Final answer:
The depression at the centre of a metal beam supported at both ends and loaded at the centre is inversely proportional to the Young's modulus (Y) of the beam's material, meaning that the correct answer is C. 1/Y.
Step-by-step explanation:
The question relates to the deflection of a metal beam when a load is applied at its centre, and how it is connected to the Young's modulus (Y) of the material.
According to the beam deflection theory for a simply supported beam with a central load, the depression at the centre (also known as the deflection) is inversely proportional to the Young's modulus of the beam's material.
Given the formula
1/F = ∆L/(A × Y × L₀),
where F is the force applied, ∆L is the change in length (deflection), A is the cross-sectional area, and L₀ is the original length, it can be seen that as Young's modulus (Y) increases, the deflection decreases, thereby indicating an inverse relationship.
Thus, the most accurate option for this question is C. 1/Y.