An aluminum bar has the desired length when at 10 degrees Celsius. How much stress is required to keep it at this length if the temperature increases to 33 degrees Celsius?

The change in L = (Initial L)x(Force)/E/Area.

The change in L also = (Alpha)x(Initial L)x(Change in Temperature)

So, (Alpha)x(Change in Temperature)x(E) = Force/Area

So, (25x10^-6)x(23)x(70x10^9) = 4.0x10^7 N/m^2

To determine the stress required to keep the aluminum bar at the desired length when the temperature increases to 33 degrees Celsius, you will need to consider the thermal expansion of aluminum. The equation to calculate the linear expansion of a material is given by:

ΔL = α * L0 * ΔT

Where:
ΔL is the change in length
α is the coefficient of linear expansion
L0 is the original length of the bar
ΔT is the change in temperature

First, you need to find the coefficient of linear expansion for aluminum. The coefficient of linear expansion for aluminum is approximately 22.2 × 10^(-6) per degree Celsius (α = 22.2 × 10^(-6) / °C).

Next, calculate the change in length (ΔL) using the equation above. The original length (L0) is not provided, so you must assume a value for it. Let's assume L0 is 1 meter.

ΔT = 33°C - 10°C = 23°C

ΔL = (22.2 × 10^(-6) / °C) * (1 meter) * (23°C)
ΔL = 0.000511 meter

The change in length is 0.000511 meters.

Finally, to calculate the stress required to keep the bar at the desired length, you need to use Hooke's Law:

Stress = (Force / Area)

Hooke's Law states that stress is directly proportional to strain (change in length divided by the original length) and the modulus of elasticity (Young's modulus). For aluminum, the modulus of elasticity is approximately 70 GPa (70 × 10^9 Pa).

Strain = ΔL / L0 = 0.000511 m / 1 m = 0.000511
Young's modulus (E) = 70 × 10^9 Pa

Stress = (Young's modulus) * (Strain)
Stress = (70 × 10^9 Pa) * (0.000511)
Stress = 35,770 Pa or 35.77 kPa

Therefore, approximately 35.77 kPa (or 35,770 Pa) of stress is required to keep the aluminum bar at the desired length when the temperature increases to 33 degrees Celsius.

To calculate the amount of stress required to keep an aluminum bar at a constant length when the temperature increases, we need to use the concept of thermal expansion. Thermal expansion refers to the tendency of materials to expand or contract in response to changes in temperature.

The equation used to calculate the change in length of a material due to temperature change is:

ΔL = α * L * ΔT

Where:
ΔL is the change in length
α (alpha) is the coefficient of thermal expansion of the material
L is the original length of the material
ΔT is the change in temperature

To calculate the stress required, we need to use Hooke's Law, which states that stress is directly proportional to strain. In this case, the strain is the change in length divided by the original length:

Strain (ε) = ΔL / L

Stress (σ) = Young's Modulus (E) * Strain

To calculate the stress, we need to know the Young's Modulus (E) for aluminum, which is a measure of its elasticity. The Young's Modulus for aluminum is typically around 70 GPa (Gigapascals).

Let's assume the original length of the aluminum bar is L = 1 meter.

Step 1: Calculate the change in length (ΔL)
ΔL = α * L * ΔT

From available data, the coefficient of linear expansion for aluminum is approximately 24 x 10^-6 per degree Celsius (α = 24 x 10^-6 / °C).

Given ΔT = 33 °C - 10 °C = 23 °C

ΔL = (24 x 10^-6 / °C) * (1 meter) * (23 °C)
ΔL = 0.000552 meters (or 0.552 mm)

Step 2: Calculate the strain (ε)
Strain (ε) = ΔL / L

ε = (0.000552 meters) / (1 meter)
ε = 0.000552

Step 3: Calculate the stress (σ)
Stress (σ) = Young's Modulus (E) * Strain

Assuming Young's Modulus (E) for aluminum is 70 GPa = 70 x 10^9 Pa

σ = (70 x 10^9 Pa) * (0.000552)
σ ≈ 38,640 Pa (or 38.64 kPa)

Therefore, the stress required to keep the aluminum bar at the desired length when the temperature increases to 33 degrees Celsius is approximately 38.64 kilopascals.