A brass lid screws tightly onto a glass jar at 15°C. To help o...

Question

A brass lid screws tightly onto a glass jar at 15°C. To help open the jar, it can be placed into a bath of hot water. After this treatment, the temperatures of the lid and the jar are both 55°C. The inside diameter of the lid is 7.0 cm. Find the size of the gap (difference in radius) that develops by this procedure.

Accepted Answer

Welcome back. Everyone in this problem. An aluminum cap fits snugly over an iron pipe at an initial temperature of negative five °C to loosen it. The assembly is warmed up to 45 °C. The internal diameter of the cap is 12.5 centimeters at negative five °C. Calculate the new gap created between the radii after heating, assume that the coefficient of linear expansion for aluminum and iron are 2.3 multiplied by 10 to the negative fifth per degree Celsius and 1.2 multiplied by a 10 to the negative fifth per degree Celsius respectively. A says that the gap is 1.2 multiplied by 10 to the negative third centimeters. B 2.3 multiplied by 10 to the negative third centimeters. C 3.4 multiplied by 10 to the negative third centimeters and D 5.4 multiplied by 10 to the negative third centimeters. No, let's make sure we understand what's going on in our problem. Let's try to sketch it. OK. So we're saying that initially, we have a, we have an aluminum cap that fits snuggly over an iron pipe. So that would suggest that the aluminum cap and the iron pipe are the same size. So in this case, then in this where our temperature is negative five °C. OK. Then that means the diameter of the pipe is the same as the diameter of the cap. And so are their radiuses or their radii? Sorry. So that means then that we could say that R represents the radius of both the cap and the pipe at negative five °C. Now what happens when we heat or warm it to 45 °C? But again, let's draw our cap and no, we know that they really are not the same. OK? Because our cap is made of aluminum and our pipe is made of iron. OK? So now in our second scenario at negative 45 °C then or iron pipe, let's call it R or it has a radius of ri OK. While our aluminum cap has a radius that we can call R, OK. Let's call that R for aluminum. And no, essentially we want to find the difference between R and R. So we're solving for the radius of the aluminum cap minus the radius of the iron pipe. No. How are we supposed to figure that out? Well, let's think about what happens here. OK. So now if we, if we want to find that change, there are few things that we can infer. Now notice that that change in radius, OK? Is going to be equal to the radius of our aluminum cap minus the radius of our iron pipe. And how did we get these new radii? Well, we added that change in the radius to our initial value of R. So really, then we could say that the change in radius equals R plus the change in radius R A minus R plus the change in radius for our iron pipe. Notice I is common to both terms. So really what we're finding then is the difference between the change in radios for our cup and the change in radios for our iron pipe. So how can we figure out what that change in radius is? What do we know that can help us to find a linear expansion for a change in temperature? Well, recall recall that for linear expansion, we know that the change in our length is equal to the initial length multiplied by the coefficient of linear expansion. OK. Multiplied by the change in temperature. In this case or change in length is the change in length for the radii. OK. So that means then that applying it to a problem or change in radius is going to be equal to the initial radius multiplied by our coefficient of linear expansion for whatever material multiplied by the change in temperature. So if we can apply this idea to both terms for the change in R A and the change in R I, then we should be able to find our change in radius. That is the new gap that has been created. No, let's apply that here. OK. Because we know that for our material, there are different changes, they have different coefficients of linear expansion. So using that idea, OK, then that means the change in radius is going to be equal to the initial length are multiplied by alpha A for aluminum multiplied by the change in temperature minus R our initial length multiplied by our coefficient for iron. Let's call that alpha I multiplied by the change in temperature. Remember that initially, both the aluminum cap and the iron pipe had the same radius since the cap fitted snugly. So that means then we could factor out I and the change in temperature because it's the same change in temperature for both uh mediums to get r multiplied by delta T multiplied by alpha alpha A minus alpha. I now we know what all of these values are. We know that initially or radius. Well, you know what, I'm skipping ahead of myself because we don't know it yet. So let's try to figure out if we can find these values. My apologies. Ok. Now, what do we know? Well, first we knew that the internal diameter of the cap is 12.5 centimeters. Ok. So we can figure out the radius from that diameter. Next, we know that initially, it was at a temperature of negative five °C. And now it's at a temperature of 45 °C. So we can find a change in temperature. Finally, we know the coefficients of linear expansion for aluminum and iron respectively. So we have all the information, we need to find the change in radio in in radius. OK. So now let's make a note of all that information. So given that we know our radius is going to be a half of 12.5 centimeters which is 6.25 centimeters. OK. Our change in temperature is equal to the difference between 45 and a negative five °C which is 50 °C. Oops, let me write it properly here and we know our coefficients of linear expansion for aluminum and iron substituting it into our formula. That means our change in radius is going to be equal to 6.25 centimeters multiplied by multiplied by a change in temperature of 50 °C multiplied by the difference between the coefficients of linear expansion, which would be 2.3 multiplied by 10 to the negative fifth per degree Celsius for aluminum and 1.2 multiplied by 10 to the negative fifth per degree Celsius for iron. When we go ahead and solve here, we put this into our calculator. Then we should find that we get our change in radius to be equal to 0.0034375 centimeters. If we write it to two significant figures like the rest of our answers. Then that means that the new gap created between the radii after heating is approximately equal to 3.4 multiplied by 10 to the negative third centimeters. If we look back at our answer choices that tells us that C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Thermal Expansion

Coefficient of Linear Expansion

Gap Calculation

Watch next