Mechanical properties of matter：練習問題

A spring has a force constant of \(k\). It is compressed by a distance \(x\) from its natural length. Which expression correctly gives the elastic potential energy stored in the spring?

A sample of a material is subjected to a tensile test. The stress-strain graph shows a linear region followed by a region where the strain increases significantly for small increases in stress, eventually leading to a fracture with no significant plastic deformation before the break. Which statement best describes the material?

A wire is stretched by a force \(F\) to produce an extension \(e\). If a second wire made of the same material has twice the length and twice the diameter of the first wire, what force is required to produce the same extension \(e\) in the second wire?

A uniform wire of length \( L \) and cross-sectional area \( A \) is made of a material with Young modulus \( E \). The wire is stretched within its limit of proportionality by a force \( F \). If a second wire of the same material has a length \( 2L \) and a radius half that of the first wire, what is the force constant \( k \) of the second wire compared to the first?

A structural wire has a diameter of \( 0.80 \text{ mm} \) and a length of \( 3.5 \text{ m} \). When a tensile force of \( 120 \text{ N} \) is applied, the wire extends by \( 1.4 \text{ mm} \). Calculate the Young modulus of the material and state whether the wire has undergone elastic or plastic deformation if it returns to its original length after the load is removed.

A student conducts an experiment to determine the Young modulus of a copper wire. The wire has an initial length of 2.40 m and a diameter of 0.56 mm. A load of 35 N is applied to the wire, causing it to extend by 1.2 mm.

(a) Calculate the tensile stress in the wire due to the 35 N load. [2]
(b) Calculate the tensile strain produced in the wire. [1]
(c) Determine the Young modulus \( E \) of copper. [2]
(d) The student replaces the wire with a different material that has the same dimensions but is more brittle. Sketch a stress-strain graph on a set of axes, labeling the original copper wire as 'D' (ductile) and the new material as 'B' (brittle). Explain the main difference between these two graphs in terms of plastic deformation. [3]