At 20.0^∘ C, an aluminum ring has an inner diameter of 5.0000 cm and a brass rod has a diameter

At 20.0^∘ C, an aluminum ring has an inner diameter of...

At $20.0^{\circ} \mathrm{C},$ an aluminum ring has an inner diameter of 5.0000 $\mathrm{cm}$ and a brass rod has a diameter of $5.0500 \mathrm{cm} .$ (a) If only the ring is warmed, what temperature must it reach so that it will just slip over the rod? (b) What If? If both the ring and the rod are warmed together, what temperature must they both reach so that the ring barely slips over the rod? (c) Would this latter process work? Explain. Hint: Consult Table 20.2 in the next chapter.

Key Concepts

Differential Expansion

Differential expansion occurs when two materials with different coefficients of thermal expansion change size by different amounts under the same temperature change. This concept is crucial in applications where parts made of different materials interact mechanically, as disparities in expansion can lead to interference or clearance issues.

Thermal Expansion

Thermal expansion refers to the tendency of materials to change their dimensions in response to a change in temperature. When a material is heated, its particles vibrate more intensely, generally causing the material's length, area, or volume to increase. This basic physical principle underlies many engineering problems where temperature variations lead to changes in the dimensions of structural components.

Coefficient of Linear Expansion

The coefficient of linear expansion is a material-specific constant that quantifies the degree to which a material expands per unit length per degree change in temperature. It allows for the calculation of dimensional changes in objects when subjected to temperature changes, and it is critical in predicting whether one component will fit with another after thermal expansion.

Interference and Clearance Fits

Interference and clearance fits describe the relationship between two parts designed to be assembled together. In the context of thermal expansion, analyzing these fits involves ensuring that temperature-induced dimensional changes will produce the desired degree of tightness or looseness between mating parts, which is especially important when using heating or cooling to facilitate assembly.

Transcript

00:01 Let's start by considering just warming the aluminum ring, essentially making its diameter equal to that of the brass rod, which is 5 .05 centimeters.

00:12 So far as we can look in our book and find the average linear expansion coefficient for aluminum, which is equal to 24 times 10 to negative 6, inverse degrees celsius.

00:30 And we also can find the desired change in diameter we want, which we know is just the diameter of the brass rod minus the diameter of the aluminum ring, which is equal to about 0 .05 centimeters or 5 times 10 to the negative 4 meters.

00:57 Okay, so now let's use the equation delta d is equal to alpha, d -i -d -t, and solve for delta t to find tf, essentially the temperature needed to expand, the aluminum ring this much.

01:14 We can plug in our numbers, 5 times 10 -negative 4 is equal to 24 times 10 -negative 6, t initial for the aluminum ring, which is just 5 centimeters or 0 .05 meters.

01:33 Just writing meters to make it extra clear, and then delta t.

01:37 Solve this for delta t and find that it is 417 degrees celsius.

01:46 It could also be kelvin because the difference doesn't really matter which one.

01:51 But this change of temperature is equal to a final temperature, t .f.

01:55 Minus our initial temperature, which you know is just 20 .0 degrees celsius.

02:00 So therefore, final temperature for this expansion is equal to 437 degrees celsius.

02:10 Okay.

02:12 Now, what if the ring and rod expand at the same time? first thing we want to do is we're going to need to find, once again, the average linear expansion coefficient but this time for the brass rod.

02:33 I'm going to write that as alpha b to make it extra clear.

02:36 And if you look that up in your book, you should find that it is equal to 19 times 10 to negative 6 inverse degrees celsius...

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