# Dimensional and Model Analysis MCQ Quiz in తెలుగు - Objective Question with Answer for Dimensional and Model Analysis - ముఫ్త్ [PDF] డౌన్‌లోడ్ కరెన్

Last updated on Feb 19, 2024

పొందండి Dimensional and Model Analysis సమాధానాలు మరియు వివరణాత్మక పరిష్కారాలతో బహుళ ఎంపిక ప్రశ్నలు (MCQ క్విజ్). వీటిని ఉచితంగా డౌన్‌లోడ్ చేసుకోండి Dimensional and Model Analysis MCQ క్విజ్ Pdf మరియు బ్యాంకింగ్, SSC, రైల్వే, UPSC, స్టేట్ PSC వంటి మీ రాబోయే పరీక్షల కోసం సిద్ధం చేయండి.

## Top Dimensional and Model Analysis MCQ Objective Questions

#### Dimensional and Model Analysis Question 1:

The number of π parameters needed to express the function F(A, V, t, μ, L) = 0 are

1. 5
2. 4
3. 3
4. 2

Option 4 : 2

#### Dimensional and Model Analysis Question 1 Detailed Solution

Buckingham's Pi Theorem:

Assume, a physical phenomenon is described by m number of independent variables like x1, x2, x3, ..., xm

The phenomenon may be expressed analytically by an implicit functional relationship of the controlling variables as:

f (x1, x2, x3, ……………, xm) = 0

Now if n be the number of fundamental dimensions like mass, length, time, temperature, etc., involved in these m variables, then according to Buckingham's Pi theorem -

The phenomenon can be described in terms of (m - n) independent dimensionless groups like π1, π2, ..., πm-n, where p terms, represent the dimensionless parameters and consist of different combinations of a number of dimensional variables out of them independent variables defining the problem.

Total no of π terms = m - n

Here, m = total parameter = 5

n = Fundamental dimensions (M, L, T) = 3

∴ no of π – terms = 5 – 3 = 2

#### Dimensional and Model Analysis Question 2:

Mach angle is denoted as

1. sin-1(M)
2. $${\sin ^{ - 1}}\left( {\frac{1}{{\rm{M}}}} \right)$$
3. $${\cos ^{ - 1}}\left( {\frac{1}{{\rm{M}}}} \right)$$
4. $${\tan ^{ - 1}}\left( {\frac{1}{{\rm{M}}}} \right)$$

Option 2 : $${\sin ^{ - 1}}\left( {\frac{1}{{\rm{M}}}} \right)$$

#### Dimensional and Model Analysis Question 2 Detailed Solution

Concepts:

Mach number

It is a dimensionless quantity and it is defined as the square root of the ratio of the inertia force of a flowing fluid to the elastic force.

It is related to the compressibility of fluid flow.

$$M = √(\frac{\text {Inertia Force}}{\text{Elastic Force}})$$

We know that

Inertia Force = ρ AV2 and Elastic Force = K A

Substituting these values, we Mach number as:

$$M = √(\frac{ρ A V^2}{KA}) = \frac{V}{√(\frac{K}{ρ})} = \frac{V}{C}$$

Where,

V is the velocity and C is the speed of sound in air

Mach angle

It is defined as half of the angle of the Mach cone. It is denoted as α  and it is given by

Sinα = C/V = 1/M

Or

α = Sin-1 (1/M)

#### Dimensional and Model Analysis Question 3:

The square root of ratio of inertia force of a flowing fluid to the surface tension force is

1. Froude's number
2. Euler's number
3. Weber's number
4. Mach's number

Option 3 : Weber's number

#### Dimensional and Model Analysis Question 3 Detailed Solution

Explanation:

Weber number

The Weber number is the ratio of dynamic pressure (i.e. inertia force) to the surface tension force.

$${W_e} = \sqrt {\frac{{inertia\;force}}{{surface\;tension}}} = \frac{V}{{\sqrt {\sigma /\rho L} }}$$

Other important dimensionless numbers are described in the table below:

 Reynold’s number $$R_e=\frac{{{inertia force}}}{{{viscous force}}} = \frac{{{\bf{\rho VL}}\;}}{{\bf{\mu }}}$$ Froude Number $${F_r} = \sqrt {\frac{{inertia\;force}}{{gravitation\;force}}} = \frac{V}{{\sqrt {{\bf{Lg}}} }}$$ Euler number $${E_u} = \sqrt {\frac{{inertia\;force}}{{pressure\;force}}} = \frac{V}{{\sqrt {P/\rho } }}$$ Mach Number $${M} = \sqrt {\frac{{inertia\;force}}{{Elastic\;force}}} = \frac{V}{{\sqrt {K/\rho } }}=~\frac VC$$

#### Dimensional and Model Analysis Question 4:

In compressible fluid flow problems at high velocities, which of the following non-dimensional number is used?

