Deformation of solids：練習問題

A wire is stretched by an increasing force until it eventually snaps. What is the name of the point on the force–extension graph beyond which the wire will no longer return to its original length when the force is removed?

A metal wire of length \(L\) and cross-sectional area \(A\) is stretched elastically by a force \(F\). A second wire made of the same material has the same total volume but has a length of \(2L\). What is the extension of the second wire when the same tensile force \(F\) is applied?

The diagram shows a mass of \(2.0\text{ kg}\) supported by a system of three springs. Two springs, each with a spring constant of \(200\text{ N m}^{-1}\), are connected in parallel. This combination is connected in series with a third spring of spring constant \(400\text{ N m}^{-1}\). Taking the acceleration of free fall \(g\) to be \(9.8\text{ m s}^{-2}\), what is the total extension of the system?

A metal spring obeys Hooke’s law. When a load of \(10\text{ N}\) is applied, the spring stretches by \(2.5\text{ cm}\). What is the extension of the same spring when the load is increased to \(20\text{ N}\), assuming the limit of proportionality is not exceeded?

Wire P has length \(L\), diameter \(d\), and Young modulus \(E\). Wire Q has length \(2L\), diameter \(d/2\), and Young modulus \(2E\). Both wires are subject to the same tensile force \(F\). If \(x_P\) and \(x_Q\) are the extensions produced in P and Q respectively, what is the ratio \(\frac{x_Q}{x_P}\)?

A specimen of a polymer is stretched beyond its elastic limit but before its breaking point. Explain, in terms of molecular structure, why the material does not return to its original length when the load is removed.

Two wires, P and Q, are made of the same metal. Wire P has length \(L\) and radius \(r\), while wire Q has length \(3L\) and radius \(2r\). If both wires are stretched by the same force \(F\), calculate the ratio of the extension of wire P to the extension of wire Q.

A copper wire of diameter \(0.80 \text{ mm}\) is subjected to a tension of \(60 \text{ N}\). Calculate the tensile stress produced in the wire.

(a) Define stress, strain, and the Young modulus.
(b) A uniform copper wire of length \(2.5 \text{ m}\) and diameter \(0.80 \text{ mm}\) is stretched by a force of \(45 \text{ N}\). The Young modulus of copper is \(1.2 \times 10^{11} \text{ Pa}\). Calculate the strain in the wire.
(c) Determine the extension produced by this force.

A composite wire consists of a steel section of length \(1.2 \text{ m}\) and a brass section of length \(0.8 \text{ m}\) joined end-to-end. Both sections have a diameter of \(1.0 \text{ mm}\). A load of \(80 \text{ N}\) is applied to the end of the composite wire.
Young modulus of steel = \(2.0 \times 10^{11} \text{ Pa}\).
Young modulus of brass = \(1.0 \times 10^{11} \text{ Pa}\).
(a) Calculate the extension of the steel section.
(b) Calculate the extension of the brass section.
(c) Determine the total elastic potential energy stored in the composite wire.