Thermal expansion/radius of curvature help

[farmerburns]

Homework Statement

A metal bar is 1.90 m long with a coefficient of thermal expansion of, 1.34e10-5 (K-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 50°C? [Hint: You can use the small-angle approximation sin θ = θ -(θ3/6).]

Homework Equations

listed in problem

The Attempt at a Solution

I have tried similar triangles, pythagorian theorem, and trig functions. I cannot see how I should use this "small angle approximation". Can someone please point me in the right direction?

 

[anathema]

I would first like to clarify the given information and assumptions. The problem states that the metal bar is rigidly held between two fixed beams, which suggests that the bar is not able to expand or contract freely in response to temperature changes. This means that the bar will experience compressive or tensile stress as it tries to change its length, and the resulting shape will be different from a simple circular arc.

However, for the purpose of this problem, we can assume that the bar is able to expand and contract freely without any external constraints. In this case, the bar will indeed take on the shape of a circular arc when it is heated, with a radius of curvature that we need to determine.

To use the given hint, we can first consider the small-angle approximation for a circular arc. This approximation states that for small angles, the length of the arc is approximately equal to the radius multiplied by the angle in radians. In this case, we can use the small-angle approximation because the temperature change of 50°C is relatively small compared to the melting point of most metals, which is typically in the range of hundreds to thousands of degrees Celsius.

Next, we can use the given coefficient of thermal expansion to calculate the change in length of the metal bar. The change in length (ΔL) is equal to the original length (L) multiplied by the temperature change (ΔT) and the coefficient of thermal expansion (α). So, we can write ΔL = L * α * ΔT. In this problem, we have all the values except for the radius of curvature, which we will denote as R.

Now, we can use the small-angle approximation for a circular arc to relate the change in length to the radius of curvature. The small-angle approximation states that for a circular arc, the change in length (ΔL) is equal to the radius (R) multiplied by the change in angle (Δθ) in radians. In this case, the change in angle is equal to the central angle of the arc, which we can determine using the given temperature change and the coefficient of thermal expansion.

Finally, we can equate the two equations for ΔL and solve for R. This will give us the radius of curvature of the circular arc formed by the metal bar when it is heated by 50°C. It is important to note that this solution is based on the assumptions made for this problem, and in a real-world scenario

 

[kerimek]

As a scientist, it is important to understand and utilize the appropriate mathematical tools to solve a problem. In this case, the small-angle approximation can be used to simplify the problem and make it more manageable. This approximation states that for small angles, the sine of the angle is approximately equal to the angle itself. In this problem, the angle θ formed by the metal bar can be considered small since the bar is rigidly held between two fixed beams. Therefore, we can use the small-angle approximation and rewrite the equation as sin θ ≈ θ.

Now, we can use the given information to solve for the radius of curvature. The length of the metal bar is 1.90 m, and when it expands due to the rise in temperature, it takes on the shape of a circular arc. This means that the length of the arc is equal to the length of the bar. We can represent this as L = rθ, where L is the length of the arc, r is the radius of curvature, and θ is the angle formed by the bar.

Using the small-angle approximation, we can rewrite this equation as L ≈ rθ. We know that the length of the bar has increased by 50°C, so we can substitute this value for θ. This gives us L ≈ r(50). Substituting the given value for the length of the bar, we get 1.90 ≈ r(50). Solving for r, we get r ≈ 0.038 m.

Therefore, the radius of curvature of the circular arc formed by the metal bar when the temperature rises by 50°C is approximately 0.038 m. This solution was made possible by utilizing the small-angle approximation, which simplified the problem and made it easier to solve. As a scientist, it is important to understand and apply mathematical concepts effectively to solve complex problems.