1. Euler number
2. Weber number
3. Mach number
4. Froude number

Option 3 : Mach number

#### Dimensional and Model Analysis Question 4 Detailed Solution

Explanation:

Mach number

Mach number is defined as the ratio of inertia force to elastic force.

$$M = \sqrt {\frac{{Inertia\;force}}{{Elastic\;force}}} = \sqrt {\frac{{\rho A{V^2}}}{{KA}}} = \sqrt {\frac{{{V^2}}}{{\frac{K}{\rho }}}} = \frac{V}{{\sqrt {\frac{K}{\rho }} }} = \frac{V}{C}\;\;\;\;\left\{ {\sqrt {\frac{K}{\rho }} = C = Velocity\;of\;sound} \right\}$$

$$M = \frac{{Velocity\;of\;body\;moving\;in\;fluid}}{{velocity\;of\;sound\;in\;fluid}}$$

For the compressible fluid flow, Mach number is an important dimensionless parameter. On the basis of the Mach number, the flow is defined.

 Mach Number Type of flow M < 0.8 Sub-sonic flow 0.8 < M <1.3 Trans-sonic M = 1 Sonic flow 1.3 < M < 5 Super-sonic flow M > 5 Hypersonic flow

 Number Definition Significance Reynolds No $$Re = \frac{{Inertia\;Force}}{{Viscous\;Force}} \Rightarrow \frac{{\rho VL}}{\mu }$$ Flow in closed conduit i.e. flow through pipes. Froude No $$Fr = \sqrt {\frac{{Inertia\;Force}}{{Gravity\;Force}}} \Rightarrow \frac{V}{{\sqrt {gL} }}$$ Where a free surface is present and gravity force is predominant. Spillway, Open Channels, waves in the ocean. Euler No. $${E_u} = \sqrt {\frac{{Inertia\;Force}}{{Pressure\;Force}}} \Rightarrow \frac{V}{{\sqrt {\frac{p}{\rho }} }}$$ In cavitation studies, where pressure force is predominant. Mach No. $$M = \sqrt {\frac{{Inertia\;Force}}{{Elastic\;Force}}} \Rightarrow \frac{V}{C}$$ Where fluid compressibility is important. Launching of rockets, airplanes and projectile moving at supersonic speed. Weber No. $${W_e} = \sqrt {\frac{{Inertia\;Force}}{{Surface\;tension\;Force}}} \Rightarrow \frac{V}{{\sqrt {\frac{\sigma }{{\rho L}}} }}$$ In Capillary studies i.e. where Surface tension is predominant.

#### Dimensional and Model Analysis Question 5:

A ship’s model of scale 1: 100 had a wave resistance of 1 N at its design speed. The corresponding wave resistance (in N) in prototype will be _____.

1. 100
2. 10000
3. 1000000
4. 1000

Option 3 : 1000000

#### Dimensional and Model Analysis Question 5 Detailed Solution

Concept:

From Froude’s law:

(Fr)m = (Fr)p

$${\left( {\frac{{{{\rm{U}}_{\rm{r}}}}}{{\sqrt {{{\rm{L}}_{\rm{g}}}} }}} \right)_{\rm{m}}} = {\left( {\frac{{\rm{V}}}{{\sqrt {{{\rm{L}}_{\rm{g}}}} }}} \right)_{\rm{p}}} \Rightarrow {{\rm{U}}_{\rm{r}}} = \sqrt {{{\rm{L}}_{\rm{r}}}} {\rm{\;or\;}}\frac{{{{\rm{V}}_{\rm{m}}}}}{{{{\rm{V}}_{\rm{p}}}}} = \sqrt {\frac{{{{\rm{L}}_{\rm{m}}}}}{{{{\rm{L}}_{\rm{p}}}}}}$$       ---(1)

Now:

$${{\rm{F}}_{\rm{r}}} = {\rm{\rho \;}}{{\rm{L}}_{\rm{r}}^2}{\rm{\;V}}_{\rm{r}}^2 = {\rm{\rho \;}}{{\rm{L}}_{\rm{r}}^2}{\rm{\;}}{\left( {\sqrt {{{\rm{L}}_{\rm{r}}}} } \right)^2} = {\rm{\rho \;L}}_{\rm{r}}^3$$ (Fr = Inertia for ratio)

$$\frac{{{{\rm{F}}_{\rm{m}}}}}{{{{\rm{F}}_{\rm{p}}}}} = {\left( {\frac{{{{\rm{L}}_{\rm{m}}}}}{{{{\rm{L}}_{\rm{p}}}}}} \right)^3}$$

