A simply-supported beam of length $$3L$$ is subjected to the loading shown in the figure.

It is given that $$P = 1$$ $$N,$$ $$L = 1$$ $$m$$ and Young’s modulus $$E = 200$$ $$GP$$a. The cross-section is a square with dimension $$100$$ $$mm$$. The bending stress (in Pa) at the point A located at the top surface of the beam at a distance of $$1.5L$$ from the left end is _____________ .

(Indicate compressive stress by a negative sign and tensile stress by a positive sign).

In the gear train shown, gear 3 is carried on arm 5. Gear 3 meshes with gear 2 and gear 4. The number of teeth on gear 2, 3, and 4 are 60, 20, and 100, respectively. If gear 2 is fixed and gear rotates with an angular velocity of 100 rpm in the counterclockwise direction, the angular speed of arm 5 (in rpm) is

A solid disc with radius a is connected to a spring at a point $$d$$ above the center of the disc. The other end of the spring is fixed to the vertical wall. The disc is free to roll without slipping on the ground. The mass of the disc is $$M$$ and the spring constant is $$K$$. The polar moment of inertia for the disc about its centre is $$J = {{M{a^2}} \over 2}$$

The natural frequency of this system in rad/s is given by

A single degree of freedom spring mass system with viscous damping has a spring constant of 10 kN/m. The system is excited by a sinusoidal force of amplitude 100 N. If the damping factor (ratio) is 0.25, the amplitude of steady state oscillation at resonance is ________mm.