To make a secure fit, rivets that are larger than the rivet hole are often used and the rivet is cooled (usually in dry ice) before it is placed in the hole. A steel rivet 1.872 cm in diameter is to be placed in a hole 1.870 cm in diameter in a metal at $22 ^\circ \textrm{C}$. To what temperature must the rivet be cooled if it is to fit in the hole?

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. So, the change in the diameter of this rivet is going to equal the coefficient of linear expansion for steel times the initial length or diameter of the root times the change in temperature. So, to solve for this Δt, change in temperature, we'll divide both sides by α l naught. And the change in temperature will be the change in diameter divided by the coefficient of linear expansion times the original diameter. So, that's 1.870 centimeters is the target diameter minus the 1.872 centimeters initial diameter of the rivet divided by 12 times 10 to the minus 6 coefficient of linear expansion for steel times 1.872 centimeters. That gives negative 89.0313 Celsius degrees. So, the final temperature will be the initial temperature plus this change in temperature. So, that's 22 plus this negative, 89.0313. And that gives negative 70 degrees Celsius is the approximate final temperature needed in order for the rivet to fit in the hole.