation due to P

V is volume

σ_{pl} is stress at proportional limit

ε_{pl} is strain due to σ_{pl}

The probable maximum relative error in u_{r} is given by the following equation:

du_{r}/u_{r} = [(dP/P)^{2} + (dδ/δ)^{2} + (2dA/A)^{2} + (dl/l)^{2}]^{1/2} --------(2)

Where each term in equation (2) represents *relative change*.

du_{r}/u_{r} = (dP/P) + (dδ/δ) + (2dA/A) + (dl/l) is obtained by taking the logarithm and partial differentiation of u_{r} = Pδ/2V.

(a) u_{r} = Pδ/2V.

So, u_{r} = 500(0.0030)/(2x.04πx1) = 6 lb-in.

So, modulus of resilience = 6 lb-in.

(b) du_{r}/u_{r} = [(dP/P)^{2} + (dδ/δ)^{2} + (2dA/A)^{2} + (dl/l)^{2}]^{1/2}

So, du_{r}/u_{r} = [(20/500)^{2} + (0.0001/0.0030)^{2} + 4(0.001/0.400)^{2} + (0.01/1)^{2}]^{1/2}

= [(0.04)^{2} + (0.033)^{2} + 4(0.0025)^{2} + (0.0001)]^{1/2} = [0.0028]^{1/2} = 0.053 = 5.3%

So, probable maximum relative error in u_{r} = 5.3%

Math

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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Derivation Of The Area Of A Trapezoid, A Rectangle And A Triangle

Derivation Of The Area Of An Ellipse

Derivation Of Volume Of A Cylinder

Derivation Of Volume Of A Sphere

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Volume Obtained By Revolving The Curve y = x^{2} About The X Axis

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Homogenizing-Non-Homogeneous-Time-Varying-IBVP-Boundary-Condition

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