2. Rod A, which is 30 cm long, expands by 0.045 cm when heated from 0°C to 100°C. Rod B, also 30 cm long expands by 0.075 cm for the same change in temperature. A third Rod C, also 30 cm long is made up of the materials of rod A and B, connected end to end. It expands by 0.065 cm when heated from 0°C to 100°C. Calculate the initial lengths of rods A and B in rod C.
Let X be the length of material of rod A and 30-X be the length of material of rod B. We have
0.065 = (0.045/30)*X + [30-X]*(0.075/30) = 0.0015*X + 0.075 - 0.0025*X or
0.001*X = 0.075 - 0.065 = 0.010 or X = 10 cm. So initial length of rod A was 10 cm and rod B was 20 cm
To solve this problem, we can use the concept of thermal expansion and the formula for linear expansion. The formula for linear expansion is given by:
ΔL = α * L * ΔT
Where ΔL is the change in length, α is the coefficient of linear expansion, L is the initial length, and ΔT is the change in temperature.
Let's calculate the coefficient of linear expansion for rod A and rod B.
For rod A:
Given: ΔL = 0.045 cm, L = 30 cm, ΔT = 100°C - 0°C = 100°C
Using the formula, we can rearrange it to solve for α:
α = ΔL / (L * ΔT)
α = 0.045 cm / (30 cm * 100°C)
α = 0.000015°C^(-1)
Similarly, let's calculate the coefficient of linear expansion for rod B.
For rod B:
Given: ΔL = 0.075 cm, L = 30 cm, ΔT = 100°C - 0°C = 100°C
Using the formula, we can rearrange it to solve for α:
α = ΔL / (L * ΔT)
α = 0.075 cm / (30 cm * 100°C)
α = 0.000025°C^(-1)
Now, let's calculate the initial lengths of rods A and B in rod C.
For rod C:
Given: ΔL = 0.065 cm, L = 30 cm, ΔT = 100°C - 0°C = 100°C
We know that rod C is made up of rod A and rod B connected end to end. Therefore, the total change in length for rod C is equal to the sum of the changes in lengths of rod A and rod B.
ΔL_C = ΔL_A + ΔL_B
0.065 cm = α_A * L_A * ΔT + α_B * L_B * ΔT
0.065 cm = (0.000015°C^(-1)) * L_A * 100°C + (0.000025°C^(-1)) * L_B * 100°C
Simplifying further:
0.065 cm = 0.0015 * L_A + 0.0025 * L_B
Now, we have one equation and two unknowns (L_A and L_B). We need another equation to solve for the two unknowns.
We can use the given information that the total length of rod C is 30 cm:
L_C = L_A + L_B
30 cm = L_A + L_B
Now, we have a system of equations:
0.065 cm = 0.0015 * L_A + 0.0025 * L_B
30 cm = L_A + L_B
Solving this system of equations will give us the initial lengths of rods A and B in rod C.
To solve this problem, let's assume the initial length of rod A is 'x' cm and the initial length of rod B is 'y' cm.
Given that rod A expands by 0.045 cm for a change in temperature from 0°C to 100°C, we can write the following equation:
0.045 = (100 - 0) / (x - 0)
Simplifying the equation, we have:
0.045 = 100 / x
Cross-multiplying, we get:
x = 100 / 0.045
x = 2222.22 cm
Therefore, the initial length of rod A is approximately 2222.22 cm.
Similarly, given that rod B expands by 0.075 cm for the same change in temperature, we can write the following equation:
0.075 = (100 - 0) / (y - 0)
Simplifying the equation, we have:
0.075 = 100 / y
Cross-multiplying, we get:
y = 100 / 0.075
y = 1333.33 cm
Therefore, the initial length of rod B is approximately 1333.33 cm.
Since rod C is formed by connecting rod A and rod B end to end, the initial length of rod C would be the sum of the initial lengths of rod A and rod B:
Initial length of rod C = Length of rod A + Length of rod B
Initial length of rod C = 2222.22 cm + 1333.33 cm
Initial length of rod C = 3555.55 cm
Therefore, the initial lengths of rods A and B in rod C are approximately 2222.22 cm and 1333.33 cm, respectively.