Question
A wire has length \(L\) and cross-sectional area \(A\). The wire is made from a metal that has Young modulus \(E\) and resistivity \(\rho\).
(a) Define the Young modulus of a material. (1 mark)
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(b) (i) State an expression, in terms of some or all of \(L\), \(A\), \(E\) and \(\rho\), for the resistance \(R_0\) of the wire. (1 mark)
\(R_0=\) __________________________
(ii) Show that the spring constant \(k_0\) of the wire is given by \(k_0=\dfrac{EA}{L}\). (2 marks)
___________________________________
(c) The wire is stretched, within the limit of proportionality, by a tensile force \(F\). Assume that any changes to the cross-sectional area of the wire are negligible.
(i) On Fig. 3.1, sketch the variation with \(F\) of the resistance \(R\) of the wire. (1 mark)
(ii) On Fig. 3.2, sketch the variation with \(F\) of the spring constant \(k\) of the wire. (1 mark)
(d) Copper has a resistivity of \(1.8\times10^{-8}\,\Omega\mathrm{m}\) and a Young modulus of \(1.3\times10^{11}\,\mathrm{Pa}\).
A copper wire of diameter \(1.6\,\mathrm{mm}\) has a resistance of \(0.034\,\Omega\).
(i) Show that the length of the wire is \(3.8\,\mathrm{m}\). (1 mark)
________________________________________________________________________________________________
(ii) Use the equation in (b)(ii) to determine the spring constant of the wire. (2 marks)
spring constant = __________________________________________ \( \mathrm{N\,m^{-1}} \)
Syllabus Topic Codes (Cambridge International AS & A Level Physics 9702):
• 6.2: Elastic and plastic behaviour — parts (b)(ii), (c)(i), (c)(ii) and (d)(ii)
• 9.3: Resistance and resistivity — parts (b)(i), (c)(i) and (d)(i)
▶️ Answer/Explanation
(a) Young modulus [1 mark]
Young modulus is the ratio of stress to strain, within the limit of proportionality.
\(\displaystyle E=\frac{\text{stress}}{\text{strain}}\)
Answer: \( \boxed{\text{ratio of stress to strain}} \)
(b)(i) Resistance of the wire [1 mark]
Using the resistance equation,
\(R=\dfrac{\rho L}{A}\)
Answer: \( \boxed{R_0=\dfrac{\rho L}{A}} \)
(b)(ii) Spring constant of the wire [2 marks]
The spring constant is
\(k=\dfrac{F}{x}\)
Young modulus is
\(E=\dfrac{FL}{Ax}\)
Since \(k=\dfrac{F}{x}\),
\(E=\dfrac{(F/x)L}{A}\)
\(E=\dfrac{kL}{A}\)
Therefore,
\(k_0=\dfrac{EA}{L}\)
Answer: \( \boxed{k_0=\dfrac{EA}{L}} \)
(c)(i) Variation of resistance with force [1 mark]
The resistance is \(R=\dfrac{\rho L}{A}\). The resistivity and cross-sectional area remain constant, while the length increases as the tensile force increases.
Therefore, \(R\) increases with \(F\). Since the wire remains within the limit of proportionality, the extension and hence the length increase linearly with force.
Required graph: a straight line with positive gradient starting at \( (0,R_0) \).
(c)(ii) Variation of spring constant with force [1 mark]
The spring constant is
\(k=\dfrac{EA}{L}\)
Within the limit of proportionality, the Young modulus \(E\) and cross-sectional area \(A\) remain constant. The increase in length is negligible for the required relationship, so the spring constant remains constant.
Required graph: a horizontal line starting at \( (0,k_0) \).
(d)(i) Length of the copper wire [1 mark]
The diameter is \(1.6\,\mathrm{mm}\), so the radius is
\(r=0.80\times10^{-3}\,\mathrm{m}\)
The cross-sectional area is
\(A=\pi r^2\)
Using \(R=\dfrac{\rho L}{A}\),
\(L=\dfrac{RA}{\rho}\)
\(L=\dfrac{0.034\times\pi(0.80\times10^{-3})^2}{1.8\times10^{-8}}\)
\(L=3.8\,\mathrm{m}\)
Answer: \( \boxed{3.8\,\mathrm{m}} \)
(d)(ii) Spring constant of the wire [2 marks]
Using the result from (b)(ii),
\(k=\dfrac{EA}{L}\)
\(k=\dfrac{1.3\times10^{11}\times\pi(0.80\times10^{-3})^2}{3.8}\)
\(k=6.9\times10^4\,\mathrm{N\,m^{-1}}\)
Answer: \( \boxed{6.9\times10^4\,\mathrm{N\,m^{-1}}} \)