A brass measuring rule is connected at 15°c. The value obtained when the length of an object is measured with the rule of 35°c appears to be 841.4cm. What is the true length of the object
To find the true length of the object, we need to account for the expansion of the brass measuring rule due to the change in temperature.
The expansion of a solid object can be calculated using the formula:
ΔL = L * α * ΔT
Where:
ΔL is the change in length
L is the original length
α is the linear expansion coefficient
ΔT is the change in temperature
Given:
Original temperature (T1) = 15°C
Measured temperature (T2) = 35°C
Length on the measuring rule at T2 (L2) = 841.4 cm
We need to find the true length at the original temperature (L1).
First, we need to calculate the change in temperature:
ΔT = T2 - T1
= 35°C - 15°C
= 20°C
Next, we need to find the linear expansion coefficient (α) for brass.
For brass, the linear expansion coefficient is typically 19 x 10^-6 per °C.
Now we can substitute the values into the formula:
ΔL = L * α * ΔT
Since we are solving for the true length at T1, we can rearrange the formula:
L1 = L2 - ΔL
Substituting the known values:
L1 = L2 - (L * α * ΔT)
Let's assume L = L1 (since we don't know the original length beforehand).
Now we can solve for L1:
L1 = L2 - (L1 * α * ΔT)
Simplifying the equation:
L1 + L1 * α * ΔT = L2
Factoring out L1:
L1 * (1 + α * ΔT) = L2
Dividing both sides by (1 + α * ΔT):
L1 = L2 / (1 + α * ΔT)
Now we can substitute the known values:
L1 = 841.4 cm / (1 + (19 x 10^-6 per °C) * 20°C)
Calculating:
L1 = 841.4 cm / (1 + 0.00038)
L1 = 841.4 cm / 1.00038
L1 ≈ 841.2387 cm
Therefore, the true length of the object is approximately 841.2387 cm.