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Question 1
ప్రశ్న 1
What are the dimensions of pressure p in the M-L-T system?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Pressure is force divided by area. Force has dimensions [M L T^-2]. Area has dimensions [L^2]. Therefore pressure has dimensions [M L^-1 T^-2]. Hence Option B is correct.
వివరణ:
Question 2
ప్రశ్న 2
Which pair correctly gives the dimensions of dynamic viscosity mu and kinematic viscosity nu?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Dynamic viscosity follows from shear stress equal to viscosity times velocity gradient. Its dimensions are [M L^-1 T^-1]. Kinematic viscosity is defined as nu = mu/rho. Dividing by density [M L^-3] gives [L^2 T^-1]. Therefore Option C is correct.
వివరణ:
Question 3
ప్రశ్న 3
What does the principle of dimensional homogeneity require for a physically meaningful equation?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A valid physical equation must remain meaningful regardless of the consistent system of units used. Quantities that are added or equated must therefore possess the same dimensional representation. For example, terms representing lengths can be added to other lengths but not directly to velocities. Dimensional homogeneity is consequently a basic check on proposed physical equations. Therefore Option A is correct.
వివరణ:
Question 4
ప్రశ్న 4
Which statement about dimensional homogeneity is most accurate?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A dimensionally inconsistent physical equation is certainly defective. However, many different mathematically incorrect relationships can still be dimensionally homogeneous. Dimensional analysis generally cannot determine pure numerical constants such as 2 or pi. It also cannot by itself establish the exact functional physics of a phenomenon. Therefore Option D is correct.
వివరణ:
Question 5
ప్రశ్న 5
Which procedure most accurately describes Rayleigh's method of dimensional analysis?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Rayleigh's method begins by identifying the variables believed to control the dependent quantity. The dependent variable is then expressed as a product of powers of those independent variables. The exponents are initially unknown. Dimensions are substituted and the powers of the fundamental dimensions are equated. Solving those equations gives the dimensional exponent relationships, so Option B is correct.
వివరణ:
Question 6
ప్రశ్న 6
Suppose a force F is assumed by Rayleigh's method to depend only on fluid density rho, velocity V and characteristic length D as F = K rho^a V^b D^c. What are a, b and c?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Force has dimensions [M L T^-2]. The product rho^a V^b D^c has dimensions [M^a L^(-3a+b+c) T^-b]. Equating mass exponents gives a = 1 and equating time exponents gives b = 2. The length equation then gives -3 + 2 + c = 1, so c = 2. Therefore F is proportional to rho V^2 D^2 and Option C is correct.
వివరణ:
Question 7
ప్రశ్న 7
If a characteristic pressure p is assumed to depend only on fluid density rho and velocity V according to p = K rho^a V^b, what exponents are obtained by Rayleigh's method?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Pressure has dimensions [M L^-1 T^-2]. The assumed product has dimensions [M^a L^(-3a+b) T^-b]. Matching the mass exponent gives a = 1. Matching time gives b = 2, and the resulting length exponent is -3 + 2 = -1 as required. Therefore p is proportional to rho V^2 and Option A is correct.
వివరణ:
Question 8
ప్రశ్న 8
The celerity C of a gravity wave is assumed to depend only on gravitational acceleration g and wavelength lambda as C = K g^a lambda^b. What values of a and b satisfy dimensional homogeneity?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Wave celerity has dimensions [L T^-1]. The right side has dimensions [L^(a+b) T^(-2a)]. Equating time exponents gives -2a = -1, so a = 1/2. The length equation then requires a + b = 1, giving b = 1/2. Therefore C is proportional to the square root of g lambda and Option D is correct.
వివరణ:
Question 9
ప్రశ్న 9
Power P delivered by a geometrically specified rotating hydraulic machine is assumed to depend only on fluid density rho, rotational speed N and diameter D as P = K rho^a N^b D^c, with [N] = [T^-1]. What are a, b and c?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Power has dimensions [M L^2 T^-3]. The assumed product has dimensions [M^a L^(-3a+c) T^-b]. Matching mass gives a = 1 and matching time gives b = 3. The length equation is -3 + c = 2, so c = 5. Thus P is proportional to rho N^3 D^5 and Option B is correct.
వివరణ:
Question 10
ప్రశ్న 10
If discharge Q from a geometrically similar rotating machine is assumed by Rayleigh's method to depend only on rotational speed N and diameter D as Q = K N^a D^b, what are a and b?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Discharge has dimensions [L^3 T^-1]. Rotational speed N has dimensions [T^-1], while diameter has dimensions [L]. Matching the time exponent gives a = 1. Matching the length exponent then gives b = 3. Therefore Q is proportional to ND^3 and Option C is correct.
వివరణ:
Question 11
ప్రశ్న 11
Rayleigh's method is applied to a problem containing more unknown dimensional exponents than can be uniquely determined from the available independent dimensional equations. What is the correct conclusion?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Each independent fundamental dimension supplies one independent dimensional equation. If there are more unknown exponents than independent equations, the algebraic system is underdetermined. One or more exponents must then be expressed in terms of free parameters. Dimensional analysis cannot supply the missing functional information or empirical constants. Therefore Option A is correct.
