A uniform light slender beam AB of section modulus EI is pinned by a frictionless joint A to the ground and supported by a light inextensible cable CB to hang a weight W as shown. If the maximum value of W to avoid buckling of the beam AB is obtained as $\beta {\pi }^{2}EI$, where $\pi$ is the ratio of circumference to diameter of a circle, then the value of $\beta$ is

The figure shows a schematic of a simple Watt governor mechanism with the spindle O1O2 rotating at an angular velocity $\omega$ 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 $\omega$. If $\omega$ is doubled, the value of h will be_________ mm.

A square threaded screw is used to lift a load W by applying a force F. Efficiency of square threaded screw is expressed as

A CNC worktable is driven in a linear direction by a lead screw connected directly to a stepper motor. The pitch of the lead screw is 5 mm. The stepper motor completes one full revolution upon receiving 600 pulses. If the worktable speed is 5 m/minute and there is no missed pulse, then the pulse rate being received by the stepper motor is

The type of fit between a mating shaft of diameter $25.0_{-0.010}^{+0.010}$ mm and a hole of diameter $25.015_{-0.015}^{+0.015}$ mm is__________.

In a linear programming problem, if a resource is not fully utilized, the shadow price of that resource is

The Clausius inequality holds good for

A tiny temperature probe is fully immersed in a flowing fluid and is moving with zero relative velocity with respect to the fluid. The velocity field in the fluid is $\vec{V}=(2x)\hat{i}+(y+3t)\hat{j}$, and the temperature field in the fluid is $T=2x^2+xy+4t$, where x and y are the spatial coordinates, and t is the time. The time rate of change of temperature recorded by the probe at (x = 1, y = 1, t = 1) is_______.

The limit $p={\lim }_{x\to \pi }\left(\frac{x^2+\alpha x+2{\pi }^2}{x-\pi }+2\sin x\right)$ has a finite value for a real $\alpha$. The value of $\alpha$ and the corresponding limit $p$ are

Given a function $\varphi =\frac{1}{2}(x^2+y^2+z^2)$ in three-dimensional Cartesian space, the value of the surface integral ${\iint }_S\hat{n}\cdot \nabla \varphi dS$, where $S$ is the surface of a sphere of unit radius and $\hat{n}$ is the outward unit normal vector on $S$, is

After playing_________ hours of tennis, I am feeling_________ tired to walk back.