Step 1: Set the surface energy balance over the diurnal cycle.
At the land surface, the net radiation \(R_n = (S_\downarrow - S_\uparrow) + (L_\downarrow - L_\uparrow)\) drives sensible (\(H\)) and latent (\(LE\)) heat fluxes and ground heat flux (\(G\)). After sunset \(S_\downarrow \approx 0\), so \(R_n<0\) due to strong longwave loss \(L_\uparrow\). Hence, \(H\) becomes negative (air is cooled by the surface), and the near‐surface layer cools continuously through the night.
Step 2: Nighttime boundary layer evolution.
Radiative cooling at the ground produces a stable surface layer (temperature increases with height). Turbulence weakens, so mixing is small and 2 m air closely follows surface cooling. This cooling proceeds monotonically through the night if sky is clear and winds are light.
Step 3: Why minimum occurs \(near\ sunrise\) and not at midnight.
Cooling continues as long as \(R_n<0\). The transition to net heating starts only after the sun rises high enough for \(S_\downarrow\) to offset longwave loss—this happens \emph{after} sunrise. Therefore the lowest 2 m temperature typically happens just before/at sunrise, with a small lag depending on clouds/wind. Midnight is earlier in the cooling period, so temperature is still dropping toward its minimum.
Step 4: Role of local factors (for completeness).
Clouds reduce longwave loss (warmer night); wind increases mixing (less cooling). Both can shift the exact timing slightly, but in tropical clear-sky land locations the canonical minimum remains just before sunrise.
Final Answer:
\[ \boxed{\text{At sunrise}} \]
Potato slices weighing 50 kg is dried from 60% moisture content (wet basis) to 5% moisture content (dry basis). The amount of dried potato slices obtained (in kg) is ............ (Answer in integer)
Two Carnot heat engines (E1 and E2) are operating in series as shown in the figure. Engine E1 receives heat from a reservoir at \(T_H = 1600 \, {K}\) and does work \(W_1\). Engine E2 receives heat from an intermediate reservoir at \(T\), does work \(W_2\), and rejects heat to a reservoir at \(T_L = 400 \, {K}\). Both the engines have identical thermal efficiencies. The temperature \(T\) (in K) of the intermediate reservoir is ........ (answer in integer). 
A bar of length \( L = 1 \, {m} \) is fixed at one end. Before heating its free end has a gap of \( \delta = 0.1 \, {mm} \) from a rigid wall as shown in the figure. Now the bar is heated resulting in a uniform temperature rise of \( 10^\circ {C} \). The coefficient of linear thermal expansion of the material is \( 20 \times 10^{-6} / \degree C \) and the Young’s modulus of elasticity is 100 GPa. Assume that the material properties do not change with temperature.
The magnitude of the resulting axial stress on the bar is .......... MPa (in integer). 
A massless cantilever beam, with a tip mass \( m \) of 10 kg, is modeled as an equivalent spring-mass system as shown in the figure. The beam is of length \( L = 1 \, {m} \), with a circular cross-section of diameter \( d = 20 \, {mm} \). The Young’s modulus of the beam material is 200 GPa.
The natural frequency of the spring-mass system is ............ Hz (rounded off to two decimal places).
A simply-supported beam has a circular cross-section with a diameter of 20 mm, area of 314.2 mm\(^2\), area moment of inertia of 7854 mm\(^4\), and a length \( L \) of 4 m. A point load \( P = 100 \, {N} \) acts at the center and an axial load \( Q = 20 \, {kN} \) acts through the centroidal axis as shown in the figure.
The magnitude of the offset between the neutral axis and the centroidal axis, at \( L/2 \) from the left, is ............ mm (rounded off to one decimal place).