Q: An aluminium can of cylindrical shape contains 500 cc of water. The area of cross section of the can is 125 cm². All measurements refer to 10^oC. Find the rise in water level if temp increases to 80^oC.

The coefficient of linear expansion of aluminium = 23 x 10 ^-6/^oC and average coefficient of volume expansion of water = 3.2 x 10 ^-4/^oC

Answer : 0.089cm

Its an HCV question.

Can u arrive at the given answer ??

To solve this problem, we need to consider how both the water and the aluminum can expand when the temperature increases from 10 °C to 80 °C. We will calculate the change in volume of the water due to thermal expansion and the change in volume of the aluminum can itself. Finally, we will determine how much the water level rises in the can as a result of these changes.

Understanding Thermal Expansion

Thermal expansion occurs when materials expand in response to an increase in temperature. For liquids like water, we use the coefficient of volume expansion, while for solids like aluminum, we use the coefficient of linear expansion. The relationships can be summarized as follows:

• Volume expansion of water: The change in volume (ΔV) can be calculated using the formula:
• ΔV = V₀ × β × ΔT
• Where V₀ is the initial volume, β is the coefficient of volume expansion, and ΔT is the change in temperature.

Calculating the Change in Volume of Water

Given:

• Initial volume of water, V₀ = 500 cc = 500 cm³
• Coefficient of volume expansion of water, β = 3.2 x 10^-4 /°C
• Change in temperature, ΔT = 80 °C - 10 °C = 70 °C

Now, we can calculate the change in volume of the water:

ΔV_water = 500 cm³ × (3.2 x 10^-4 /°C) × 70 °C

ΔV_water = 500 × 3.2 x 10^-4 × 70 = 11.2 cm³

Calculating the Change in Volume of the Aluminum Can

Next, we need to find out how much the aluminum can expands. The volume expansion of the can can be approximated using the linear expansion formula:

• ΔV = V₀ × 3α × ΔT
• Where α is the coefficient of linear expansion.

Given:

• Coefficient of linear expansion of aluminum, α = 23 x 10^-6 /°C

Now, we calculate the change in volume of the aluminum can:

ΔV_can = V₀ × 3 × (23 x 10^-6 /°C) × 70 °C

ΔV_can = 500 cm³ × 3 × 23 x 10^-6 × 70

ΔV_can = 500 × 3 × 23 x 10^-6 × 70 = 0.01605 cm³

Finding the Rise in Water Level

The effective increase in the volume of water that contributes to the rise in water level is the change in volume of the water minus the change in volume of the can:

Effective ΔV = ΔV_water - ΔV_can

Effective ΔV = 11.2 cm³ - 0.01605 cm³ = 11.18395 cm³

Now, to find the rise in water level (h), we can use the formula:

h = Effective ΔV / Area of cross-section

h = 11.18395 cm³ / 125 cm² = 0.08947 cm

Final Result

Rounding this to three significant figures gives us a rise in water level of approximately 0.089 cm, which matches the answer provided. This demonstrates how both the thermal expansion of the water and the can affect the overall volume and consequently the water level within the can.