cm2
A steel ring with a hole having area of 3.99
is to be placed on an aluminum rod with cross-sectional area of
cm2.
4.00
Both rod and ring are initially at a temperature of 35.0�C. At what common temperature can the steel ring
be slipped onto one end of the aluminum rod?
To find the common temperature at which the steel ring can be slipped onto one end of the aluminum rod, we can use the principle of thermal expansion.
The formula for thermal expansion is:
ΔL = α * L * ΔT
Where:
ΔL = Change in length
α = Coefficient of linear expansion
L = Initial length
ΔT = Change in temperature
For the steel ring:
ΔL1 = α1 * L1 * ΔT
For the aluminum rod:
ΔL2 = α2 * L2 * ΔT
Since both the steel ring and aluminum rod have the same change in temperature, ΔT, we can set their respective change in lengths equal to each other:
ΔL1 = ΔL2
α1 * L1 * ΔT = α2 * L2 * ΔT
ΔT cancels out:
α1 * L1 = α2 * L2
Now we can solve for the common temperature by rearranging the equation:
α1 * L1 / α2 = L2
Substituting the given areas for the hole and cross-sectional area:
(α1 * L1) / α2 = (3.99 cm²) / (4.00 cm²)
Now we need to find the coefficient of linear expansion for steel and aluminum. Let's assume the coefficient of linear expansion for steel is α1 and for aluminum is α2.
Using online sources, we can find approximate values for the coefficients of linear expansion:
For steel: α1 ≈ 11.7 x 10^-6 1/°C
For aluminum: α2 ≈ 23.1 x 10^-6 1/°C
Substituting these values into the equation:
(11.7 x 10^-6 1/°C * L1) / (23.1 x 10^-6 1/°C) = (3.99 cm²) / (4.00 cm²)
Simplifying the equation:
11.7/23.1 = (3.99/4)
Cross-multiplying:
(11.7 * 4) = (23.1 * 3.99)
Simplifying further:
46.8 = 92.229
Therefore, the equation is not balanced and there is no common temperature at which the steel ring can be slipped onto one end of the aluminum rod.
To find the common temperature at which the steel ring can be slipped onto the aluminum rod, we need to consider the thermal expansion of both materials.
Let's assume the coefficient of linear expansion for steel is α₁ and for aluminum is α₂. We also need to consider the change in area for both the ring and the rod.
The formula for the change in length due to thermal expansion is given by:
ΔL = α * L * ΔT
For the ring, the change in length is negligible since it has a hole. However, the change in area is significant. The change in area for the ring can be calculated as follows:
ΔA₁ = α₁ * A₁ * ΔT
For the rod, both the change in length and the change in area will be considered. The change in length can be calculated as follows:
ΔL₂ = α₂ * L * ΔT
The change in area for the rod can be calculated as follows:
ΔA₂ = α₂ * A₂ * ΔT
Since the final area after slipping the ring onto the rod will be the sum of the original area of the rod and the ring, we can represent it as:
(A₂ + ΔA₂) - (A₁ + ΔA₁) = ΔA₂ - ΔA₁
Now, let's solve for the common temperature at which the change in area for both the ring and the rod is equal. Substituting the equations for ΔA₁ and ΔA₂, we get:
(α₂ * A₂ * ΔT) - (α₁ * A₁ * ΔT) = 0
Given the values of α₁ (coefficient of linear expansion for steel), A₁ (area of the ring), α₂ (coefficient of linear expansion for aluminum), A₂ (area of the rod), and ΔT (change in temperature), we can solve for the common temperature using the equation above.