Bulk properties of solids：練習問題

A metal wire of length \(L\) and diameter \(d\) is suspended vertically and a load is applied to its lower end, resulting in an extension \(e\). If a second wire made of the same metal with length \(2L\) and diameter \(2d\) is subjected to the same load, what will be the extension?

A metal wire of length \(L\) and diameter \(d\) has a breaking force \(F\). What is the breaking force for a wire made of the same metal with length \(2L\) and diameter \(3d\)?

A spring with spring constant \(k\) is stretched by a distance \(x\) within its elastic limit. A second identical spring is then connected in series with the first spring, and the combination is stretched by the same total distance \(x\). What is the ratio of the elastic potential energy stored in the single spring to the total energy stored in the series combination?

A rigid horizontal bar of length \(1.2\text{ m}\) is supported by two vertical wires, one at each end. Wire 1 is made of steel (Young modulus \(200\text{ GPa}\)) and Wire 2 is made of copper (Young modulus \(120\text{ GPa}\)). Both wires have the same initial length and the same cross-sectional area. A weight \(W\) is hung from the bar at a distance \(x\) from Wire 1 such that the bar remains horizontal. Calculate \(x\).

Two solids, X and Y, have densities \(\rho_X\) and \(\rho_Y\) respectively. A composite object is formed by joining equal masses of X and Y. What is the average density of the composite object?

A wire is stretched elastically such that the tensile force increases linearly from \( 0 \) to \( 60 \text{ N} \), producing an extension of \( 4.0 \text{ mm} \). Calculate the elastic strain energy stored in the wire.

A metal specimen is stretched beyond its elastic limit. It follows Hooke's law up to \( 200 \text{ N} \) with an extension of \( 2.0 \text{ mm} \). From \( 2.0 \text{ mm} \) to \( 5.0 \text{ mm} \), the force stays constant at \( 200 \text{ N} \). Calculate the total work done during the entire extension.

Two identical springs, each with a spring constant \( k = 250 \text{ N m}^{-1} \), are connected in parallel to support a weight of \( 15 \text{ N} \). Calculate the total extension of the system.

A copper sphere of radius \(2.5 \text{ cm}\) is suspended from a vertical spring. The spring has a spring constant \(k = 180 \text{ N m}^{-1}\). The density of copper is \(8960 \text{ kg m}^{-3}\).
(a) Calculate the weight of the copper sphere.
(b) Calculate the extension of the spring when the sphere is in equilibrium in air (ignore the buoyancy of air).
(c) The sphere is then completely immersed in a beaker of oil with density \(850 \text{ kg m}^{-3}\). Calculate the new extension of the spring.

A vertical spring is used to launch a small mass of \(0.15 \text{ kg}\). The spring has a natural length of \(0.12 \text{ m}\). When the mass is placed on the spring and allowed to reach equilibrium, the length of the spring becomes \(0.10 \text{ m}\). The mass is then pushed down further until the spring length is \(0.04 \text{ m}\) and then released from rest.
(a) Show that the spring constant \(k\) is \(73.6 \text{ N m}^{-1}\).
(b) Calculate the total elastic potential energy stored in the spring just before the mass is released.
(c) Calculate the maximum height reached by the mass above its release position. Assume that the spring does not detach from the base and air resistance is negligible.