Thursday, April 30, 2015

H C Verma solutions, Introduction to Physics, EXERCISES (1-10), Chapter-1, Concepts of Physics, Part-I

Introduction to Physics

EXERCISES (Q-1 to Q-10)

1. Find the dimensions of

(a) linear momentum,

(b) frequency and

(c) pressure.        


ANSWER: (a) Linear momentum = mass x velocity

= mass x distance/time
→p = m*l/t

The dimension of mass = [M]

The dimension of velocity = [L/T] = [LT⁻¹] 

Hence the dimensions of linear momentum,

[p] = [M]*[LT⁻¹] = [MLT⁻¹]   


(b) Frequency = 1/Time period 

→f = 1/T

The dimension of frequency [f] = [1/T] = [T⁻¹] 


(c) Pressure = Force per unit area
→ P = Force/Area = Mass*acceleration/Area
The dimension of mass = [M]
The dimension of acceleration = [LT⁻²]
The dimension of area = [L²]
Hence the dimension of pressure,
[P] = [M]*[LT⁻²]/[L²] =[ML⁻¹T⁻²]

             


 

2. Find the dimensions of

(a) angular speed ⍵,

(b) angular acceleration α

(c) Torque Г and

(d) moment of inertia I.

Some of the equations involving these quantities are

⍵ = (θ₂-θ₁)/(t₂-t₁), α = (⍵₂-⍵₁)/(t₂-t₁),

Г = F*r and I = mr².

The symbols have standard meanings.       


ANSWER: (a) Angular speed ⍵ = (θ₂ - θ₁)/(t₂ - t₁)

The dimension of [θ₂ - θ₁] =Dimension of angle which is dimensionless 

The dimension of [t₂ - t₁] = Dimension of time =[T]

Hence [⍵] = 1/[T] = [T⁻¹]   


(b) Angular acceleration α = (⍵₂ - ⍵₁)/(t₂ - t₁)           The dimension of (⍵₂ - ⍵₁) = The dimension of angular speed = [T⁻¹] 

The dimension of (t₂ - t₁) = The dimension of time =[T]

Hence the dimension of angular acceleration,

[α] = [T⁻¹]/[T] = [T⁻²]


(c) Torque = Force *Arm (distance),

i.e. Г = F*r

The dimension of force [F] = mass*acceleration =[M]*[LT⁻²] =[MLT⁻²]
The dimension of arm, [r] = [L]
Hence the dimension of torque,
[Г] = [MLT⁻²]*[L] =[ML²T⁻²]

(d) The moment of inertia, I =mr²
The dimension of mass, [m] = [M]
The dimension of radius/distance, [r] = [L]
Hence the dimension of the moment of inertia,
[I] = [M]*[L]² =[ML²]


 


3. Find the dimensions of 

(a) electric field E,

(b) magnetic field B and 

(c) magnetic permeability µ₀.

The relevant equations are

F =qE, F = qvB, and B = µ₀I/2πa

where F is a force, q is a charge, v is speed, I is current, and a is distance.       


ANSWER: (a) F = qE, hence

Electric field E = F/q

The dimension of force [F] = [MLT⁻²] 

The dimension of charge, [q] = [TI] 

{Since charge = current*time}

Hence the dimension of electric field,

[E] = [MLT⁻²]/[TI] = [MLT⁻³I⁻¹]


(b) Since F = qvB,

Magnetic field, B = F/qv

Now, [F] = [MLT⁻²]

[q] = [TI] 

Speed, [v] = [LT⁻¹]

Hence the dimension of magnetic field,

[B] = [MLT⁻²]/[TI]*[LT⁻¹] = [MT⁻²I⁻¹]


(c) Given, B = µ₀I/2πa

→µ₀ =2πaB/I

The dimension of distance, [a] =[L]

The dimension of the magnetic field, [B] =[MT⁻²I⁻¹]

The dimension of current, [I] = [I], and π is dimensionless. Hence the dimension of magnetic permeability,

[µ₀] = [L][MT⁻²I⁻¹]/[I] =[MLT⁻²I⁻²]            



 

4. Find the dimensions of 

(a) electric dipole moment p and

(b) magnetic dipole moment M.

The defining equations are p = q.d, and M = IA

where d is distance, A is area, q is a charge and I is current.         


