Question

The linear expansion coefficient of aluminium is 24 × 10^-6 (°C)^-1. When thetemperature is 33 °C, a spherical aluminium ball has a diameter of 11.21 cm.A) Calculate the diameter (in cm) of the aluminium ball when the temperature is raisedto 257° C? (Round your answer to 3 decimal places.)B) Calculate the change in temperature (in °C) needed to increase the volume of thealuminium ball by 3.24%.

Answer 1

A) Use the following formula for the linear expansion, in this case, of the diameter of the spherical ball:

$d=d_o\lbrack1+\alpha(T_2-T_1)\rbrack$

where,

do: initial diameter = 11.21cm

d: final diameter = ?

α: linear expansion coefficient = 24*10^6 /°C

T2: final temperature = 257°C

T1: intial temperature = 33°C

Replace the previous values of the parameters into the formula for d and simplify:

Hence, the diameter of the spherical ball is 11.27cm

B) First, consider what is the change in the radius of the sphere, as follow:

$\Delta V=0.0324V_o$

Now, use the following formula for the change in volume with the temperature:

$\Delta V=\beta V_o\Delta T=3\alpha V_o\Delta T$

where you have used the equivalence β = 3α.

Solve the equation above for the change in temperature and replace the values of the other parameters:

$\Delta T=\frac{\Delta V}{3\alpha V_o}=\frac{0.0324V_o}{3(24\cdot10^{-6}(\degree C)^{-1})V}=450\degree C$

Hence, the change in temperature is 450°C