Show that the Young's modulus Y, modulus of rigidity n and Poisson's ratio o are related by the equation Y=2n (1+ o).
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- Consider the schematic of the single pendulum. M The kinetic energy T and potential energy V may be written as: T = ²m²²8² V = -gml cos (0) аас dt 80 The Lagrangian L is given by L=T-V, and the Euler-Lagrange equations for the motion of the pendulum are given by the following second order differential equation in : ас 80 = 11 = 0 Write down the second order ODE using the specific T and V defined above. Please write this ODE in the form = f(0,0). Notice that this ODE is not linear! Now you may assume that l = m = g = 1 for the remainder of the problem. You may still suspend variables to get a system of two first order (nonlinear) ODEs by writing the ODE as: w = f(0,w) What are the fixed points of this system where all derivatives are zero? Write down the linearized equations in a neighborhood of each fixed point and determine the linear stability. You may formally linearize the nonlinear ODE or you may use a small angle approximation for sin(0); the two approaches are equivalent.Calculate the energy, corrected to first order, of a harmonic oscillator with potential:A high-carbon steel with a fully pearlitic microstructure was used to form a high-strength bolt (H.-C. Lee et al., J. Mater. Proc. Tech. 211, 1044 (2011)). It was found that the bolt head had an average interlamellar spacing of 257 nm whereas the average spacing in the body of the bolt was 134 nm. Assuming that dislocation pileup is the primary mechanism responsible for the strength of this alloy, what ratio of strength (or hardness) might be expected in the head and body of the bolt?
- Find the centroid of the homogeneous lamina. Assume r = 4. 2r- 2r (x, y) = X (Hint: The moments of the union of two or more nonoverlapping regions equ regions.) (Use symbolic notation and fractions where needed. Give your answer as poiA particle of mass m described by one generalized coordinate q movesunder the influence of a potential V(q) and a damping force −2mγq˙ proportional to its velocity. Show that the following Lagrangian gives the desired equation of motion: L = e2γt(1/2 * mq˙2 − V (q))