(II) To make a secure fit, rivets that are larger than the...

(II) 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 $\mathrm{cm}$ in diameter is to be placed in a hole 1.870 $\mathrm{cm}$ in diameter in a metal at $20^{\circ} \mathrm{C}$ . To what temperature must the rivet be cooled if it is to fit in the hole?

Key Concepts

Thermal Expansion and Contraction

Thermal expansion and contraction refer to the change in a material's dimensions as its temperature varies. When a material is heated, it expands; conversely, when it is cooled, it contracts. This principle is fundamental in materials science for predicting and controlling dimensional changes in manufacturing, construction, and mechanical design.

Coefficient of Thermal Expansion

The coefficient of thermal expansion is a property of a material that quantifies the extent to which the material expands or contracts per unit change in temperature. It is a crucial parameter for determining how temperature variations affect the size of components, which is particularly important when ensuring that parts will fit together after undergoing thermal changes.

Interference Fit

An interference fit is a method of assembling components where one part is intentionally made slightly larger than the mating part. This results in a tight, secure match once the parts are joined. By using thermal contraction—such as cooling a larger component to temporarily reduce its size—a secure fit can be achieved when the parts are assembled and then return to ambient temperature.

Transcript

00:01 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.

00:10 A still rivet, 1 .872 centimetre in diameter, is to be placed in a hole that is 1 .87 centimetres in a metal at 20 degrees celsius.

00:21 To what temperature must the rivet be cooled if it is to fit into the hole? so we're going to start off from our standard formula for length expansion.

00:36 So change in length is equal to original length times by change in temperature and the coefficient of expansion length out of route.

00:50 So rearranging that to make delta t the subject because we're looking for a change in, in temperature, we can say that delta t is equal to delta l divided by l alpha.

01:08 So this is just obtained by dividing both sides of the equation by l alpha l.

01:16 So to get delta t, we observed that our coefficient of expansion of a still, so working out these coefficients, so delta l in this case is equal to 1 .872 subtract 1 .870 subtract 1 .870.

01:36 So delta l is equal to minus 0 .0 .02.

01:46 Our current l is going to be given by our current at the current temperature, so is 1 .870.

01:54 2, l is equal to 1 .872, and alpha l is to be given by, so the coefficient of expansion for steel is given by 12 times 10 to the minus 6.

02:19 Now, because we're dividing through the two lengths, our units don't matter here as long as they're the same...

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