List of practice Questions

The velocity vector of a 2D flow field is given by \(\vec{V} = 2y^2 \hat{i} + x^2t \hat{j}\).
The acceleration is
Note: There's a typo in the OCR of the question (î twice) and options. The solution assumes the standard form V = uî + vĵ.

Water is flowing with a free stream velocity of 0.25 m/s around a submerged flat plate of 2 m length (in the direction of flow) and 1 m width. The local shear stress at a distance \(x\) from the leading edge of the plate is given by \[ \tau = \frac{0.332\rho u^2}{\sqrt{Re_x}} \] where \(\rho = 1000\) kg/m³ is the density of the water, \(u\) is the free stream velocity and \(Re_x\) is the Reynolds number at \(x\). Assume that the flow is laminar, and the kinematic viscosity of water is \(10^{-6}\) m²/s. The drag force (in Newton) acting on one side of the plate lies between

A ship of length 125 m has a design speed of 25 knots (1 knot = 0.5144 m/s). A 5.0 m long geometrically similar model with wetted surface area of 4 m\(^2\) has a coefficient of residuary resistance of \(1.346 \times 10^{-3}\) at the corresponding speed. The ship's residuary resistance in kN (in sea water of density 1025 kg/m\(^3\)), and the model speed in knots (round off to the nearest integer) respectively are

A solid cantilever shaft of diameter 0.1 m and length 2 m is subjected to a torque of 10 kN-m at the free end (shear modulus is 82 GPa). The maximum induced shear stress is .................... N/mm\(^2\) (round off to the nearest integer).

The value of the surface integral \( \iint_S (x^2 \,dydz + y^2 \,dzdx + z^2 \,dxdy) \) over the surface of the cube given by \( 0 \le x \le 2, 0 \le y \le 2, 0 \le z \le 2 \), is

An oil tanker of breadth 20 m and having a displacement of 24000 tonnes in sea water (density of sea water = 1025 kg/m\(^3\)) is carrying oil of relative density 0.8 in 9 longitudinally distributed tanks which are all half-filled. Each longitudinal tank is 12 m long and 16 m wide. The apparent change in vertical center of gravity, due to the presence of oil in the tanks is .................... m (round off to one decimal place)

For a regular sinusoidal wave propagating in deep water having wave height of 3.5 m and wave period of 9 s, the wave steepness is .................... (round off to three decimal places)

A ship of length 180 m has a displacement of 14400 tonnes and is floating on an even keel in sea water of density 1025 kg/m\(^3\). The trim changes by 0.18 m when a weight of 120 tonnes that is already onboard, is shifted 24 m forward. The longitudinal metacentric height is .................... m.

A piezometer and a pitot tube measure the static and the total pressure of a fluid in a pipe flow respectively. The piezometer reads 100 kPa and the pitot tube shows 200 kPa. The density of the fluid is 1000 kg/m\(^3\). The velocity of the flow is .................... m/s (round off to one decimal place)

A Carnot heat engine operates between two reservoirs of temperatures 900 \(^{\circ}\)C (T\(_H\)) and 30 \(^{\circ}\)C (T\(_L\)). If the heat transferred during one cycle to the engine from T\(_H\) is 150 kJ, then the energy rejected to T\(_L\) is .................... kJ (round off to the nearest integer)

A stationary ship has longitudinal symmetry. The surge, sway and heave motions are represented by indices 1-2-3, respectively and roll, pitch and yaw motions are represented by indices 4-5-6, respectively. Which of the following are TRUE about the added mass (\(A_{ij}\))?

The failure modes that may be observed in a riveted joint to fasten two plate members, subjected to shear load are

Let X be any random variable and \( Y = -2X + 3 \). If E[Y] = 1 and E[Y\(^2\)] = 9, then which of the following are TRUE?

Consider the contour integral \( \oint_C \frac{dz}{z^4 + z^3 - 2z^2} \), along the curve \( |z| = 3 \) oriented in the counterclockwise direction. If Res[f, z\(_0\)] denotes the residue of f(z) at the point z\(_0\), then which of the following are TRUE?

A submerged cylinder of diameter 1 m is rotating clockwise at 100 rpm, in a flow with a free stream velocity of 10 m/s. Assuming ideal flow, the number of stagnation points on the cylinder is

From the following, choose the offshore platform that can be used ONLY for offshore drilling purpose.

Which method among the following is based on the strain energy principle?

In dimensional analysis, according to Buckingham's \(\pi\)-theorem, if n is the total number of variables and m is the number of independent dimensions, then the maximum number of independent dimensionless \(\pi\)-groups will be

In a marine steam power cycle employing regeneration, the feed water heater for waste heat recovery is placed after the

Comparing a catamaran (with a separation between demi-hulls) and a mono-hull craft of the same displacement and water plane area, the initial metacentric radius of the catamaran will be

The vapor quality of steam in the turbine of a Rankine cycle can be improved by employing

Consider the function \( z = \tan^{-1}\left(\frac{y}{x}\right) \), where \( x = u \sin v \) and \( y = u \cos v \). The partial derivative, \( \frac{\partial z}{\partial v} \) is

Consider the function \( z = x^3 - 2x^2y + xy^2 + 1 \). The directional derivative of z at the point (1, 2) along the direction \( 3\hat{i} + 4\hat{j} \) is

The coefficient of \(x^4\) in the polynomial \((x - 1)^3 (x - 2)^3\) is equal to .....................

Which one of the following shapes can be used to tile (completely cover by repeating) a flat plane, extending to infinity in all directions, without leaving any empty spaces in between them? The copies of the shape used to tile are identical and are not allowed to overlap.