Ali needs to fix plug inside the wall, he then buy a steel plug that is going to be placed in a ring made of aluminium. At 15 degree celcius, the diameter of the plug is 10.765 cm and that of the inside of the ring is 10.432 cm. What common temperature must both steel and aluminium be brought in order to fit?

To determine the common temperature needed for both the steel plug and the aluminum ring to fit, we can use the concept of thermal expansion. The expansion or contraction of a material due to changes in temperature is often given by its coefficient of linear expansion (α).

Let's break down the problem step by step:

Step 1: Find the difference in size between the plug and the ring at the given temperature:

- Diameter of the plug at 15 degrees Celsius = 10.765 cm
- Diameter of the inside of the ring at 15 degrees Celsius = 10.432 cm

The difference in size between the plug and the inside of the ring at 15 degrees Celsius is:
10.765 cm - 10.432 cm = 0.333 cm

Step 2: Find the coefficients of linear expansion for steel and aluminum:

- The coefficient of linear expansion for steel (α_steel) is typically around 12 × 10^-6 per degree Celsius.
- The coefficient of linear expansion for aluminum (α_aluminum) is typically around 22 × 10^-6 per degree Celsius.

Step 3: Use the formula for linear expansion to find the common temperature:

delta_L = L_0 * α * delta_T

where:
- delta_L is the change in length,
- L_0 is the original length,
- alpha (α) is the coefficient of linear expansion, and
- delta_T is the change in temperature.

We can modify the formula for our purposes:

delta_D = D_0 * α * delta_T

delta_D is the change in diameter, and D_0 is the original diameter.

For the steel plug:

delta_D_steel = D_0_steel * α_steel * delta_T

For the aluminum ring:

delta_D_aluminum = D_0_aluminum * α_aluminum * delta_T

Step 4: Solve for the common temperature (delta_T):

Since we want the steel plug and aluminum ring to fit, the change in diameter for the steel plug (delta_D_steel) and the aluminum ring (delta_D_aluminum) should be equal.

So we have:

D_0_steel * α_steel * delta_T = D_0_aluminum * α_aluminum * delta_T

We can cancel out the delta_T term:

D_0_steel * α_steel = D_0_aluminum * α_aluminum

Now we can solve for delta_T:

delta_T = (D_0_aluminum * α_aluminum) / (D_0_steel * α_steel)

Step 5: Plug in the values and calculate:

D_0_aluminum = 10.432 cm
α_aluminum = 22 × 10^-6 per degree Celsius
D_0_steel = 10.765 cm
α_steel = 12 × 10^-6 per degree Celsius

Substituting the values into the formula:

delta_T = (10.432 cm * 22 × 10^-6 per degree Celsius) / (10.765 cm * 12 × 10^-6 per degree Celsius)

Calculating:

delta_T = (0.229504 × 10^-4 cm² per degree Celsius) / (0.12918 × 10^-4 cm² per degree Celsius)

delta_T = 1.77928992258

Therefore, the common temperature for both the steel plug and the aluminum ring to fit is approximately 1.779 degrees Celsius.