A steel rod is 3000 cm in diameter at 25°C. a brass ring has an interior diameter of 2.992 cm at 25°C. At what common temperature will the ring just slide onto the rod?

To determine the common temperature at which the brass ring will just slide onto the steel rod, we need to consider how the dimensions of both materials change with temperature. This phenomenon is due to thermal expansion, which affects solids differently based on their material properties.

Understanding Thermal Expansion

Thermal expansion refers to the way materials expand when heated. Each material has a coefficient of linear expansion, which quantifies how much it expands per degree change in temperature. For steel, the coefficient is approximately 11 x 10^-6 °C^-1, while for brass, it is about 19 x 10^-6 °C^-1.

Initial Dimensions

Let's start with the initial dimensions:

• The diameter of the steel rod at 25°C is 3.000 cm.
• The interior diameter of the brass ring at 25°C is 2.992 cm.

Setting Up the Equation

We want to find the temperature at which the diameter of the brass ring equals the diameter of the steel rod. We can express the diameters as functions of temperature:

• Diameter of the steel rod at temperature T: D_steel(T) = 3.000 + (3.000 × 11 x 10^-6 × (T - 25))
• Diameter of the brass ring at temperature T: D_brass(T) = 2.992 + (2.992 × 19 x 10^-6 × (T - 25))

Finding the Common Temperature

To find the temperature at which the ring just fits over the rod, we set the two diameter equations equal to each other:

3.000 + (3.000 × 11 x 10^-6 × (T - 25)) = 2.992 + (2.992 × 19 x 10^-6 × (T - 25))

Simplifying the Equation

Now, let's simplify this equation:

1. Distribute the terms:

• 3.000 + 3.000 × 11 x 10^-6 × (T - 25)
• 2.992 + 2.992 × 19 x 10^-6 × (T - 25)

2. Rearranging gives us:

3.000 - 2.992 = (2.992 × 19 x 10^-6 - 3.000 × 11 x 10^-6) × (T - 25)

3. Calculate the left side:

0.008 = (2.992 × 19 x 10^-6 - 3.000 × 11 x 10^-6) × (T - 25)

4. Calculate the coefficients:

2.992 × 19 = 0.000056784

3.000 × 11 = 0.000033

So, 0.000056784 - 0.000033 = 0.000023784

5. Now we have:

0.008 = 0.000023784 × (T - 25)

6. Solving for T:

T - 25 = 0.008 / 0.000023784

T - 25 ≈ 336.5

T ≈ 361.5°C

Final Result

The common temperature at which the brass ring will just slide onto the steel rod is approximately 361.5°C. This temperature allows for the expansion of both materials to match, enabling the ring to fit over the rod seamlessly.