Steel Cables
Consider a 200 ft long steel cable is used to help hikers climb up Half Dome in Yosemite National Park, CA. During the summer, the ambient temperature can reach up to 95°F. If the cable shrinks by a max of 1.2 inches in the winter due to thermal deformation, how cold does Half Dome get in °F? Note the coefficient of thermal expansion of steel is 0.000007 ft/ft°F.
Expand Hint
$$$\delta_t=\alpha L(T-T_0)$$$
where
$$\delta_t$$
is the deformation caused by a temp change,
$$\alpha$$
is the temp coefficient of expansion,
$$L$$
is the member length,
$$T$$
is the final temp, and
$$T_0$$
is the initial temp.
Hint 2
Solve for
$$T$$
:
$$$T=\frac{\delta_t}{\alpha L}+T_0$$$
The equation for thermal deformation is:
$$$\delta_t=\alpha L(T-T_0)$$$
where
$$\delta_t$$
is the deformation caused by a temp change,
$$\alpha$$
is the temp coefficient of expansion,
$$L$$
is the member length,
$$T$$
is the final temp, and
$$T_0$$
is the initial temp.
Solving for
$$T$$
:
$$$T=\frac{\delta_t}{\alpha L}+T_0$$$
Remember,
$$\delta_t$$
is negative because the cables shrink in the winter and to convert the 1.2 inch shrinkage to feet. Thus,
$$$T=\frac{-0.1ft(ft^{\circ}F)}{0.000007ft (200ft)}+95^{\circ}F=\frac{-14,285.74^{\circ}F}{200}+95^{\circ}F$$$
$$$=-71.43^{\circ}F+95^{\circ}F=23.6^{\circ}F$$$
23.6°F
Time Analysis
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Similar Problems from FE Section: Thermal Deformations
169. Thermal Deformation
231. Subway Hand Rail
268. Installing a Bearing
403. Train Tracks