వివరణ:
Question 12
ప్రశ్న 12
According to the Buckingham Pi theorem, if a physical problem contains n dimensional variables and the dimensional matrix has rank r, how many independent dimensionless Pi groups are required?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Buckingham's theorem reduces a dimensional relationship to a smaller set of independent dimensionless products. The number of such groups equals the number of variables minus the number of dimensionally independent reference dimensions. More generally, r is the rank of the dimensional matrix. Thus the required number is n - r. Therefore Option D is correct.
వివరణ:
Question 13
ప్రశ్న 13
A fluid-mechanics phenomenon is described by seven relevant dimensional variables involving the three independent fundamental dimensions M, L and T. How many independent Pi terms are expected?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The number of variables is n = 7. The three independent fundamental dimensions give r = 3. Buckingham's theorem therefore predicts n - r = 7 - 3 dimensionless groups. This gives four independent Pi terms. Hence Option B is correct.
వివరణ:
Question 14
ప్రశ్న 14
Which requirement is appropriate when selecting repeating variables in the Buckingham Pi method?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The repeating variables provide the dimensional basis used to nondimensionalize the remaining variables. They must therefore collectively represent all independent fundamental dimensions appearing in the problem. They must also be dimensionally independent of one another. A dimensionless product should not be obtainable solely from the repeating set. Therefore Option C is correct.
వివరణ:
Question 15
ప్రశ్న 15
For a problem involving rho, mu, nu, V and L, why is the set rho, mu and nu unsuitable as three independent repeating variables?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Dynamic viscosity, density and kinematic viscosity are related by mu = rho nu. Consequently the product mu/(rho nu) is already dimensionless and equals one by definition. The three quantities therefore do not form a dimensionally independent repeating set. Selecting them would make the exponent equations dependent. Hence Option A is correct.
వివరణ:
Question 16
ప్రశ్న 16
Which statement regarding the uniqueness of Pi terms is correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Buckingham's theorem fixes the required number of independent dimensionless groups but not a unique algebraic form for them. Different acceptable choices of repeating variables can yield different-looking Pi groups. Products, quotients and powers of an existing independent set can often generate another equivalent set. The new groups must still be dimensionless and collectively independent. Therefore Option D is correct.
వివరణ:
Question 17
ప్రశ్న 17
The drag force F on a smooth sphere is taken to depend on density rho, velocity V, sphere diameter D and dynamic viscosity mu. Using M, L and T, which pair is a valid set of dimensionless groups?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Five variables described by three independent dimensions produce two Pi groups. The quantity F/(rho V^2 D^2) is dimensionless because rho V^2 D^2 has the dimensions of force. The quantity rho VD/mu is also dimensionless and is the Reynolds number based on D. These two independent groups can describe the drag problem. Therefore Option C is correct.
వివరణ:
Question 18
ప్రశ్న 18
Let G = Delta p/L be the pressure drop per unit pipe length. If G depends on pipe diameter D, density rho, dynamic viscosity mu and mean velocity V, which pair forms a valid set of Pi groups?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Pressure gradient G has dimensions [M L^-2 T^-2]. Multiplying G by D gives dimensions of pressure, which match those of rho V^2. Thus GD/(rho V^2) is dimensionless. The second group rho VD/mu is the Reynolds number and is also dimensionless. Therefore Option A is correct.
వివరణ:
Question 19
ప్రశ్న 19
Why is the Reynolds number Re = rho V L/mu dimensionless?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Density has dimensions [M L^-3]. Multiplying by velocity [L T^-1] and length [L] gives [M L^-1 T^-1]. Dynamic viscosity mu has exactly the same dimensions. Their ratio therefore has dimensions M^0 L^0 T^0. Hence Reynolds number is dimensionless and Option B is correct.
వివరణ:
Question 20
ప్రశ్న 20
Which statement verifies that the Froude number Fr = V/sqrt(gL) is dimensionless?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Gravitational acceleration has dimensions [L T^-2]. Multiplication by characteristic length L gives [L^2 T^-2]. Taking the square root produces [L T^-1]. These are exactly the dimensions of velocity V. Consequently their ratio is dimensionless and Option C is correct.
వివరణ:
Question 21
ప్రశ్న 21
Which of the following expressions is NOT dimensionless?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The first expression is Reynolds number and is dimensionless. The second is Froude number and is dimensionless. Since surface tension has dimensions [M T^-2], rho V^2 L/sigma is also dimensionless and corresponds to a Weber-number form. However, mu/(rho V) has dimensions of length. Therefore Option D is correct.
