What Temperature Causes a Steel Ball to Loosen in a Pyrex Cube?

[Lance WIlliam]

Homework Statement

A steel ball bearing is encased in a Pyrex glass cube 1.0 cm on a side. At 390K , the ball bearing fits tightly inside the cube.

At what temperature will it have a clearance of 0.90pico meters all around?

Homework Equations

Coefficent of linear expansion:
[tex]\propto[/tex] =(Change in L / L) / change in T

Im not sure what my Delta L should be...
L= .01 m
T initial is 390K
the [tex]\propto[/tex] is 3.2e-6

The Attempt at a Solution

I just need delta L in order to solve for T final

 

[Doc Al]

How does the expansion coefficient of Pyrex compare to that of steel?

 

[baba89]

.

Based on the given information, we can use the coefficient of linear expansion, \propto, to solve for the change in length, \Delta L, of the Pyrex glass cube. The equation for coefficient of linear expansion is \propto = (\Delta L / L) / \Delta T, where \Delta T is the change in temperature and L is the original length of the material.

In this case, we know that the original length, L, is 1.0 cm or 0.01 m. We also know that the coefficient of linear expansion for Pyrex is 3.2e-6. We can rearrange the equation to solve for \Delta T:

\Delta T = (\Delta L / L) / \propto

Since we are looking for the temperature at which the ball bearing will have a clearance of 0.90 picometers, we can plug in \Delta L = 0.90 pm or 9.0e-13 m.

\Delta T = (9.0e-13 m / 0.01 m) / 3.2e-6 = 0.0281 K

Therefore, at a temperature of 390 K + 0.0281 K = 390.0281 K, the ball bearing will have a clearance of 0.90 picometers all around inside the Pyrex glass cube.