A glass window pane is exactly 23 cm by 38 cm at 10oC. By how much has its area increased when it temperature is 49oC, assuming that it can expand freely? Take the coefficient of thermal expansion for the glass to be exactly 9 x 10-6/Co
Area increases by a factor equal to the square of the length increase factor.
Old area = 23 x 38 = 874 cm^2
New area = (Old area)*(1 + 39*9*10^-6)^2
= 874*(1.000351)^2
= 874.61 cm^2
For small fractional increases such as we have here, the Area increase factor is very nearly twice the Length increase factor.
Anew/Aold = (1 + 2*39*9*10^-6)
To determine by how much the area of the glass window pane has increased, we need to consider the change in temperature and the coefficient of thermal expansion for the glass.
The formula to calculate the change in length due to thermal expansion is given by:
ΔL = L * α * ΔT
Where:
ΔL is the change in length,
L is the original length,
α is the coefficient of thermal expansion,
ΔT is the change in temperature.
Since we are interested in the change in area, we need to square both sides of the formula:
(ΔL)^2 = (L * α * ΔT)^2
The original length and coefficient of thermal expansion are given as 23 cm and 9 x 10^-6/°C, respectively. To calculate the change in length, we first need to determine the change in temperature.
The change in temperature is given by:
ΔT = final temperature - initial temperature
Substituting the given temperatures, we have:
ΔT = 49°C - 10°C
= 39°C
Now we can substitute the values into the formula to find ΔL:
(ΔL)^2 = (23 cm * 9 x 10^-6/°C * 39°C)^2
Calculating the expression, we get:
(ΔL)^2 = (0.000207 cm)^2
Taking the square root of both sides, we find:
ΔL = 0.000207 cm
Since the area of a rectangle is given by length multiplied by width, we can calculate the change in area by multiplying the change in length by the original width:
ΔA = ΔL * initial width
Substituting the values, we get:
ΔA = 0.000207 cm * 38 cm
Calculating this expression, we find:
ΔA ≈ 0.00786 cm^2
Therefore, the area of the glass window pane increases by approximately 0.00786 cm^2 when its temperature increases from 10°C to 49°C.