A hole in an aluminum plate has a diameter of 1.465 cm at 22.80° C.
(a) What is the diameter of the hole at 199.0° C (to four significant digits)?
(b) At what temperature is the diameter of the hole equal to 1.463 cm (to three significant digits)?
the diameter expands by ∆L=L0.α. ∆T
To solve this problem, we can use the coefficient of linear expansion for aluminum. The coefficient of linear expansion (α) represents the fractional change in length of a material per degree Celsius of temperature change.
(a) To find the diameter of the hole at 199.0° C, we need to calculate the change in diameter due to the change in temperature. The formula to calculate the change in length is:
ΔL = L * α * ΔT
Where:
ΔL is the change in length,
L is the original length,
α is the coefficient of linear expansion, and
ΔT is the change in temperature.
For a hole with diameter D, the change in diameter (ΔD) can be calculated using the same formula:
ΔD = D * α * ΔT
Given: D = 1.465 cm, α (for aluminum) = 0.000022/°C, ΔT = 199.0°C - 22.80°C
Substituting the values into the formula, we have:
ΔD = 1.465 cm * 0.000022/°C * (199.0°C - 22.80°C)
ΔD = 0.0006413 cm
To find the diameter at 199.0°C, we add the change in diameter to the original diameter:
D_199.0°C = D + ΔD
D_199.0°C = 1.465 cm + 0.0006413 cm
D_199.0°C = 1.4656413 cm
Therefore, the diameter of the hole at 199.0°C is approximately 1.4656 cm (to four significant digits).
(b) To find the temperature at which the diameter of the hole is equal to 1.463 cm, we can rearrange the formula:
ΔT = ΔD / (D * α)
Given: D = 1.465 cm, α (for aluminum) = 0.000022/°C, ΔD = 1.463 cm - 1.465 cm
Substituting the values into the formula, we have:
ΔT = (1.463 cm - 1.465 cm) / (1.465 cm * 0.000022/°C)
ΔT = -0.002 cm / (1.465 cm * 0.000022/°C)
ΔT = -64.55°C
To find the temperature at which the diameter is 1.463 cm, we subtract the change in temperature from the original temperature:
T = T_original + ΔT
T = 22.80°C - 64.55°C
T ≈ -41.75°C
Since the temperature cannot be negative in this context, it means that the diameter of the hole will not be equal to 1.463 cm with the given temperature range.