a metal rod A of 25 cm length expands by 0.050 cm when its temperature is rasied from 0 deg to 100 deg. another rod B of a different metal of length 40 cm expands by 0.040 cm for the same rise in temp. A third rod C of 50 cm length is made up of pieces of rods A and B placed end to end expands by 0.03 cm on heating from 0 deg to 50 deg. Find the lengths of each portion of the composite rod.

To solve this problem, we need to understand how thermal expansion works in different materials. The expansion of a rod when heated can be described using the formula:

Understanding Thermal Expansion

The linear expansion of a material can be expressed as:

ΔL = L₀ * α * ΔT

Where:

• ΔL = change in length
• L₀ = original length of the rod
• α = coefficient of linear expansion of the material
• ΔT = change in temperature

From the problem, we have two rods, A and B, with known expansions when heated from 0°C to 100°C. We can use this information to find the coefficients of linear expansion for each rod.

Calculating Coefficients of Linear Expansion

For rod A:

• Original length (L₀) = 25 cm
• Change in length (ΔL) = 0.050 cm
• Change in temperature (ΔT) = 100°C

Using the formula:

0.050 = 25 * α_A * 100

Solving for α_A gives:

α_A = 0.050 / (25 * 100) = 0.00002 cm/cm/°C

For rod B:

• Original length (L₀) = 40 cm
• Change in length (ΔL) = 0.040 cm
• Change in temperature (ΔT) = 100°C

Using the same formula:

0.040 = 40 * α_B * 100

Solving for α_B gives:

α_B = 0.040 / (40 * 100) = 0.00001 cm/cm/°C

Analyzing the Composite Rod C

Now, we have a composite rod C made up of pieces of rods A and B. The total length of rod C is 50 cm, and it expands by 0.03 cm when heated from 0°C to 50°C. We can denote the lengths of the portions of rods A and B in rod C as x and y, respectively.

Since the total length of rod C is 50 cm, we have:

x + y = 50

Now, the change in length for rod C can be expressed as:

ΔL_C = ΔL_A + ΔL_B

For rod A in C:

ΔL_A = x * α_A * ΔT

For rod B in C:

ΔL_B = y * α_B * ΔT

Substituting the values for ΔT (50°C) and the coefficients:

0.03 = (x * 0.00002 * 50) + (y * 0.00001 * 50)

This simplifies to:

0.03 = 0.001 * x + 0.0005 * y

Setting Up the Equations

Now we have two equations:

• 1. x + y = 50
• 2. 0.001x + 0.0005y = 0.03

We can solve these equations simultaneously. From the first equation, we can express y in terms of x:

y = 50 - x

Substituting this into the second equation:

0.001x + 0.0005(50 - x) = 0.03

0.001x + 0.025 - 0.0005x = 0.03

0.0005x = 0.03 - 0.025

0.0005x = 0.005

x = 0.005 / 0.0005 = 10 cm

Now substituting back to find y:

y = 50 - 10 = 40 cm

Final Results

The lengths of each portion of the composite rod C are:

• Length of rod A in C: 10 cm
• Length of rod B in C: 40 cm

This analysis shows how different materials expand differently and how we can use their properties to solve composite problems effectively.