cm2

Question

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?

Answer 1

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.