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 unit.              


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 in 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 taking k as 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 the 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 it's 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 bothe 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 right side
=[ML⁻¹T⁻²][L]⁴[T]/[ML⁻¹T⁻¹][L]
=[ML³T⁻¹]/[MT⁻¹]
=[L³]
Dimensions of both sides are 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 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



CHAPTER- 20 - Dispersion and Spectra


CHAPTER- 19 - Optical Instruments

CHAPTER- 18 - Geometrical Optics



CHAPTER- 17 - Light Waves




CHAPTER- 14 - Fluid Mechanics



CHAPTER- 13 - Fluid Mechanics


CHAPTER- 12 - Simple Harmonic Motion








CHAPTER- 11 - Gravitation




CHAPTER- 10 - Rotational Mechanics




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


CHAPTER- 8 - Work and Energy

Click here for → Question for Short Answers

Click here for → OBJECTIVE-I

Click here for → OBJECTIVE-II

Click here for → Exercises (1-10)

Click here for → Exercises (11-20)

Click here for → Exercises (21-30)

Click here for → Exercises (31-42)

Click here for → Exercise(43-54)

CHAPTER- 7 - Circular Motion

Click here for → Questions for Short Answer 

Click here for → OBJECTIVE-I

Click here for → OBJECTIVE-II

Click here for → EXERCISES (1-10)

Click here for → EXERCISES (11-20)

Click here for → EXERCISES (21-30)

CHAPTER- 6 - Friction

Click here for → Questions for Short Answer

Click here for → OBJECTIVE-I

Click here for → Friction - OBJECTIVE-II

Click here for → EXERCISES (1-10)

Click here for → Exercises (11-20)

Click here for → EXERCISES (21-31)

For more practice on problems on friction solve these- "New Questions on Friction".

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CHAPTER- 5 - Newton's Laws of Motion


Click here for → QUESTIONS FOR SHORT ANSWER

Click here for→Newton's Laws of Motion,Exercises(Q.No. 13 to 27)

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CHAPTER- 4 - The Forces

The Forces-

"Questions for short Answers"    


Click here for "The Forces" - OBJECTIVE-I


Click here for "The Forces" - OBJECTIVE-II


Click here for "The Forces" - Exercises


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CHAPTER- 3 - Kinematics - Rest and Motion

Click here for "Questions for short Answers"


Click here for "OBJECTIVE-I"


Click here for EXERCISES (Question number 1 to 10)


Click here for EXERCISES (Question number 11 to 20)


Click here for EXERCISES (Question number 21 to 30)


Click here for EXERCISES (Question number 31 to 40)


Click here for EXERCISES (Question number 41 to 52)


CHAPTER- 2 - "Physics and Mathematics"

Click here for "Questions for Short Answers"


Click here for "OBJECTIVE-II"

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