An L-shaped elastic member ABC with slender arms AB and BC of uniform crosssection is clamped at end A and connected to a pin at end C. The pin remains in continuous contact with and is constrained to move in a smooth horizontal slot. The section modulus of the member is same in both the arms. The end C is subjected to a horizontal force P and all the deflections are in the plane of the figure. Given the length AB is 4a and length BC is 𝑎, the magnitude and direction of the normal force on the pin from the slot, respectively, are

A thin-walled cylindrical pressure vessel has mean wall thickness of t and nominal radius of r. The Poisson’s ratio of the wall material is 1/3. When it was subjected to some internal pressure, its nominal perimeter in the cylindrical portion increased by 0.1% and the corresponding wall thickness became t̅. The corresponding change in the wall thickness of the cylindrical portion, i.e. 100 × (t̅ − t)/t, is ________% (round off to 3 decimal places).

The figure shows a schematic of a simple Watt governor mechanism with the spindle O₁O₂ rotating at an angular velocity ω about a vertical axis. The balls at P and S have equal mass. Assume that there is no friction anywhere and all other components are massless and rigid. The vertical distance between the horizontal plane of rotation of the balls and the pivot O₁ is denoted by h. The value of h = 400 mm at a certain ω. If ω is doubled, the value of h will be _________ mm.

A rigid uniform annular disc is pivoted on a knife edge A in a uniform gravitational field as shown, such that it can execute small amplitude simple harmonic motion in the plane of the figure without slip at the pivot point. The inner radius r and outer radius 𝑅 are such that r² = R²/2, and the acceleration due to gravity is g. If the time period of small amplitude simple harmonic motion is given by $T = β π \sqrt{R/g} $ where π is the ratio of circumference to diameter of a circle, then β = ________ (round off to 2 decimal places).