The figure shows three glasses P, Q and R with water and floating ice cube. Glass P has a solid ice cube, glass Q has an ice cube with a small solid steel ball embedded in it and glass R has an ice cube with an air bubble. After the ice cube melts, the level of water in glasses P, Q and R, respectively:
Step 1: Glass P — pure ice cube.
A floating ice cube displaces water equal to its weight. When it melts, it simply becomes the same mass of water. Thus the water level remains the same.
Step 2: Glass Q — ice cube with steel ball.
The steel ball is denser than water. While the ice floats, part of the ball's weight is supported by the ice, causing extra water displacement. After melting, the steel ball sinks and displaces only its own volume, which is less water than before. Hence the water level decreases.
Step 3: Glass R — ice cube with air bubble.
The air bubble makes the ice less dense overall. While floating, the ice displaces water equal to its weight, which is more volume than the water produced after melting (because melting releases water but the bubble disappears). Hence the water level decreases.
Final Answer: (C) remains same, decreases, and decreases
A flexible chain of mass $m$ is hanging as shown. Find tension at the lowest point. 

Two designs A and B, shown in the figure, are proposed for a thin-walled closed section that is expected to carry only torque. Both A and B have a semi-circular nose, and are made of the same material with a wall thickness of 1 mm. With strength as the only criterion for failure, the ratio of maximum torque that B can support to the maximum torque that A can support is _________ (rounded off to two decimal places).
A thin flat plate is subjected to the following stresses: \[ \sigma_{xx} = 160 \, {MPa}; \, \sigma_{yy} = 40 \, {MPa}; \, \tau_{xy} = 80 \, {MPa}. \] Factor of safety is defined as the ratio of the yield stress to the applied stress. The yield stress of the material under uniaxial tensile load is 250 MPa. The factor of safety for the plate assuming that material failure is governed by the von Mises criterion is _________ (rounded off to two decimal places).
A prismatic vertical column of cross-section \( a \times 0.5a \) and length \( l \) is rigidly fixed at the bottom and free at the top. A compressive force \( P \) is applied along the centroidal axis at the top surface. The Young’s modulus of the material is 200 GPa and the uniaxial yield stress is 400 MPa. If the critical value of \( P \) for yielding and for buckling of the column are equal, the value of \( \frac{l}{a} \) is __________ (rounded off to one decimal place).
A uniform rigid bar of mass 3 kg is hinged at point F, and supported by a spring of stiffness \( k = 100 \, {N/m} \), as shown in the figure. The natural frequency of free vibration of the system is ___________ rad/s (answer in integer).
A jet-powered airplane is steadily climbing at a rate of 10 m/s. The air density is 0.8 kg/m³, and the thrust force is aligned with the flight path. Using the information provided in the table below, the airplane’s thrust to weight ratio is ___________ (rounded off to one decimal place). 