A point mass of $$100$$ kg is dropped onto a massless elastic bar (cross-sectional area = $$100$$ $$mm$$² ,length = $$1$$ m, Young's modulus = $$100$$ GPa) from a height $$H$$ of $$10$$ $$mm$$ as shown (Figure is not to scale). If g = $$10$$ $$m/s$$² , the maximum compression of the elastic bar is _______ $$mm.$$

A horizontal bar, fixed at one end (x = 0), has a length of 1 m, and cross-sectional area of 100 mm². Its elastic modulus varies along its length as given by E(x) = 100 e^-x GPa, Where x is the length coordinate (in m) along the axis of the bar. An axial tensile load of 10 kN is applied at the free end (x=1). The axial displacement of the free end is _______ mm.

Consider a beam with circular cross-section of diameter $$d.$$ The ratio of the second moment of area about the neutral axis to the section modulus of the area is

An initially stress-free massless elastic beam of length $$L$$ and circular cross-section with diameter $$d (d < < L)$$ is held fixed between two walls as shown. The beam material has Young's modulus $$E$$ and coefficient of thermal expansion $$\alpha .$$

If the beam is slowly and uniformly heated, the temperature rise required to cause the beam to buckle is proportional to