Deformation of solids：練習問題

A spring has a spring constant of \( 250 \text{ N m}^{-1} \). What is the elastic potential energy stored in the spring when it is extended by \( 4.0 \text{ cm} \) within its limit of proportionality?

A wire of length \( L \) and cross-sectional area \( A \) is made of a material with Young modulus \( E \). A second wire is made of the same material but has length \( 2L \) and diameter twice that of the first wire.
What is the ratio of the extension of the second wire to the extension of the first wire when both are subjected to the same tensile force \( F \)?

A copper wire is stretched by a load such that it undergoes plastic deformation. The area under the force-extension graph during the loading process is \( W_1 \). When the load is gradually removed, the area under the force-extension graph during the unloading process is \( W_2 \).
Which expression represents the energy per unit volume that is dissipated as heat within the wire during this cycle, if the wire has a volume \( V \)?

Which statement best defines the limit of proportionality for a material being stretched?

A metal wire of length 2.0 m and cross-sectional area \( 4.0 \times 10^{-7} \text{ m}^2 \) is stretched by a force of 80 N. The Young modulus of the metal is \( 2.0 \times 10^{11} \text{ Pa} \).
What is the strain energy stored in the wire, assuming it obeys Hooke's law?

A spring with a spring constant of \( 250 \text{ N m}^{-1} \) is compressed by \( 4.0 \text{ cm} \).
Calculate the force required to produce this compression, assuming the spring obeys Hooke's law.

A wire of original length \( 1.5 \text{ m} \) and cross-sectional area \( 2.0 \times 10^{-7} \text{ m}^2 \) is stretched within its limit of proportionality. A tensile force of \( 40 \text{ N} \) produces an extension of \( 3.0 \text{ mm} \).
Calculate the Young modulus of the material of the wire.

A rubber band is stretched within its limit of proportionality. A force-extension graph for the band shows a linear relationship. When the force is 10.0 N, the extension is 0.20 m.

(a) State Hooke's law.
(b) Calculate the elastic potential energy (strain energy) stored in the rubber band when the force is 10.0 N.

A steel wire has a diameter of 0.50 mm and an initial length of 2.5 m. The Young modulus of steel is \( 2.0 \times 10^{11} \text{ Pa} \).

(a) Calculate the cross-sectional area of the wire.
(b) Calculate the weight that must be suspended from the wire to produce an extension of 1.2 mm. Assume the deformation is entirely elastic.