The Eiffel Tower is a steel structure whose height increases by 19.5 cm when the temperature changes from -9 to +41 oC. What is the approximate height (in meters) at the lower temperature?
To find the approximate height of the Eiffel Tower at the lower temperature, we can use the concept of thermal expansion. The change in height (∆H) can be calculated using the formula:
∆H = α * H * ∆T
Where:
∆H = change in height
α = coefficient of linear expansion for steel (approximately 0.000012)
H = initial height
∆T = change in temperature
In this case, the change in temperature (∆T) is (-9 - (-41)) = 32°C.
The change in height (∆H) is given as 19.5 cm.
So, we can rewrite the formula as:
19.5 = 0.000012 * H * 32
Now we can solve for H, the initial height:
H = 19.5 / (0.000012 * 32)
H ≈ 53,125 meters
Therefore, the approximate height of the Eiffel Tower at the lower temperature is 53,125 meters.
To find the approximate height of the Eiffel Tower at the lower temperature, we can use the concept of linear thermal expansion.
Here's how we can calculate it:
1. Determine the change in temperature: The temperature changes from -9 to +41 oC, so the temperature difference is (41 - -9) = 50 oC.
2. Find the linear expansion coefficient for steel: Different materials have different thermal expansion coefficients. For steel, the coefficient is approximately 12 x 10^(-6) per degree Celsius (12 μm/m•°C).
3. Calculate the change in height: The change in height can be calculated using the formula:
Δh = α * L * ΔT
where Δh is the change in height, α is the linear expansion coefficient, L is the original length, and ΔT is the change in temperature.
Since we want to find the height at the lower temperature, Δh is given as 19.5 cm (or 0.195 m), α is 12 x 10^(-6) per degree Celsius, and ΔT is 50 oC.
Substituting the values, we can rearrange the formula to solve for L, the original length (which represents the height at the lower temperature):
L = Δh / (α * ΔT)
L = 0.195 m / (12 x 10^(-6) per °C * 50 °C)
L ≈ 0.325 m
Therefore, the approximate height of the Eiffel Tower at the lower temperature is 0.325 meters.
ΔL=L₁-L = L+ΔL -L = αLΔt
L= ΔL/ αΔt =0.195 m /12•10⁻⁶•50 =325 m