the length of an aluminum is 76.5cm and that of an iron ruler is 80cm at 0¡ãc. at what temperature will the length of the rules become

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To determine at what temperature the length of the rulers will become equal, we need to use the principle of thermal expansion. Materials expand or contract with changes in temperature.

The formula for linear thermal expansion is given by:

ΔL = α * L₀ * ΔT

where:
ΔL is the change in length
α is the coefficient of linear expansion
L₀ is the initial length
ΔT is the change in temperature

Given:
Length of aluminum ruler (L_aluminum) = 76.5 cm
Length of iron ruler (L_iron) = 80 cm
Initial change in temperature (ΔT_initial) = 0 °C

We can rearrange the formula to solve for the change in temperature (ΔT):

ΔT = ΔL / (α * L₀)

Now, let's calculate the values for each ruler:

For the aluminum ruler:
ΔL_aluminum = L_aluminum - L_iron = 76.5 cm - 80 cm = -3.5 cm (negative value indicates contraction)
α_aluminum = coefficient of linear expansion for aluminum = 0.000022 cm/°C (approximately)
L₀_aluminum = initial length of aluminum ruler = 76.5 cm

For the iron ruler:
ΔL_iron = L_iron - L_aluminum = 80 cm - 76.5 cm = 3.5 cm (positive value indicates expansion)
α_iron = coefficient of linear expansion for iron = 0.000012 cm/°C (approximately)
L₀_iron = initial length of iron ruler = 80 cm

Now, we can calculate the change in temperature for both rulers:

ΔT_aluminum = ΔL_aluminum / (α_aluminum * L₀_aluminum)
ΔT_iron = ΔL_iron / (α_iron * L₀_iron)

Since we want the rulers to have the same length, set ΔL_aluminum equal to ΔL_iron:

ΔL_aluminum = ΔL_iron
-3.5 cm = 3.5 cm

Substitute the values into the equations:

-3.5 cm / (0.000022 cm/°C * 76.5 cm) = 3.5 cm / (0.000012 cm/°C * 80 cm)

Simplify the equation:

-3.5 * (1 / (0.000022 * 76.5)) = 3.5 * (1 / (0.000012 * 80))

Calculate the change in temperature:

ΔT_aluminum = -2670.5679 °C
ΔT_iron = 2583.3333 °C

The negative value for ΔT_aluminum indicates that the aluminum ruler would need to be cooled to reach the desired length, while the positive value for ΔT_iron indicates that the iron ruler needs to be heated.

Therefore, to equalize the lengths, the aluminum ruler needs to be cooled to approximately -2670.57 °C, and the iron ruler needs to be heated to approximately 2583.33 °C.