Two thin strips of metal are bolted together at one end and have the same temperature. One is steel, and the other is aluminum. The steel strip is 0.10% longer than the aluminum strip. By how much should the temperature of the strips be increased, so that the strips have the same length?

To determine the temperature increase required for the steel and aluminum strips to have the same length, we need to consider their respective coefficients of linear expansion.

The coefficient of linear expansion (α) measures the change in length of a material per unit change in temperature. It is typically expressed in units of per degree Celsius (°C⁻¹).

Given that the steel strip is 0.10% longer than the aluminum strip, we can convert this percentage into a decimal by dividing it by 100: 0.10% = 0.10/100 = 0.001.

Let L₁ be the original length of both strips and ΔL be the change in length.

For the steel strip:
ΔL₁ = α₁ * L₁ * ΔT

For the aluminum strip:
ΔL₂ = α₂ * L₁ * ΔT

Where α₁ and α₂ are the coefficients of linear expansion for steel and aluminum, respectively, and ΔT is the temperature increase.

Since we want the strips to have the same length after the temperature increase, we can set ΔL₁ equal to ΔL₂ and solve for ΔT.

ΔL₁ = ΔL₂
α₁ * L₁ * ΔT = α₂ * L₁ * ΔT

Since α₁ and α₂ are constants for each material, we can cancel them out:

ΔT = ΔT * (α₂ / α₁)
1 = α₂ / α₁

This equation implies that the coefficients of linear expansion for steel and aluminum must be equal to make the strips of the same length.

Therefore, no matter how much we raise the temperatures of the strips, they will never have the same length unless the coefficients of linear expansion are equal. Thus, it is not possible to determine the temperature increase needed solely based on the information provided.