Calculation:

Given,

Fm = 1 N

$$Scale = \frac{{{L_m}}}{{{L_p}}} = \frac{1}{{100}}$$

$$\frac{{{{\rm{F}}_{\rm{m}}}}}{{{{\rm{F}}_{\rm{p}}}}} = {\left( {\frac{{{{\rm{L}}_{\rm{m}}}}}{{{{\rm{L}}_{\rm{p}}}}}} \right)^3}$$

$$\frac{1}{{{{\rm{F}}_{\rm{p}}}}} = {\left( {\frac{1}{{100}}} \right)^3} \Rightarrow {{\rm{F}}_{\rm{p}}} = {10^6}{\rm{\;N}}$$

#### Dimensional and Model Analysis Question 6:

With respect to fluid flow, match the following in Column X with Column Y:

 Column X Column Y A. Viscosity 1. Mach number B. Gravity 2. Reynolds number C. Compressibility 3. Euler number D. Pressure 4. Froude number

1. A - 2, B - 4, C - 1, D - 3
2. A - 3, B - 4, C - 1, D - 2
3. A - 4, B - 2, C - 1, D - 3
4. A - 2, B - 4, C - 3, D - 1

Option 1 : A - 2, B - 4, C - 1, D - 3

#### Dimensional and Model Analysis Question 6 Detailed Solution

Concepts:

Mach number is dimensionless quantity  and it is defined as the ratio of flow velocity past a boundary to the local speed of sound. It is related to compressibility of fluid flow.

Reynolds Number is dimensionless quantity  and it is defined as ratio of inertia force to viscous force in fluid flow. Based on this flow can be considered as either laminar or turbulent.

Froude Number is dimensionless quantity  and it is defined as square root of ratio of inertia force to gravitational force. It governed the fluid flow in open channels.

Euler Number is dimensionless quantity  and it is defined as square root of ratio of inertia force to pressure force. It governs the fluid flow in pipes.

#### Dimensional and Model Analysis Question 7:

Which one of the dimensionless number identifies the compressibility effect of fluid.

1. Euler number
2. Froude number
3. Mach number
4. Weber number

Option 3 : Mach number

#### Dimensional and Model Analysis Question 7 Detailed Solution

Concept:

Mach number

Mach number is defined as the ratio of inertia force to elastic force.

$$M = \sqrt {\frac{{Inertia\;force}}{{Elastic\;force}}} = \sqrt {\frac{{\rho A{V^2}}}{{KA}}} = \sqrt {\frac{{{V^2}}}{{\frac{K}{\rho }}}} = \frac{V}{{\sqrt {\frac{K}{\rho }} }} = \frac{V}{C}\;\;\;\;\left\{ {\sqrt {\frac{K}{\rho }} = C = Velocity\;of\;sound} \right\}$$

$$M = \frac{{Velocity\;of\;body\;moving\;in\;fluid}}{{velocity\;of\;sound\;in\;fluid}}$$

If Mach number is less than 0.3, fluids are said to be incompressible.

For the compressible fluid flow, Mach number is an important dimensionless parameter. On the basis of the Mach number, the flow is defined.

 Mach Number Type of flow M < 0.8 Sub-sonic flow 0.8 < M <1.3 Trans-sonic M = 1 Sonic flow 1.3 < M < 5 Super-sonic flow M > 5 Hypersonic flow

Other important dimensionless numbers are described in the table below

 Reynold’s number $$R_e=\frac{{{inertia force}}}{{{viscous force}}} = \frac{{{\bf{\rho VL}}\;}}{{\bf{\mu }}}$$ Froude Number $${F_r} = \sqrt {\frac{{inertia\;force}}{{gravitation\;force}}} = \frac{V}{{\sqrt {{\bf{Lg}}} }}$$ Euler number $${E_u} = \sqrt {\frac{{inertia\;force}}{{pressure\;force}}} = \frac{V}{{\sqrt {P/\rho } }}$$ Weber Number $${W_e} = \sqrt {\frac{{inertia\;force}}{{surface\;tension}}} = \frac{V}{{\sqrt {\sigma /\rho L} }}$$

#### Dimensional and Model Analysis Question 8:

A tidal model has been made with horizontal scale ratio of 1 ∶ 7000 and vertical scale ratio of 1 ∶ 350. What is the period in which a tide of its natural period 8 hour can be simulated in the model?

