A circular steel plate of radius 15 cm is cooled from 300∘C to 80∘C.
Part A
By what percentage does the plate's area decrease?
for stell α=(11-13) •10⁻⁶ K⁻¹
take α=12•10⁻⁶ K⁻¹
ΔA=A₀2 αΔT
ΔA•100%/A₀=2αΔT•100%=
=2•12•10⁻⁶ 220•100%= 0.528%
To calculate the percentage decrease in the plate's area, we need to find the difference between the initial and final areas and express it as a percentage of the initial area.
The formula for the area of a circle is given by:
A = πr^2
where A is the area and r is the radius.
Let's first calculate the initial and final areas:
Initial area (A₁) = π(15 cm)^2
Final area (A₂) = π(15 cm)^2
Now, let's calculate the percentage decrease by using the following formula:
Percentage decrease = ((A₁ - A₂) / A₁) * 100
Substituting the values:
Percentage decrease = ((π(15 cm)^2 - π(15 cm)^2) / (π(15 cm)^2)) * 100
Simplifying:
Percentage decrease = 0 / (π(15 cm)^2) * 100
Since the numerator is 0, the percentage decrease is 0%.
Therefore, the circular plate's area does not decrease when cooled from 300∘C to 80∘C.
To find the percentage decrease in the plate's area, we need to calculate the initial and final areas of the plate and then calculate the percentage difference between them.
Step 1: Calculate the initial area of the plate.
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius. In this case, the radius is 15 cm. Substituting this value into the formula, we have A_initial = π(15 cm)^2.
Step 2: Calculate the final area of the plate.
To calculate the final area, we need to use the final radius of the plate. Since the plate is cooled, the radius will shrink. To find the final radius, we need to calculate the change in temperature and use the coefficient of linear expansion of steel. However, since the question does not provide the coefficient of linear expansion, we can assume that the radius decreases proportionally to the change in temperature.
The change in temperature is from 300°C to 80°C. Thus, the change in temperature is ΔT = 300°C - 80°C = 220°C.
Assuming a linear relationship, we can calculate the change in radius using the formula:
ΔR = (ΔT / T_initial) * R_initial,
where ΔR is the change in radius, ΔT is the change in temperature, T_initial is the initial temperature, and R_initial is the initial radius.
Substituting the given values, we have ΔR = (220°C / 300°C) * 15 cm.
Now, the final radius is given by R_final = R_initial - ΔR.
Substituting the values, we get R_final = 15 cm - (220°C / 300°C) * 15 cm.
Finally, we calculate the final area using the formula A = πr^2: A_final = π(R_final)^2.
Step 3: Calculate the percentage decrease.
To find the percentage decrease, we use the formula:
Percentage decrease = ((A_initial - A_final) / A_initial) * 100.
Now, you can substitute the values calculated earlier and find the percentage decrease in the plate's area.