Assuming that the cylinder is fixed at both ends and that its wall thickness is 7.5 mm, what force would be exerted by the cylinder during this change in temperature?
(Note: the Young’s modulus for aluminium is 70 GPa.)
To calculate the force exerted by the cylinder during a change in temperature, we need to consider two factors: the change in length of the cylinder and the stiffness of the material.
First, let's calculate the change in length (∆L) of the cylinder due to the change in temperature. We can use the thermal expansion formula:
∆L = α * L0 * ∆T
Where:
α is the coefficient of linear expansion of the material (for aluminum, it is typically around 22 × 10^(-6) per degree Celsius).
L0 is the original length of the cylinder.
∆T is the change in temperature.
Next, we need to consider the stiffness or Young's modulus (E) of the material. For aluminum, the Young's modulus is stated as 70 GPa (Gigapascals).
The formula for stress (σ) is:
σ = E * (∆L / L0)
Where:
σ is the stress experienced by the material.
E is the Young's modulus.
∆L is the change in length.
L0 is the original length.
Finally, to find the force (F) exerted by the cylinder, we use the formula:
F = A * σ
Where:
F is the force exerted by the cylinder.
A is the cross-sectional area of the cylinder.
σ is the stress experienced by the material.
To calculate the cross-sectional area, we need to consider the radius (r) and the thickness (t) of the cylinder. Since the thickness is given as 7.5 mm, we can calculate the radius as r = (r_outer - r_inner)/2.
Now that we have all the necessary formulas, we can plug in the given values and calculate the force exerted by the cylinder during the change in temperature.