FrontierScience / 94c55a69-1e8d-4934-ae59-2cf0401e1b01 / A horizontal metal rod has density ρ, length L, circular cross-section radius r≪L, Young's modulus Y≫ρgL³/r² in a gravitational field of strength g. It is fixed at one end to a wal…

A horizontal metal rod has density ρ, length L, circular cross-section radius r≪L, Young's modulus Y≫ρgL³/r² in a gravitational field of strength g. It is fixed at one end to a wall, which provides a constant vertical force and torque. Define coordinates as follows: x is the distance along the rod from the fixed end at the wall, and y is the vertical displacement from the fixed point, y>0 is upwards. Using suitable approximations, find the function y(x) which represents the rod's vertical displacement along it, expressed in terms of ρ,g,Y,r,x. Also give the angle, α<0, between the x-axis and the rod at x=L, in terms of ρ,g,L,Y,r as a single approximated fraction.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.

A horizontal metal rod has density \(\rho\), length \(L\), circular cross-section radius \(r \ll L\), Young's modulus \(Y \gg \rho g L^3 / r^2\) in a gravitational field of strength \(g\). It is fixed at one end to a wall, which provides a constant vertical force and torque. Define coordinates as follows: \(x\) is the distance along the rod from the fixed end at the wall, and \(y\) is the vertical displacement from the fixed point, \(y > 0\) is upwards. Using suitable approximations, find the function \(y(x)\) which represents the rod's vertical displacement along it, expressed in terms of \(\rho, g, Y, r, x\). Also give the angle, \(\alpha < 0\), between the \(x\)-axis and the rod at \(x=L\), in terms of \(\rho, g, L, Y, r\) as a single approximated fraction.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.