楊氏模量：練習題

Two wires, P and Q, are made of the same material. Wire P has length \( L \) and diameter \( d \). Wire Q has length \( 2L \) and diameter \( 2d \). Both wires are subjected to the same tensile force. What is the ratio of the extension of P to the extension of Q?

Two wires of the same length \( L \) and cross-sectional area \( A \) are connected in series to support a load. The first wire is made of a material with Young modulus \( E \) and the second wire has a Young modulus of \( 2E \). What is the effective Young modulus of the combined system?

In a stress-strain graph for a typical metal wire, what physical quantity does the gradient of the linear section represent?

A student performs an experiment to determine the Young modulus of a wire. The following measurements are taken:
Force \( F = 50 \pm 1 \text{ N} \)
Original length \( L = 2.000 \pm 0.005 \text{ m} \)
Diameter \( d = 0.50 \pm 0.02 \text{ mm} \)
Extension \( \Delta L = 4.0 \pm 0.2 \text{ mm} \)
What is the percentage uncertainty in the calculated value of the Young modulus?

A metal wire of length 2.5 m and cross-sectional area \( 1.2 \times 10^{-7} \text{ m}^2 \) is stretched by a force of 60 N. If the Young modulus of the metal is \( 2.0 \times 10^{11} \text{ Pa} \), what is the extension of the wire, assuming the elastic limit has not been exceeded?

A composite rod is formed by joining a \( 0.50 \text{ m} \) aluminum rod to a \( 0.50 \text{ m} \) steel rod, both of area \( 2.0 \times 10^{-4} \text{ m}^2 \). Calculate the total extension when a tensile force of \( 20 \text{ kN} \) is applied. (\( E_{al} = 7.0 \times 10^{10} \text{ Pa} \), \( E_{st} = 2.0 \times 10^{11} \text{ Pa} \)).

Two wires, P and Q, are made of the same material. Wire P has twice the length and three times the diameter of wire Q. Calculate the ratio of the extension of P to the extension of Q when both are subjected to the same tensile force.

In an experiment to determine the Young modulus of a wire, the following percentage uncertainties are recorded: Force \( 2.0\% \), Length \( 0.8\% \), Diameter \( 1.5\% \), and Extension \( 4.0\% \). Calculate the total percentage uncertainty in the calculated value of the Young modulus.

A student carries out an experiment to determine the Young modulus of a metal wire. The wire has an initial length of \(2.40 \text{ m}\) and a diameter of \(0.56 \text{ mm}\). A load of \(45 \text{ N}\) is applied to the wire, causing it to extend by \(1.10 \text{ mm}\).
(a) Calculate the tensile stress in the wire.
(b) Calculate the tensile strain in the wire.
(c) Determine the Young modulus of the metal.
(d) Calculate the elastic strain energy stored in the wire at this extension, assuming the limit of proportionality has not been exceeded.

A horizontal rigid bar of mass \(2.0 \text{ kg}\) and length \(0.80 \text{ m}\) is supported at its ends by two vertical wires, X and Y, of the same initial length \(1.5 \text{ m}\). Wire X is made of steel (Young modulus \(2.0 \times 10^{11} \text{ Pa}\)) and has a diameter of \(0.50 \text{ mm}\). Wire Y is made of brass (Young modulus \(1.0 \times 10^{11} \text{ Pa}\)) and has a diameter of \(0.80 \text{ mm}\).
(a) A weight \(W\) is hung from the center of the bar. Calculate the ratio of the extension of wire X to the extension of wire Y.
(b) Calculate the magnitude of weight \(W\) that would cause the bar to tilt by an angle of \(0.10^{\circ}\). Assume the bar remains rigid and the extensions are small.