Friday, May 15, 2015

H C Verma solutions, Introduction to Physics, EXERCISES (11-19), Chapter-1, Concepts of Physics, Part-I

Introduction to Physics

EXERCISES (Q-11 to Q-19)

11. Express the power of a 100-watt bulb in CGS units.              


ANSWER: (a) 1 watt = 1 joule/second,

                               = 1 N-m/s

But 1 N = 1 kg-m/s² =1000*100 g-cm/s²                              = 10⁵ dyne

Hence 1 watt = 10⁵*100 dyne-cm/s =10⁷ erg/s

So, 100 watt = 100*10⁷ erg/s           

                   = 10⁹ erg/s

 



12. The normal duration of I.Sc. physics practical period in Indian colleges is 100 minutes. Express this period in microcenturies. 1 microcentury = 10⁻⁶x100 years. How many microcenturies did you sleep yesterday?              


ANSWER: (a) 10⁻⁶x100 years = 1 microcentury

{1 year = 365*24*60 mimutes}

So, 10⁻⁶x100*365*24*60 minutes = 1 microcentury

→100 minutes = 10⁶/(365*24*60) microcenturies

                      = 1.90 microcenturies.      


I slept for 6 hours yesterday, i.e. =6*60 min =360 minutes = 360*1.9/100 micocenturies
           =6.84 microcenturies

 



13. The surface tension of water is 72 dyne/cm. Convert it into SI unit.              


ANSWER: (a) The surface tension of water =72 dyne/cm 

{But 1 dyne = 1 g-cm/s² =(1/1000*100) kg-m/s² =10⁻⁵ N}

=72*10⁻⁵*100 N/m =0.072 N/m      


 


14. The kinetic energy K of a rotating body depends on its moment of inertia I and its angular speed ⍵. Assuming the relation to be K = kIaωb where k is a dimensionless constant, find a and b. Moment of inertia of a sphere about its diameter is (2/5)Mr².   


                      


ANSWER: (a) The dimensions of the moment of inertia, I = [ML²], and of angular speed, ⍵ =[T⁻¹]

The dimensions of KE, K =[ML²T⁻²]

For the given relation to be dimensionally correct,

 [ML²T⁻²] = [ML²]a[T⁻¹]b = [MaL²aT⁻b]

Equating the dimensions on both sides,

a = 1
And, -b = -2
→b = 2

 



15. Theory of relativity reveals that mass can be converted into energy. The energy E so obtained is proportional to certain powers of mass m and the speed c of light. Guess a relation among the quantities using the method of dimensions.              


ANSWER: (a) E ∝ ma cb

→E = k ma cb   

Where k is a dimensionless constant. Writing the dimensions on both sides,

[ML²T⁻²] = [M]a[LT⁻¹]b =[MaLbT⁻b]

Equating the dimensions on both sides, we have,

a = 1 and b = 2
Hence the relation among the quantities may be,

E = kmc²



16. Let I = current through a conductor, R = its resistance and V = potential difference across its ends. According to Ohm's law, the product of two of these quantities equals the third. Obtain Ohm's law from dimensional analysis. Dimensional formulae for R and V are ML²I⁻²T⁻³ and ML²T⁻³I⁻¹ respectively.              


ANSWER: (a) The dimensions of the current, I = [I]

Since the product of two of these given quantities equals the third, we need to know whether the product of V and R gives [I] or their ratio. 

It is clear that the product will not give [I]. If we divide V by R we get,

[ML²T⁻³I⁻¹]/[ML²I⁻²T⁻³] = [I]

So we get the desired relation,

V/R = I

→V = IR

Hence, it is Ohm's law.    


 


17. The frequency of vibration of a string depends on the length L between the nodes, the tension F in the string, and its mass per unit length m. Guess the expression for its frequency from dimensional analysis.              


ANSWER: (a) From the given condition,

frequency, 𝜈 ∝   La Fb mc

→𝜈 = k   La Fb mc  

{Where k is a dimensionless constant}

Putting the dimensions on both sides,
[T⁻¹] =    [L]a [MLT⁻²]b [ML⁻¹]c
→[T⁻¹] =  [Mb+c La+b-c  T⁻²b ]
 Equating the powers of M, L and T on both sides,
b+c = 0, a+b-c = 0  and -2b = -1
It gives, b = 1/2, c = -1/2 and a+b-c = 0
→a+1/2-(-1/2) =0
→a + 1 =0
→a = -1
Now the expression for the frequency becomes,
𝜈 = k L⁻¹ √F*(1/√m)
→𝜈 = (k/L)√(F/m)



 

18. Test if the following equations are dimensionally correct:

(a) h = 2S Cosθ/ρrg

(b) v = √(P/ρ)

(c) V = πPr⁴t/8ηl

(d) 𝞶 = (1/2π)√(mgl/I) 

where h = height, S = surface tension, ρ = density, P = pressure, V = volume, η = coefficient of viscosity, 𝞶 = frequency and I = moment of inertia.           


