Physics Help Forum Three Rods, Forming a Right-Angled Triangle

 Thermodynamics and Fluid Mechanics Thermodynamics and Fluid Mechanics Physics Help Forum

 Apr 10th 2018, 02:00 AM #1 Junior Member   Join Date: Apr 2018 Posts: 1 Three Rods, Forming a Right-Angled Triangle Three rods, with lengths: $\displaystyle L_1$, $\displaystyle L_2$ and $\displaystyle L_3$ form a right-angled triangle in which $\displaystyle L_1$ and $\displaystyle L_2$ are the right-angle's sides (the legs) and $\displaystyle L_3$ is the hypotenuse. The rods' coefficient of thermal expansion are, $\displaystyle \alpha_1$, $\displaystyle \alpha_2$ and $\displaystyle \alpha_3$, respectively. Is it possible to define the $\displaystyle \alpha_3$ based on $\displaystyle \alpha_1$ and $\displaystyle \alpha_2$, such that these rods form a right-angled triangle at all temperatures?How?
 Apr 10th 2018, 05:46 AM #2 Senior Member   Join Date: Aug 2010 Posts: 434 At temperature t, the lengths of the three sides are $\displaystyle (1+ \alpha_1t)L_1$, $\displaystyle (1+ \alpha_2t)L_2$, and $\displaystyle (1+ \alpha_3t)L_3$. In order that this be a right triangle for all t we must have (Pythagorean theorem) $\displaystyle (1+ \alpha_1t)^2L_1^2+ (1+ \alpha_2t)^2L_2^2= (1+ \alpha_3t)^2L_3^2$ To solve that equation for $\displaystyle \alpha_3$, first divide both sides by $\displaystyle L_3^2$: $\displaystyle (1+ \alpha_1t)^2\frac{L_1^2}{L_3^2}+ (1+ \alpha_2t)^2\frac{L_2^2}{L_3^2}= 1+ \alpha_3t$ now subtract 1 from both sides: $\displaystyle (1+ \alpha_1t)^2\frac{L_1^2}{L_3^2}+ (1+ \alpha_2t)^2\frac{L_2^2}{L_3^2}- 1= \alpha_3t$ and, finally, divide that by t: $\displaystyle (1+ \alpha_1t)^2\frac{L_1^2}{tL_3^2}+ (1+ \alpha_2t)^2\frac{tL_2^2}{tL_3^2}- \frac{1}{t}= \alpha_3$. Since the original configuration was a right triangle we can replace [math]L_3^2[/math with $\displaystyle L_1^2+ L_2^2$: $\displaystyle \alpha_3= (1+ \alpha_1t)^2\frac{L_1^2}{t(L_1^2+ L_2^2)}+ (1+ \alpha_2t)^2\frac{tL_2^2}{t(L_1^2+ L_2^2)}- \frac{1}{t}$. If, for given $\displaystyle \alpha_1$, $\displaystyle \alpha_2$, $\displaystyle L_1$, and $\displaystyle L_2$, that is independent of t, then the answer is "yes", otherwise, "no". Beeroonee likes this.

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