A steel ball bearing is encased in a Pyrex glass cube 1.0 cm on a side. At 380 K, the ball bea

step 1 In this problem, we're considering a Pyrex cube. That's one centimeter on each side. And we have

A steel ball bearing is encased in a Pyrex glass cube 1.0...

Key Concepts

Thermal Expansion

Thermal expansion is the phenomenon where the dimensions of a material change with temperature. As temperature increases, most materials expand due to increased atomic vibrations, and the change in any linear dimension is generally proportional to the temperature change. This concept is fundamental in understanding how components in structures or machinery will behave under varying thermal conditions.

Coefficient of Linear Expansion

The coefficient of linear expansion is a material-specific property that quantifies the degree to which a material's length changes per unit change in temperature. It is expressed in units such as per degree Kelvin or Celsius and is essential for calculating how much an object expands or contracts in response to temperature variations.

Differential Expansion and Clearance

Differential expansion refers to the situation where two materials with different coefficients of expansion change their dimensions by different amounts under the same temperature change. In engineering design, incorporating a clearance—or intentional gap—between two interfacing parts accounts for this difference, ensuring that mechanical components do not bind or experience undue stress when subjected to temperature fluctuations.

Transcript

00:01 In this problem, we're considering a pyrex cube.

00:05 That's one centimeter on each side.

00:09 And we have a steel ball inside of the cube.

00:12 When the temperature is equal to 380 kelvin, the ball just fits inside the cube.

00:20 And the question is, what should the temperature be if we want there to be a space of about one micron on each side of the ball? so we know that things expand when they warm.

00:35 In this case, we want things to contract as a cool to give some more space in the cube for the ball.

00:44 But both the pyrex and the ball will change.

00:49 And so we need to keep both in mind when trying to solve this problem.

00:54 Because this is a thermal expansion problem, i'm going to go ahead.

00:58 And before i even jump in, write down the equations and look up a few values.

01:04 So the equation will be mostly dealing with is that for linear expansion, so the coefficient of linear expansion is the change in length divided by the original link over the change in temperature.

01:21 And for pyrex, the coefficient is 3 .2 times 10 to the negative 6.

01:31 Inverse kelvin, and for steel, it is 12 times 10 to the negative 6 inverse kelvin.

01:46 Now that we have that information, we can go ahead and think about solving this problem.

01:54 So we want the change in length.

01:58 That's the thing that we're interested in.

02:00 And so delta l generally is going to be equal to alpha delta t where l is the original length delta t is the change in temperature and alpha is the coefficient but we want that two micron on each if it's one micron on each side and we're imagining a ball inside of the cube then if that's one micron and that's one micron, we need a total gap of two microns.

02:33 So we want two microns to be equal to the difference in length.

02:44 But because both are going to change, we're going to have the shrinkage from the steel ball.

02:51 So we'll have delta l for the steel plus or i guess minus the steel.

03:03 Shrinkage of the pyrex cube.

03:06 So delta l for the pyrex.

03:09 So, of course, the ball is going to shrink.

03:13 There will be some amount of space left because of that...