ANSWER: (a) Left side:-

The dimension of h = [L]

Right side:-

Dimensions of S = Force/Length =[MLT⁻²]/[L] =[MT⁻²]

Cosθ is a ratio, hence dimensionless. 

The dimensions of ρ = [ML⁻³]   

The dimensions of r = [L]

The dimensions of g = [LT⁻²]

Hence the dimensions of 2S cosθ/ρrg 
=[MT⁻²]/[ML⁻³][L][LT⁻²]
=[MT⁻²]/[ML⁻¹T⁻²]
=[L]
Hence the dimensions of both sides are the same. So it is dimensionally correct.

(b) v = √(P/ρ)
Left side:-
Dimensions of velocity, v = [LT⁻¹]

Right side:-
Dimensions of pressure P = Force/area =[MLT⁻²]/[L²] =[ML⁻¹T⁻²]
Dimensions of ρ =[ML⁻³]
Hence, the dimensions of √(P/ρ)
=√[ML⁻¹T⁻²]/[ML⁻³]
=√[L²T⁻²] 
=[LT⁻¹]
Same dimensions on both sides. Hence, it is dimensionally correct.
    

(c) V = πPr⁴t/8ηl    
Left side:-
Dimensions of volume, V = [L³]
Right side:-
π is dimensionless. Dimensions of P =[ML⁻¹T⁻²]
Dimensions of r =[L]
Dimensions of t =[T]
Dimensions of l =[L]
Dimensions of η =Force/area =[ML⁻¹T⁻¹]   
Hence the dimensions of the right side
=[ML⁻¹T⁻²][L]⁴[T]/[ML⁻¹T⁻¹][L]
=[ML³T⁻¹]/[MT⁻¹]
=[L³]
Dimensions of both sides are the same. Hence, it is dimensionally correct.


(d) 𝞶 = (1/2π)√(mgl/I)
 Left side:-
Dimensions of frequency, 𝜈 = [T⁻¹]

Right side:-
π is dimensionless.
Dimensions of m =[M]
Dimensions of g =[LT⁻²]
Dimensions of l =[L]
Dimensions of I =[ML²]
Dimensions of the right side,
=√{[M][LT⁻²][L]}/√{[ML²]}
=√[T⁻²]
=[T⁻¹]
Dimensions of both sides are the same; hence it is dimensionally correct.
  So, all are dimensionally correct.




19. Let x and a stand for distance. Is ∫dx/√(a²-x²) = (1/a)*sin⁻¹(a/x) dimensionally correct?              


ANSWER: (a) Since both x and a are distances, their dimensions are = [L]. dx is an infinitesimally small distance; hence its dimension is also =[L]. Inverse sine is an angle; hence it is also dimensionless. So the dimensions of the left side,

      = [L]/√[L²] =[L]/[L] = L⁰ =Dimensionless

The dimensions of the right side = 1/[L] =[L⁻¹]

We see that the dimensions of both sides are not the same. Hence, the equation is not dimensionally correct.     

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Links to the Chapters





Links to Chapters:-

CHAPTER-1 - Introduction to Physics

Questions for Short Answers

OBJECTIVE - I 

OBJECTIVE - II

EXERCISES (1-10)

EXERCISES (11-19)

CHAPTER-2 - Physics, and Mathematics

Questions for Short Answers

Objective - I

Objective - II

Exercises (1 - 19)

Exercises (20 - 35)

CHAPTER-3 - Kinematics - Rest and Motion

Questions for Short Answers

Objective - I

Objective - II

Exercises - Q 1 to Q 10

Exercises - Q 11 to Q 20

Exercises - Q 21 to Q 30

Exercises - Q 31 to Q 40

Exercises - Q 41 to Q 52

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CHAPTER-4 - The Forces

Questions for Short Answers

Objective - I

Objective - II

Exercises

CHAPTER-5 - Newton's Laws of Motion

Questions for Short Answers

Objective - I

Objective - II

Exercises - Q 1 to Q 12

Exercises - Q 13 to Q 27

Exercises - Q 28 to Q 42


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CHAPTER-6 - Friction

Questions for Short Answers

Objective - I

Objective - II

Exercises - Q 1 to Q 10

Exercises - Q 11 to Q 20

Exercises - Q 21 to Q 31

CHAPTER-7 - Circular Motion

Questions for Short Answers

Objective - I

Objective - II

Exercises - Q 1 to Q 10

Exercises - Q 11 to Q 20

Exercises - Q 21 to Q 30

CHAPTER-8 - Work and Energy

Questions for Short Answers

Objective - I

Objective - II

Exercises - Q 1 to Q 10

Exercises - Q 11 to Q 20

Exercises - Q 21 to Q 30

Exercises - Q 31 to Q 42

Exercises - Q 43 to Q 54

Exercises - Q 55 to Q 64


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CHAPTER- 9 - Center of Mass, Linear Momentum, Collision

