Calculating Radius of Curvature for Metal Bar

[brad sue]

Hi guys ,I need help to find the radius of curvature for this exercice:

A metal bar is 1.75m long with a coefficient of thermal expansion of 1.34*10^-5K^-1. It is rigidly held between two fixed beams. When the temperature rises, the metal bar takes on the shape of the arc of a circle.
What is the radius of curvature of the circle when the temperature rises by 40 degrees celcius?

hint:for small angle: sin(x)=x-x³/6.

thankx for your help!

 

[durt]

I'm confident you can find the length of the heated bar. Then, just draw an arc and relate these lengths to the angle of the arc and the radius of curvature. You'll need to use the law of sines, and then you can use the "hint" and solve for the radius.

 

[quicknote]

I would approach this problem by using the formula for calculating the radius of curvature, which is R = L/2sinθ, where R is the radius, L is the length of the bar, and θ is the angle of curvature. In this case, the angle of curvature can be approximated using the small angle approximation, θ = x - x^3/6, where x is the change in temperature in radians.

Using the given information, we can calculate the change in temperature in radians by converting the temperature difference of 40 degrees Celsius to radians: x = (40 degrees * π/180 degrees) = 0.698 radians.

Substituting this value into the small angle approximation, we get θ = 0.698 - (0.698)^3/6 = 0.698 - 0.004 = 0.694 radians.

Next, we can plug in the values for L and θ into the formula for radius of curvature, R = L/2sinθ, to get R = (1.75m)/2sin(0.694) = 1.75m/2(0.678) = 1.75m/1.356 = 1.29m.

Therefore, the radius of curvature for the metal bar when the temperature rises by 40 degrees Celsius is approximately 1.29 meters. It is important to note that this calculation is an approximation and may not be completely accurate due to the use of the small angle approximation. Other factors such as the rigidity of the beams and the material properties of the bar may also affect the actual radius of curvature.