Answer to Question #287740 in Mechanics | Relativity for Teejay

Question

(a) Derive the formula for elastic potential energy, PEE, in terms of applied force F, stiffness constant k, and extension ΔL of a wire. State any assumptions made. (b) Figure 4.6a shows a F versus ΔL graph for a wire of length 2.0 m and diameter 1.2 mm. (i) Determine the stiffness constant of the wire from the graph. (ii) Determine the Young’s modulus for the material of the wire. (iii)What is a practical advantage of knowing the Young’s modulus as compared with the stiffness constant of the wire?

Answer 1

(a) The formula for elastic potential energy (assuming the wire is stretched within its elastic limit):

$$PE_E=\dfrac 12 k\Delta L^2.$$

(b) (i) Determine the stiffness constant of the wire from the graph: $k=F/\Delta L.$

(ii) Determine the Young’s modulus for the material of the wire: $E=(FL)/(A\Delta L).$

(iii) Knowing the Young’s modulus (compared with the stiffness constant of the wire) allows designing wires of other dimensions, and, along with the info on dimensions of the wire, gives much more information: stress, maximum force before breaking, etc.