Math Problem Statement
A copper rod of length 2l is connected to two plastic rods, each of length l + ∆l, forming a triangle of height h. The ends of each rod are fixed to the others by a hinge, so that the angles between them can change as the rods expand or contract with changing temperature, and each rod has the same cross-sectional area. At 300 K, the copper rod is 2.000 m long, and each plastic rod is 1.020 m. Assuming that ∆l ≪ l, show that h ≃ 2l ∆l. Calculate h when all rods are at 300 K.
Solution
We are tasked with solving for the height of a triangle formed by a copper rod of length and two plastic rods, each of length , at a temperature of 300 K.
Step 1: Understand the geometric setup
- The copper rod forms the base of the triangle, with length .
- The two plastic rods form the other two sides, each with length .
- is the height of the triangle, perpendicular to the copper rod, and intersects it at its midpoint.
Given that , we can assume the change in length is very small, and we will make approximations based on this fact.
Step 2: Use the properties of an isosceles triangle
We can split the triangle into two right triangles by drawing a vertical line from the midpoint of the copper rod (of length ) to the vertex where the two plastic rods meet. This vertical line has length and divides the base into two segments of length .
Each right triangle has:
- A base of length ,
- A hypotenuse of length ,
- A height .
Using the Pythagorean theorem, we write:
Step 3: Solve for
Expand :
Since , we can neglect the term (as it's very small compared to the other terms):
Now subtract from both sides:
Finally, solve for :
For small , this can be approximated as: where when the approximation holds for small values of .
Step 4: Calculate when all rods are at 300 K
At 300 K, we are given:
- The copper rod has a length of 2.000 m, so , meaning .
- The plastic rods each have a length of 1.020 m, so .
Thus, we can find :
Now, substitute and into the formula for :
Final Answer:
The height of the triangle at 300 K is approximately .
Do you want more details on any specific part of this solution, or do you have any questions?
Here are 5 related questions for further exploration:
- What would happen to the height if the temperature increased and the rods expanded further?
- How does the assumption affect the accuracy of the height calculation?
- How would the calculation change if the copper rod were also allowed to expand with temperature?
- Could this method be applied to non-isosceles triangles? How would the equations change?
- What physical properties of copper and plastic cause their lengths to change with temperature?
Tip: Small approximation techniques like neglecting higher-order terms () are very useful in simplifying complex equations in physics and engineering.
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Math Problem Analysis
Mathematical Concepts
Geometry
Pythagorean Theorem
Approximations
Formulas
Pythagorean Theorem: a^2 + b^2 = c^2
h ≃ 2l ∆l
l^2 + 2l∆l ≈ l^2 + h^2
Theorems
Pythagorean Theorem
Suitable Grade Level
Grades 10-12 (High School Physics/Math)
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