వివరణ:
Question 22
ప్రశ్న 22
Water has rho = 1000 kg/m³ and dynamic viscosity mu = 1.0 × 10^-3 Pa·s. For V = 2.0 m/s and L = 0.10 m, what is the Reynolds number?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Reynolds number is Re = rho V L/mu. The numerator is 1000 × 2.0 × 0.10 = 200 in SI units. Dividing by 1.0 × 10^-3 gives 200000. Thus Re = 2.0 × 10^5. Therefore Option B is correct.
వివరణ:
Question 23
ప్రశ్న 23
For a drag problem, F = 100 N, rho = 1000 kg/m³, V = 2.0 m/s and D = 0.10 m. What is the value of the dimensionless group Pi = F/(rho V^2 D^2)?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The denominator is rho V^2 D^2. Numerically it is 1000 × 4 × 0.01 = 40 N. The dimensionless group is therefore 100/40. This gives Pi = 2.50. Hence Option C is correct.
వివరణ:
Question 24
ప్రశ్న 24
A pipe-flow experiment has pressure gradient G = Delta p/L = 800 Pa/m, D = 0.050 m, rho = 1000 kg/m³ and V = 2.0 m/s. What is Pi = GD/(rho V^2)?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The numerator GD is 800 × 0.050 = 40 Pa. The denominator rho V^2 is 1000 × 4 = 4000 Pa. Their ratio is 40/4000. Hence Pi equals 0.0100. Therefore Option A is correct.
వివరణ:
Question 25
ప్రశ్న 25
For a geometrically similar rotating hydraulic machine, the flow coefficient is Pi_Q = Q/(ND^3). If Q = 0.050 m³/s, N = 10 s^-1 and D = 0.20 m, what is Pi_Q?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The cube of diameter is D^3 = 0.20^3 = 0.008 m^3. Thus ND^3 = 10 × 0.008 = 0.080 m^3/s. The flow coefficient is 0.050/0.080. This evaluates to 0.625. Therefore Option D is correct.
వివరణ:
Answer Key సమాధానాల పట్టిక
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Question 1 ప్రశ్న 1Answer: B. [M L^-1 T^-2] సమాధానం: B.
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Question 2 ప్రశ్న 2Answer: C. mu: [M L^-1 T^-1], nu: [L^2 T^-1] సమాధానం: C.
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Question 3 ప్రశ్న 3Answer: A. All additive terms in the equation must have the same dimensions సమాధానం: A.
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Question 4 ప్రశ్న 4Answer: D. Dimensional homogeneity is necessary for a physical equation but is not by itself sufficient to prove that the equation is physically correct సమాధానం: D.
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Question 5 ప్రశ్న 5Answer: B. Express the dependent variable as a product of powers of the governing independent variables and determine exponents by dimensional homogeneity సమాధానం: B.
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Question 6 ప్రశ్న 6Answer: C. a = 1, b = 2, c = 2 సమాధానం: C.
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Question 7 ప్రశ్న 7Answer: A. a = 1, b = 2 సమాధానం: A.
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Question 8 ప్రశ్న 8Answer: D. a = 1/2, b = 1/2 సమాధానం: D.
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Question 9 ప్రశ్న 9Answer: B. a = 1, b = 3, c = 5 సమాధానం: B.
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Question 10 ప్రశ్న 10Answer: C. a = 1, b = 3 సమాధానం: C.
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Question 11 ప్రశ్న 11Answer: A. Dimensional analysis alone cannot determine a unique power-law dependence; one or more exponents or functional relationships remain undetermined సమాధానం: A.
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Question 12 ప్రశ్న 12Answer: D. n - r సమాధానం: D.
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Question 13 ప్రశ్న 13Answer: B. 4 సమాధానం: B.
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Question 14 ప్రశ్న 14Answer: C. The repeating variables should be dimensionally independent and collectively contain all fundamental dimensions involved సమాధానం: C.
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Question 15 ప్రశ్న 15Answer: A. Because mu = rho nu, so the three are dimensionally dependent సమాధానం: A.
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Question 16 ప్రశ్న 16Answer: D. Different algebraically equivalent sets of independent dimensionless Pi terms may represent the same physical problem సమాధానం: D.
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Question 17 ప్రశ్న 17Answer: C. F/(rho V^2 D^2) and rho VD/mu సమాధానం: C.
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Question 18 ప్రశ్న 18Answer: A. GD/(rho V^2) and rho VD/mu సమాధానం: A.
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Question 19 ప్రశ్న 19Answer: B. Because rho V L and mu both have dimensions [M L^-1 T^-1] సమాధానం: B.
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Question 20 ప్రశ్న 20Answer: C. sqrt(gL) has the dimensions of velocity [L T^-1] సమాధానం: C.
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Question 21 ప్రశ్న 21Answer: D. mu/(rho V) సమాధానం: D.
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Question 22 ప్రశ్న 22Answer: B. 2.0 × 10^5 సమాధానం: B.
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Question 23 ప్రశ్న 23Answer: C. 2.50 సమాధానం: C.
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Question 24 ప్రశ్న 24Answer: A. 0.0100 సమాధానం: A.
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Question 25 ప్రశ్న 25Answer: D. 0.625 సమాధానం: D.