Q1. (2 points)

Imagine that I have just started working with mechanical engineers in the automotive sector for the company Delta Motor, S.C.P. I have been commissioned to design pistons for different steel motor cylinders. These motors operate at temperatures between 20ºC and 150ºC. Assume that the coefficients of expansion for brass (α_brass = 2.0 × 10⁻⁵ C⁻¹) and steel (α_steel = 1.2 × 10⁻⁵ C⁻¹) are constants within this temperature range.

a) If the piston fits tightly inside the cylinder at 20ºC, will the motors be able to operate at higher temperatures? Create a schematic of the situation and provide reasoning for your answer.

b) If the pistons have a diameter of 25.00 cm at 20ºC, what minimum diameter should the cylinders have at this temperature for the pistons to work at 150ºC?

The motors will be able to operate at higher temperatures if the expansion of the cylinder is greater than the expansion of the piston. The minimum diameter of the cylinders at 20ºC should be 25.003 cm to ensure the pistons work at 150ºC.

Explanation:

Situation:

We have been commissioned to design pistons for different cylinders of steel motors. The motors operate at temperatures between 20ºC and 150ºC. We assume that the coefficients of thermal expansion for aluminum (αllautó=2.0 10-5 C-1) and steel (αacer=1.2 10-5 C -1) are constants in this temperature range.

a) Can the motors operate at higher temperatures?

To determine if the motors can operate at higher temperatures, we need to compare the expansion of the piston and the cylinder.

At 20ºC, the piston with a thin cap is inside the cylinder. If the motors operate at higher temperatures, both the piston and the cylinder will expand. If the expansion of the cylinder is greater than the expansion of the piston, the motors will be able to operate at higher temperatures.

b) Minimum diameter of the cylinders at 20ºC:

To calculate the minimum diameter of the cylinders at 20ºC, we need to consider the expansion of the piston from 20ºC to 150ºC.

Given that the pistons have a diameter of 25.00 cm at 20ºC, we can use the coefficient of thermal expansion for steel (αacer=1.2 10-5 C -1) to calculate the expansion of the piston.

ΔL = α * L * ΔT

ΔL = (1.2 10-5 C -1) * 25.00 cm * (150ºC - 20ºC)

ΔL = 0.003 cm

The piston will expand by 0.003 cm from 20ºC to 150ºC. Therefore, the minimum diameter of the cylinders at 20ºC should be 25.003 cm to ensure the pistons work at 150ºC.

Learn more about expansion of materials due to temperature change here:

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