Question

3. At 20°C, an aluminum ring has an inner diameter of 1.995 cm and an outer diameter of 3.00 cm. The ring needs to fit over a stainless steel rod that has a diameter of 2.00 cm at 20°C. Assume that the ring and rod are situated such that they both need to be heated by the same flame (i.e., we can't just heat up the ring alone but they both have the same temperature). To what temperature in °C must the rod and ring be heated so that the ring just barely slips over the rod? Use coefficients of linear expansion: $\alpha_{Al} =$ $2.3 \times 10^{-5} (°C)^{-1}$ and $\alpha_{st} = 1.7 \times 10^{-5} (°C)^{-1}$. a. Treat this as a one dimensional problem along the diameter of the objects, i.e., the diameter of each object increases as temperature increases. For each object, write down an expression for the final length in terms of the initial length, coefficient of linear expansion, and change in temperature (use symbols). b. At the proper final temperature, the inner diameter of the ring should equal the diameter of the rod. Use this fact to set the two expressions you found in step 1 equal. Solve for $\Delta T$ in terms of $L_0$ and $\alpha$ for each object. c. Plug in numbers for $\Delta T$. To what temperature do the rod and ring need to be raised so that the ring slips over the rod?

          3. At 20°C, an aluminum ring has an inner diameter of 1.995 cm and an outer diameter of
3.00 cm. The ring needs to fit over a stainless steel rod that has a diameter of 2.00 cm
at 20°C. Assume that the ring and rod are situated such that they both need to be
heated by the same flame (i.e., we can't just heat up the ring alone but they both have
the same temperature). To what temperature in °C must the rod and ring be heated so
that the ring just barely slips over the rod? Use coefficients of linear expansion: $\alpha_{Al} =$
$2.3 \times 10^{-5} (°C)^{-1}$ and $\alpha_{st} = 1.7 \times 10^{-5} (°C)^{-1}$.
a. Treat this as a one dimensional problem along the diameter of the objects, i.e., the
diameter of each object increases as temperature increases. For each object, write
down an expression for the final length in terms of the initial length, coefficient of
linear expansion, and change in temperature (use symbols).
b. At the proper final temperature, the inner diameter of the ring should equal the
diameter of the rod. Use this fact to set the two expressions you found in step 1
equal. Solve for $\Delta T$ in terms of $L_0$ and $\alpha$ for each object.
c. Plug in numbers for $\Delta T$. To what temperature do the rod and ring need to be raised
so that the ring slips over the rod?

3. At 20°C, an aluminum ring has an inner diameter of 1.995 cm and an outer diameter of
3.00 cm. The ring needs to fit over a stainless steel rod that has a diameter of 2.00 cm
at 20°C. Assume that the ring and rod are situated such that they both need to be
heated by the same flame (i.e., we can't just heat up the ring alone but they both have
the same temperature). To what temperature in °C must the rod and ring be heated so
that the ring just barely slips over the rod? Use coefficients of linear expansion: αAl =
2.3 × 10^-5 (°C)^-1 and αst = 1.7 × 10^-5 (°C)^-1.
a. Treat this as a one dimensional problem along the diameter of the objects, i.e., the
diameter of each object increases as temperature increases. For each object, write
down an expression for the final length in terms of the initial length, coefficient of
linear expansion, and change in temperature (use symbols).
b. At the proper final temperature, the inner diameter of the ring should equal the
diameter of the rod. Use this fact to set the two expressions you found in step 1
equal. Solve for Δ T in terms of L0 and α for each object.
c. Plug in numbers for Δ T. To what temperature do the rod and ring need to be raised
so that the ring slips over the rod?

Added by Pamela M.

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