At room temperature (20°C), a 5.0-cm-long brass rod is 20 μm too long to fit into a slot. To what temperature should you cool the rod so that it just barely fits?

Question

Accepted Answer

Hey, everyone. So this problem is dealing with thermal expansion. Let's see what it's asking us. We have a 15 centimeter stick of aluminum placed in temperature control chamber at 15 degrees Celsius. That stick is placed between two fixed points and it's 35 micrometers shorter than required. So what temperature should this room be? So that the stick fits between those two fixed jaws. Our multiple choice answers are a 20 degrees Celsius B 25 degrees Celsius, C 30 degrees Celsius or D 35 degrees Celsius. So this is a pretty straightforward problem. As long as we recall that our equation for linear thermal expansion is given by delta L is equal to alpha L. Delta T for alpha is our linear expansion coefficient L is our total length. Delta T is our change in temperature. And then delta L is there a change in length due to that thermal expansion. So we are looking for a temperature, we're actually looking for the final temperature given an initial temperature, initial conditions at 15 degrees Celsius. So we can rearrange this problem simplifying out delta T to be T final minus T initial is equal to S L divided by alpha. And so solving for T F, we know from the problem that delta L is 35 micrometers. We wanna keep everything in standard units. So that's going to be 35 times 10 to the negative six m divided by alpha, which is our linear expansion coefficient. This is a material dependent number. And so for aluminum, we can recall that that is 2.3 times 10 to the negative five first C oh sorry, four degrees Celsius. And then our original length centimeters again, keeping in standard units will write that as 150.15 m and then all of that plus our initial temperature of degrees Celsius. And so when we plug that into our calculator, we get 25 degrees Celsius as the temperature that that room should be to make that aluminum stick fit. And that aligns with answer choice B so B is the correct answer for this problem. So that's all we have for this one. We'll see you in the next video.

At room temperature (20°C), a 5.0-cm-long brass rod is 20 μm too long to fit into a slot. To what temperature should you cool the rod so that it just barely fits?

Verified Solution

Key Concepts

Thermal Expansion

Coefficient of Linear Expansion

Temperature Change Calculation