A brass lid screws tightly onto a glass jar at 20 oC. To help open the jar, it can be

placed into a bath of hot water. After this treatment, the temperature of the lid and the
jar are both 60 oC. The inside diameter of the lid is 8.0 cm at 20 oC. Find the size of the
gap (difference in radius) that develops by this procedure.

Solutions please?

To solve this problem, we can use the concept of thermal expansion. When an object is heated, it expands in size. In this case, both the lid and the jar are heated, so we need to consider the expansion of both.

We are given the inside diameter of the lid at 20°C, which is 8.0 cm. Let's call this diameter D₀. We also know the final temperature Tf, which is 60°C. We need to find the difference in radius, which we will call Δr.

To find the difference in radius, we first need to find the change in diameter due to the temperature increase. We can use the formula for the linear expansion of a solid:

ΔL = α * L₀ * ΔT

Where:
ΔL is the change in length
α is the coefficient of linear expansion (which we can assume to be the same for both brass and glass)
L₀ is the initial length
ΔT is the change in temperature

Since we want the change in diameter, we can rearrange the formula:

ΔD = α * D₀ * ΔT

Substituting the given values, we have:

ΔD = α * 8.0 cm * (60°C - 20°C)

Now, let's consider the fact that the increase in diameter is divided equally between the lid and the jar. So, the change in radius for each is:

Δr = (1/2) * ΔD

Finally, we can evaluate the expression to find the size of the gap:

Δr = (1/2) * α * D₀ * (Tf - Ti)

To find the coefficient of linear expansion α, you may need to refer to a table or use the material properties of brass. The specific value for α will depend on the type of brass used.

Once you have the value of α, D₀, Tf, and Ti, you can substitute them into the equation to calculate the size of the gap Δr.

I am not going to do it for you. I will be happy to critique your work. I am getting the idea that you are answer grazing.