1. 4.4 minutes
2. 82.3 seconds
3. 77 seconds
4. 160 seconds

Option 3 : 77 seconds

#### Dimensional and Model Analysis Question 8 Detailed Solution

9m/sec)Concept:

Tidal Model:

In tidal model the most dominating factors of all the forces is gravity force and hence Froude's law will be applicable in such cases.

As per this law, Froude's number of models should be equal to Froude's number of prototypes i.e.  (Fr)m = (Fr)p

⇒ $$V_m\over \sqrt( gy_m)$$ = $$V_p\over \sqrt gy_p$$

$$V_m \over V_p$$ = $$\sqrt{ y_m \over { y_p}}$$

⇒ Vr = $$\sqrt L_r$$v

Thus, Time ratio (Tm)

$${L_m \over T_m} \over {L_p\over T_p}$$$$\sqrt L_{rv}$$

⇒ $${L_m\over L_p } × {T_p\over T_m} = {\sqrt L_{rv}}$$

where

Vm = velocity of flow in model (m/sec)

Vp = velocity of flow in prototype (m/sec)

ym = depth of flow in model(m)

yp = depth of flow in prototype(m)

g= acceleration due to gravity(m/sec2)

Tm = Time period in model

Tp = Time period in prototype

Lm = length of model (m)

Lp = length pf prototype(m)

Calculation:

Given:

$$L_m\over L_p$$ = $$1\over 7000$$  = LrH

LrV = $$1\over 350$$

Tp = 8 hrs = 8×60×60 secs

$${L_m\over L_p } * {T_p\over T_m} = {\sqrt L_{rv}}$$   putting values

⇒ $$1 \over 7000$$ × $$8×60×60 \over T_m$$ =  $$\sqrt {1\over 350}$$

Tm = 77 secs

#### Dimensional and Model Analysis Question 9:

Mach angle α and Mach number M are related as

1. $$M = {\sin ^{ - 1}}\left( {\frac{1}{\alpha }} \right)$$
2. $$\alpha = {\cos ^{ - 1}}\left( {\sqrt {\frac{{{M^2} - 1}}{{{M^2}}}} } \right)$$
3. $$\tan \alpha \left( {\sqrt {{M^2} - 1} } \right)$$
4. $$\alpha = cose{c^{ - 1}}\left( {\frac{1}{M}} \right)$$

Option 2 : $$\alpha = {\cos ^{ - 1}}\left( {\sqrt {\frac{{{M^2} - 1}}{{{M^2}}}} } \right)$$

#### Dimensional and Model Analysis Question 9 Detailed Solution

Explanation:
Mach number:

Mach number has been defined as the square root of the ratio of the inertia force of a flowing fluid to the elastic force.

Mach Number (M) = $$\sqrt{\frac{Inertia\;Force}{Elastic\;Force}}=\sqrt{\frac{\rho AV^2}{KA}}=\frac{V}{\sqrt{K/\rho}}=\frac{V}{C}$$

Mach Angle:

It is defined as the half of the angle of the Mach cone. It is given by

$$\sinα=\frac{C}{V}=\frac{1}{M}$$

cos2α = 1 - sin2α

$$\cos^2\alpha=1-\frac{1}{M^2}$$

$$\cos^2\alpha=\frac{M^2-1}{M^2}$$

$$\cos\alpha=\sqrt{\frac{M^2-1}{M^2}}$$

$$\alpha = {\cos ^{ - 1}}\left( {\sqrt {\frac{{{M^2} - 1}}{{{M^2}}}} } \right)$$

Flow can be classified on the basis of Mach number:

 Flow Mach number Subsonic M < 1 Sonic M = 1 Supersonic 1 < M < 3 Hypersonic M > 3

#### Dimensional and Model Analysis Question 10:

Specific speed indicates:

1. type of pump
2. rated rpm
3. RPM at maximum efficiency point

Option 2 : rated rpm

#### Dimensional and Model Analysis Question 10 Detailed Solution

Explanation:

Specific speed (NS) is the rated speed at some specified operating conditions.

For Pump:

Specific speed is defined as the speed of a geometrically similar pump that would deliver unit discharge in m3/s against a unit head in m.

$${N_S} = \frac{{N\sqrt Q }}{{{H^{\frac{3}{4}}}}}$$

For Turbine:

Specific speed is defined as the speed of a geometrical similar turbine that would deliver unit power in kW against a unit head in m.

$${N_S} = \frac{{N\sqrt P }}{{{H^{\frac{5}{4}}}}}$$

Important Points

Specific speed and shape number are two different parameters. A specific speed is a dimensional parameter whereas a shape number is a dimensionless parameter.

Shape number $$S = \frac{{N\sqrt {P/\rho } }}{{\left( {g{H^{}}}^{} \right)}{{5\over4}}}$$.
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