To tackle the problem of how the dimensions and properties of a copper penny change when its temperature is raised, we need to understand the concept of thermal expansion. When materials are heated, they tend to expand. In this case, we know that the diameter of the penny increases by 0.18% when the temperature rises by 100°C. Let's break down the calculations step by step.
Understanding Linear Expansion
The linear expansion of a material can be described by the formula:
ΔL = L₀ * α * ΔT
Where:
- ΔL = change in length (or diameter in this case)
- L₀ = original length (or diameter)
- α = coefficient of linear expansion
- ΔT = change in temperature
Calculating Changes in Area, Thickness, Volume, and Mass
(a) Change in Area
The area of a face of the penny can be calculated using the formula for the area of a circle:
A = π * (d/2)²
When the diameter increases by 0.18%, the new diameter (d') can be expressed as:
d' = d * (1 + 0.0018)
The new area (A') becomes:
A' = π * (d'/2)² = π * (d * (1 + 0.0018)/2)²
Using the binomial expansion for small changes, we can approximate:
A' ≈ A * (1 + 0.0018)² ≈ A * (1 + 0.0036)
Thus, the percentage increase in area is approximately 0.36%.
(b) Change in Thickness
Assuming the penny maintains its shape and volume, the thickness will also expand. If we denote the original thickness as t, the new thickness (t') can be approximated as:
t' = t * (1 + 0.0018)
Thus, the percentage increase in thickness is also 0.18%.
(c) Change in Volume
Volume expansion can be calculated using the formula:
ΔV = V₀ * β * ΔT
Where β (the coefficient of volumetric expansion) is approximately three times the coefficient of linear expansion (3α). The volume of the penny can be expressed as:
V = A * t
Using the area increase and thickness increase, the new volume (V') can be approximated as:
V' ≈ V * (1 + 0.0036) * (1 + 0.0018)
Thus, the percentage increase in volume is approximately 0.36% + 0.18% = 0.54%.
(d) Change in Mass
The mass of the penny is related to its volume and density. Assuming the density remains constant, the mass increase can be calculated as:
Δm = m₀ * (1 + 0.0054)
Thus, the mass will increase by approximately 0.54%.
(e) Coefficient of Linear Expansion
To find the coefficient of linear expansion (α), we can rearrange the linear expansion formula:
α = ΔL / (L₀ * ΔT)
Given that ΔL is 0.18% of the original diameter (d), we can express it as:
α = 0.0018 / (1 * 100) = 0.000018
This gives us a coefficient of linear expansion of approximately 18 x 10-6 °C-1.
In summary, when the temperature of the copper penny increases by 100°C, the area increases by about 0.36%, the thickness by 0.18%, the volume by 0.54%, and the mass by 0.54%. The coefficient of linear expansion for the penny is approximately 18 x 10-6 °C-1.
