Temperature (Thermal Expansion) problem

[bcd201115]

Homework Statement

Two metal bars are made of invar and a third is made of aluminum. At 0°C, each of the 3 bars is drilled with two holes 40cm apart. The bars are assembled into an equilateral triangle.

Homework Equations

a)ignoring expansion of the invar, find the angles between the invar bars as a function of Celsius temperature
b)is the answer accurate for negative as well as positive temps and c) for 0°C
d)assuming the tabulated expansion coefficients are constant, solve again including expansion of the invar
e and f) find the greatest and smallest attainable angles between the invar bars

I know this is a long problem with a lot of work but can anyone at least get me started, I am stuck as far as how to even start

3. The Attempt at a Solution [/b
aluminum melts at 660°C and invar at 1427°C

 

[CWatters]

Write an equation for the length of Ali bar when hot.
Draw a triangle.
Write an equation for the angle opposite the Ali bar.

Hint: It might be easier to write an equation for "half the angle" then double it... Draw a line from the corner where the invar bars meet to the middle of the Ali bar. Due to symetry it meets the Ali bar at 90 degrees.

 

[bcd201115]

what kind of equation do i write?

 

[CWatters]

An equation for the length as a function of temperature. See "Linear Expansion" down here

http://en.wikipedia.org/wiki/Thermal_expansion

It's been a while since I did this but...

L_hot = L_initial + DeltaL

DeltaL = L_initial * AlphaL * DeltaT

So

L_hot = L_initial + L_initial * AlphaL * DeltaT

L_hot = L_initial * (1+ AlphaL*DeltaT)

 

[bcd201115]

okay so the first one says find the angle between the invar bars ignoring expansion, so how would i start that? sorry I am just really confused

 

[CWatters]

At 0C all bars have same length => triangle with equal length sides => 180/3 = 60 degrees.

 

[bcd201115]

wow i was making that a lot harder than it was. it says as a function of celsius temperature so would it just be 0°C?

 

[CWatters]

Err no. That's just the "angle between the invar bars ignoring expansion".

The question says ignore expansion of the invar but account for expansion of the Ali. Try this...

Draw a triangle with the base horizontal. Label the base "Ali" and the other two sides "Invar".

Draw a construction line from the corner at the top of the triangle to the middle of the Ali bar. Note it meets at a right angle.

Now imagine that the middle of the ali bar is fixed to the page with the right and left ends able to move away from the middle to the left and right as the ali expands. Note how the symetry is preserved. The angle where the construction line meets the Ali bar is allways 90 degrees.

Now look at one half of the triangle only. Let's say the right hand half. Label the angle between the right hand invar bar and the vertical construction line. The question actually asks for the angle between the two invar bars but you can multiply by 2 later right?

Now that angle you labeled..

Sin(angle) = Opposite over Hypotenuse

In this case the "opposite" is half the length of the Ali rod. The Hypotenuse is the length of the invar rod. So

Sin(angle) = 0.5 * length of ali rod/(length of invar rod)
or
Angle = Sin^-1(0.5 * length of ali rod/(length of invar rod)

So now find an equation for the length of an ali bar vs temperature. Substitute it into the equation above.

Remember to multiply the answer by two.

Check your answer by setting a temperature of 0C. The angle should be 30 x 2 = 60 degrees.

Job done... or at least part a).