Question 11.2: Stressing a Strut Goal Use the energy transfer equation in t...
Stressing a Strut
Goal Use the energy transfer equation in the context of linear expansion and compressional stress.
Problem A steel strut near a ship’s furnace is 2.00 \mathrm{~m} long, with a mass of 1.57 \mathrm{~kg} and cross-sectional area of 1.00 \times 10^{-4} \mathrm{~m}^{2}. During operation of the furnace, the strut absorbs thermal energy in a net amount of 2.50 \times 10^{5} \mathrm{~J}. (a) Find the change in temperature of the strut. (b) Find the increase in length of the strut. (c) If the strut is not allowed to expand because it’s bolted at each end, find the compressional stress developed in the strut.
Strategy This problem can be solved by substituting given quantities into three different equations. In part (a), the change in temperature can be computed by substituting into Equation 11.3,
Q=m c\Delta T (11.3)
which relates temperature change to the energy transferred by heat. In part (b), substituting the result of part (a) into the linear expansion equation yields the change in length. If that change of length is thwarted by poor design, as in part (c), the result is compressional stress, found with the compressional stress-strain equation.
(a) Find the change in temperature.
Solve Equation 11.3 for the change in temperature and substitute:
\begin{aligned} Q & =m_{s} c_{s} \Delta T \rightarrow \Delta T=\frac{Q}{m_{s} c_{s}} \\ \Delta T & =\frac{\left(2.50 \times 10^{5} \mathrm{~J}\right)}{(1.57 \mathrm{~kg})\left(448 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\right)}=355^{\circ} \mathrm{C} \end{aligned}
(b) Find the change in length of the strut if it’s allowed to expand.
Substitute into the linear expansion equation:
\begin{aligned} \Delta L & =\alpha L_{0} \Delta T=\left(11 \times 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right)(2.00 \mathrm{~m})\left(355^{\circ} \mathrm{C}\right) \\ & =7.8 \times 10^{-3} \mathrm{~m} \end{aligned}
(c) Find the compressional stress in the strut if it is not allowed to expand.
Substitute into the compressional stress-strain equation:
\begin{aligned} \frac{F}{A} & =Y \frac{\Delta L}{L_{0}}=\left(2.00 \times 10^{11} \mathrm{~Pa}\right) \frac{7.8 \times 10^{-3} \mathrm{~m}}{2.01 \mathrm{~m}} \\ & =7.8 \times 10^{8} \mathrm{~Pa} \end{aligned}
Remarks Notice the use of 2.01 \mathrm{~m} in the denominator of the last calculation, rather than 2.00 \mathrm{~m}. This is because, in effect, the strut was compressed back to the original length from the length to which it would have expanded. (The difference is negligible, however.) The answer exceeds the ultimate compressive strength of steel and underscores the importance of allowing for thermal expansion. Of course, it’s likely the strut would bend, relieving some of the stress (creating some shear stress in the process). Finally, if the strut is attached at both ends by bolts, thermal expansion and contraction would exert sheer stresses on the bolts, possibly weakening or loosening them over time.