A bronze sleeve with an inside diameter of 99.8 mm is to be placed over a solid steel cylinder that has an outside diameter of 100 mm.

Given:
- Coefficient of linear expansion for bronze, [α_B = 16.9 \times 10^{-6} \, ^\circ \text{C}^{-1}]
- Coefficient of linear expansion for steel, [α_s = 11.9 \times 10^{-6} \, ^\circ \text{C}^{-1}]

If the temperatures of the cylinder and sleeve remain equal, how much must the temperature be increased in order for the bronze sleeve to slip over the steel cylinder?

Question

28-06-2025
Engineering

College

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A bronze sleeve with an inside diameter of 99.8 mm is to be placed over a solid steel cylinder that has an outside diameter of 100 mm.

Given:
- Coefficient of linear expansion for bronze, [α_B = 16.9 \times 10^{-6} \, ^\circ \text{C}^{-1}]
- Coefficient of linear expansion for steel, [α_s = 11.9 \times 10^{-6} \, ^\circ \text{C}^{-1}]

If the temperatures of the cylinder and sleeve remain equal, how much must the temperature be increased in order for the bronze sleeve to slip over the steel cylinder?

Answer :

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marisht marisht · Answer 1 · 2023-12-16T16:18:14-05:00

Final answer:

The temperature does not need to be increased for the bronze sleeve to slip over the steel cylinder, as long as the temperatures of the two remain equal.

Explanation:

To determine how much the temperature must be increased for the bronze sleeve to slip over the steel cylinder, we need to consider the thermal expansion of both materials. The change in diameter of the bronze sleeve can be calculated using the formula:

ΔD = αB × D × ΔT

Where ΔD is the change in diameter, αB is the coefficient of linear expansion for bronze, D is the initial inside diameter of the sleeve, and ΔT is the temperature change. Similarly, the change in diameter of the steel cylinder can be calculated using the formula:

ΔD = αâ‚› × D × ΔT

Where αâ‚› is the coefficient of linear expansion for steel. Since the two diameters must be equal for the sleeve to slip over the cylinder, we can set the two formulas equal to each other and solve for ΔT:

αB × D × ΔT = αâ‚› × D × ΔT

ΔT = 0

This means that the temperature does not need to be increased for the bronze sleeve to slip over the steel cylinder, as long as the temperatures of the two remain equal.