Question

A circular hole in an aluminum plate is 3.231 cm in diameter at 0.000°C. What is its diameter when the temperature of the plate is raised to 64.91°C? The linear expansion coefficient of aluminum is 23.00 × 10-6 /C°.

Answer 1

3.236 cm

Step-by-step explanation:

diameter at 0°C, do = 3.231 cm

Initial temperature, To = 0° C

Final temperature, T = 64.91° C

ΔT = T - To = 64.91

Coefficient of linear expansion, α = 23 x 10^-6 / °C

Let d be the diameter when the temperature is raised.

Coefficient of areal expansion = 2 x α = 46 x 10^-6 / °C

Use the formula for the areal expansion

A = Ao(1 + βΔT)

where, ΔT be the rise in temperature.

`(\pi d^(2))/(4)=(\pi d_(0)^(2))/(4)\left ( 1+\beta \Delta T \right )`

`d^(2)=d_(0)^(2)\left ( 1+\beta \Delta T \right )`

`d^(2)=3.231^(2)\left ( 1+46*10^(-6)* 64.91\right )`

`d^(2)=10.44\left ( 1.00298\right )=10.471`

d = 3.236 cm

Thus, the diameter of the circular hole is 3.236 cm after heating at the given temperature.