When the temperature of a metal cylinder is raised from 60 to 100°C its length increase 0.092%.

(a) Find the percent change in density.

(b) Identify the metal.

To tackle the problem of how the density of a metal cylinder changes with temperature, we need to consider the relationships between temperature, volume, and density. Let's break this down step by step.

Understanding the Relationship Between Temperature and Density

When a metal is heated, it expands. This expansion affects both its volume and density. Density is defined as mass per unit volume, represented by the formula:

Density (ρ) = Mass (m) / Volume (V)

As the temperature of the metal increases, its volume expands, while the mass remains constant. Therefore, the density will change as follows:

Step 1: Calculate the Change in Volume

The problem states that the length of the cylinder increases by 0.092% when the temperature rises from 60°C to 100°C. For a cylindrical object, the volume can be expressed in terms of its length (L) and cross-sectional area (A):

Volume (V) = A × L

When the length increases by 0.092%, the new length (L') can be calculated as:

L' = L + (0.00092 × L) = 1.00092 × L

The new volume (V') will then be:

V' = A × L' = A × (1.00092 × L) = 1.00092 × V

Step 2: Calculate the Change in Density

Since the mass of the cylinder remains unchanged, we can express the new density (ρ') as:

ρ' = m / V' = m / (1.00092 × V) = ρ / 1.00092

Now, to find the percent change in density, we can use the formula:

Percent Change in Density = [(ρ - ρ') / ρ] × 100%

Substituting ρ' into this formula gives:

Percent Change in Density = [(ρ - ρ / 1.00092) / ρ] × 100%

This simplifies to:

Percent Change in Density = [1 - (1 / 1.00092)] × 100%

Calculating this yields:

Percent Change in Density ≈ [1 - 0.99908] × 100% ≈ 0.092%

Identifying the Metal

Now that we have the percent change in density, we can identify the metal. Different metals have specific coefficients of linear expansion, which is the amount they expand per degree of temperature change. The coefficient of linear expansion (α) can be estimated using the formula:

α = ΔL / (L × ΔT)

Where ΔL is the change in length, L is the original length, and ΔT is the change in temperature. In this case, we know:

• ΔL = 0.092% of L = 0.00092 × L
• ΔT = 100°C - 60°C = 40°C

Substituting these values into the formula gives:

α = (0.00092 × L) / (L × 40) = 0.00092 / 40 = 0.000023

Converting this to a more standard form gives:

α ≈ 23 × 10^-6 / °C

Now, comparing this value with known coefficients of linear expansion for various metals, we find that aluminum has a coefficient of approximately 23 × 10^-6 / °C, which matches our calculation closely.

Final Thoughts

In summary, the percent change in density of the metal cylinder when heated from 60°C to 100°C is approximately 0.092%. Based on the coefficient of linear expansion, the metal is likely aluminum. This example illustrates how temperature affects physical properties and helps us identify materials based on their thermal characteristics.