A 5 cm long metal rod AB was initially at a uniform temperature of T0°C. Thereafter, temperature at both the ends are maintained at 0°C. Neglecting the heat transfer from the lateral surface of the rod, the heat transfer in the rod is governed by the one-dimensional diffusion equation $\rm\frac{\partial T}{\partial t}=D\frac{\partial^2T}{\partial x^2}$, where D is the thermal diffusivity of the metal, given as 1.0 cm2/s.

The temperature distribution in the rod is obtained as

$\rm T(x,t)=\Sigma_{n=1,3,5...}^{\infty}C_n\sin\frac{n\pi x}{5}e^{-\beta n^2t}$,

where x is in cm measured from A to B with 𝑥 = 0 at A, t is in s, 𝐶𝑛 are constants in °C, T is in °C, and β is in s−1 .

The value of β (in 𝑠−1 , rounded off to three decimal places) is ________.

A beam is subjected to a system of coplanar forces as shown in the figure. The magnitude of vertical reaction at Support P is ______ N (round off to one decimal place). 

A 2D thin plate with modulus of elasticity, 𝐸 = 1.0 N/m², and Poisson’s ratio, 𝜇 = 0.5, is in plane stress condition. The displacement field in the plate is given by 𝑢 = 𝐶𝑥²𝑦 and 𝑣 = 0, where 𝑢 and 𝑣 are displacements (in m) along the 𝑋 and 𝑌 directions, respectively, and 𝐶 is a constant (in m⁻²). The distances x and y along X and Y, respectively, are in m. The stress in the 𝑋 direction is 𝜎_𝑋𝑋 = 40𝑥𝑦 N/m², and the shear stress is 𝜏_𝑋𝑌 = 𝛼𝑥² N/m². What is the value of 𝛼 (in N/m⁴, in integer)? ________

Muller-Breslau principle is used in analysis of structures for ________.