A brass ring of diameter 10.00 cm at 22.2°C is heated and slipped over an aluminum rod of diameter 10.01 cm at 22.2°C. Assume the average coefficients of linear expansion are constant.
(b) What if the aluminum rod were 9.16 cm in diameter?
nitial area:A=πd24 =π(10)24 =25π cm2Coefficient of areal expansion:β=2α =2×1.8 =3.6×10−5Change in temperature:△t=50oCIncrease in area:△A=βA△t =(3.6×10−5)(25π)(5
To calculate the final diameter of the brass ring when it is heated and slipped over an aluminum rod with a diameter of 9.16 cm, we can use the formula for the coefficient of linear expansion:
ΔL = α * L * ΔT
Where:
ΔL = change in length
α = coefficient of linear expansion
L = initial length
ΔT = change in temperature
Since the problem states that the average coefficients of linear expansion are constant, we can assume α is constant for both brass and aluminum.
Given that the initial diameter of the brass ring is 10.00 cm and the initial diameter of the aluminum rod is 9.16 cm, we can calculate the initial lengths using the formula:
L = π * d
where d is the diameter.
For the brass ring:
Initial length of the brass ring = π * 10.00 cm
For the aluminum rod:
Initial length of the aluminum rod = π * 9.16 cm
Next, we need to calculate the change in temperature (ΔT). The problem doesn't provide this information, so you will need to use the specific temperature change you want to calculate the final diameter for.
Once you have ΔT, you can calculate ΔL for both the brass ring and the aluminum rod using the formula mentioned earlier.
Finally, calculate the final diameter of the brass ring by adding ΔL and the initial length of the aluminum rod.
Let me know if you need any further clarification!
To find the answer, we'll need to use the concept of thermal expansion. The equation for linear expansion is ΔL = αLΔT, where ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature.
For the brass ring:
- Diameter before heating, D1 = 10.00 cm
- Diameter after heating, D2
For the aluminum rod:
- Diameter before heating, d1 = 10.01 cm
- Diameter after heating, d2
We can calculate the change in diameter for each material using the equation ΔD = αDΔT.
Since the average coefficients of linear expansion are constant for both materials and the initial and final temperatures are the same, the change in diameter for both will be the same.
Let's use the given information to solve for ΔD.
Given:
Initial diameter of the brass ring (D1) = 10.00 cm
Initial diameter of the aluminum rod (d1) = 10.01 cm
To find:
Change in diameter (ΔD) when the aluminum rod is 9.16 cm in diameter
We can set up a proportion to solve for ΔD:
(D1/D2) = (d1/d2)
Substituting the given values:
(10.00 cm / D2) = (10.01 cm / 9.16 cm)
Cross-multiplying:
10.00 cm * 9.16 cm = D2 * 10.01 cm
91.6 cm^2 = 10.01 * D2 cm^2
Dividing both sides by 10.01 cm:
D2 = (91.6 cm^2) / (10.01 cm)
D2 ≈ 9.15 cm
Therefore, when the aluminum rod is 9.16 cm in diameter, the brass ring will have a diameter of approximately 9.15 cm after heating.