Questions for Short Answers

Objective - I

Objective - II

EXERCISES Q-1 TO Q-10

EXERCISES Q-11 TO Q-20

EXERCISES Q-21 TO Q-30

EXERCISES Q-31 TO Q-42

EXERCISES Q-43 TO Q-54

EXERCISES Q-55 TO Q-64

CHAPTER- 10 - Rotational Mechanics

Questions for Short Answers

OBJECTIVE - I

OBJECTIVE - II

EXERCISES Q-01 TO Q-15

EXERCISES Q-16 TO Q-30

EXERCISES Q-31 TO Q-45

EXERCISES Q-46 TO Q-60

EXERCISES Q-61 TO Q-75

EXERCISES Q-76 TO Q-86


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CHAPTER- 11 - Gravitation


Questions for Short Answers

OBJECTIVE - I

OBJECTIVE - II

EXERCISES Q-01 TO Q-10

EXERCISES Q-11 TO Q-20

EXERCISES Q-21 TO Q-30

EXERCISES Q-31 TO Q-39 (With Extra 40th problem)

CHAPTER- 12 - Simple Harmonic Motion 


Questions for Short Answers 


OBJECTIVE - I

OBJECTIVE-II

EXERCISES - Q-1 TO Q-10

EXERCISES - Q-11 TO Q-20

EXERCISES - Q-21 TO Q-30

EXERCISES - Q-31 TO Q-40

EXERCISES - Q-41 TO Q-50

EXERCISES - Q-51 TO Q-58 with EXTRA QUESTIONS Q-59 and Q-60 


CHAPTER- 13 - Fluid Mechanics 


Questions for Short Answers

OBJECTIVE-I

OBJECTIVE-II

EXERCISES - Q-1 TO Q-10

EXERCISES - Q-11 TO Q-20

EXERCISES - Q-21 TO Q-30

EXERCISES - Q-31 TO Q-35



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CHAPTER- 14 - Some Mechanical Properties of Matter 


Questions for Short Answers

OBJECTIVE-I

OBJECTIVE-II

EXERCISES - Q-1 TO Q-10

EXERCISES - Q -11 TO Q -20

EXERCISES - Q -21 TO Q -32

CHAPTER- 15 - Wave Motion and Waves on a String

Questions for Short Answers

OBJECTIVE-I

OBJECTIVE-II

EXERCISES - Q-1 TO Q-10

EXERCISES - Q-11 TO Q-20

EXERCISES - Q-21 TO Q-30

EXERCISES - Q-31 TO Q-40

EXERCISES - Q-41 TO Q-50

EXERCISES - Q-51 TO Q-57

CHAPTER- 16 - Sound Waves

Questions for Short Answers

OBJECTIVE-I

OBJECTIVE-II

EXERCISES - Q-1 TO Q-10

EXERCISES - Q-11 TO Q-20

EXERCISES - Q-21 TO Q-30

EXERCISES - Q-31 TO Q-40

EXERCISES - Q-41 TO Q-50

EXERCISES - Q-51 TO Q-60

EXERCISES - Q-61 TO Q-70 

EXERCISES - Q-71 TO Q-80

EXERCISES - Q-81 TO Q-89

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CHAPTER- 17 - Light Waves


CHAPTER- 18 - Geometrical Optics

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Part-II

Solutions - "Concepts of Physics" Part-II, by H C Verma

CHAPTER- 23 - Heat and Temperature

Questions for Short Answer

OBJECTIVE - I

OBJECTIVE - II

EXERCISES - Q-1 TO Q-10

EXERCISES - Q-11 TO Q-20

EXERCISES - Q-21 TO Q-34


CHAPTER- 24 - Kinetic Theory of Gases

Questions for Short Answer

OBJECTIVE - I

OBJECTIVE - II

EXERCISES - Q1 to Q10

EXERCISES - Q-11 to Q-20

EXERCISES - Q-21 to Q-30


EXERCISES - Q-31 to Q-40

EXERCISES - Q-41 to Q-50

EXERCISES - Q-51 to Q-62




CHAPTER- 25 - Calorimetry

Questions for Short Answer

OBJECTIVE - I

OBJECTIVE - II

EXERCISES - Q-1 to Q-10

EXERCISES - Q11 to Q-18 




CHAPTER- 26 - Laws of Thermodynamics

Questions for Short Answer

OBJECTIVE - I

OBJECTIVE - II

EXERCISES - Q-1 to Q-10

EXERCISES - Q-11 to Q-22




CHAPTER- 27 - Specific Heat Capacities of Gases

Questions for Short Answer

OBJECTIVE - I

OBJECTIVE - II

EXERCISES - Q-1 to Q-10 




CHAPTER- 28 - Heat Transfer


Questions for Short Answer

OBJECTIVE - I

OBJECTIVE - II

EXERCISES - Q-1 to Q-10











  

















    



























































































 





































































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