ANSWER: (a) The electric dipole moment, p = q*d

The dimensions of charge, [q] =[TI]

The dimensions of distance, [d] =[L]   

Hence the dimensions of electric dipole moment,

[p] = [TI][L] = [LTI] 


(b) The magnetic dipole moment, M = IA

The dimension of the area, [A] =[L²] 

Hence the dimensions of the magnetic dipole moment, [M] = [I][L²] =[L²I]             



 

5. Find the dimensions of Planck's constant h from the equation E = h𝞶 where E is the energy and 𝞶 is the frequency.        


ANSWER: Since E = h𝞶 

→h = E/𝞶  

Energy = Force*distance {same as work done}

→[E] = [MLT⁻²][L] =[ML²T⁻²]

The dimensions of frequency, [𝞶] = [T⁻¹] 

Hence the dimensions of Planck's constant,

[h] = [ML²T⁻²]/[T⁻¹] = [ML²T⁻¹]             




 

6. Find the dimensions of 

(a) specific heat capacity c,

(b) the coefficient of linear expansion α and

(c) the gas constant R. 

Some of the equations involving these quantities are 

Q =mc(T₂-T₁), Lₜ = L₀[1+α(T₂-T₁)] and PV = nRT.        


ANSWER: (a) Since Q = mc(T₂ - T₁)

→c = Q/m(T₂ - T₁)
The dimension of (T₂-T₁) = The dimension of temperature = [K]  
The dimension of mass, [m] =[M]
The dimensions of heat, Q = the dimension of energy =[ML²T⁻²] 
Hence the dimensions of specific heat capacity,
[c] = [ML²T⁻²]/[M][K] =[L²T⁻²K⁻¹]    

(b) Since Lₜ = L₀ {1+α(T₂-T₁)}
→Lₜ - L₀ = L₀α(T₂-T₁)
→α = (Lₜ - L₀)/{L₀(T₂-T₁)}
The dimensions of (Lₜ-L₀) and L₀ = the dimensions of length =[L]
The dimensions of (T₂-T₁) = dimension of temperature =[K]
Hence the dimensions of the coefficient of linear expansion, [α] =[L]/[L][K] =[K⁻¹]


(c) Since PV = nRT

→R = PV/nT

The dimensions of pressure, [P] =Force/area =[MLT⁻²]/[L²] =[ML⁻¹T⁻²]

The dimensions of volume, [V] =[L³]

The dimensions of temperature, T = [K] 

The dimensions of amount of gas, [n] = [(mol)]

Hence the dimensions of the gas constant,

[R] = [ML⁻¹T⁻²][L³]/[(mol)][K] 

      =[ML²T⁻²K⁻¹(mol)⁻¹]                 



 

7. Taking force, length and time to be the fundamental quantities, find the dimensions of

(a) density,

(b) pressure

(c) momentum, and

(d) energy.        


ANSWER:  Note that instead of mass, force is taken as a fundamental quantity. Hence, we shall express the mass in terms of force wherever it comes. For example, Force= mass*acceleration

→mass = Force/acceleration 

(a) Density = mass per unit volume = mass/volume

                 = Force/(acceleration*volume)

 Dimension of force = [F]

 Dimensions of acceleration = [LT⁻²]
 Dimensions of volume = [L³]
Hence the dimensions of density = [F]/{[LT⁻²][L³]}
          =[FL⁻⁴T²]  

              

(b) Pressure = Force/Area = [F]/[L²] =[FL⁻²] 


(c) Momentum = Mass*velocity 

     ={Force/(Acceleration)}*{Length/Time}

     ={[F]/[LT⁻²]}*{[L]/[T]}

     =[FL⁻¹T²]*[LT⁻¹] 

     =[FT] 

(d) Energy = Force*Distance =[F]*[L] =[FL] 


  

8. Suppose the acceleration due to gravity at a place is 10 m/s². Find its value in cm/(minute)².        


ANSWER: 1 m = 100 cm, and 1 s = 1/60 minute

Putting these in the units,

10 m/s² = 10*(100 cm)/(1/60 minute)²

             = 1000*60² cm/(minute)²

             = 1000*3600 cm/(minute)²

             = 3.6x10⁶ cm/(minute)²                 



 

9. The average speed of a snail is 0.020 miles/hour and that of a leopard is 70 miles/hour. Convert these speeds in SI units.       


ANSWER: The SI unit of length is m (meter), and of time is s (seconds).

1 mile = 1.6 km = 1600 m, 1 hour = 3600 s

Hence 1 mile/hour = (1600 m)/(3600 s)
                            = 4/9 m/s
So, average speed of a snail = 0.020 miles/hour 
= 0.020*4/9 m/s =0.08/9 m/s =0.00889 m/s 
= 8.89x10⁻³ m/s


And the average speed of a leopard =70 miles/hour

=70*4/9 m/s =280/9 m/s =31.11 m/s

                  


 

10. The height of the mercury column in a barometer in a Calcutta laboratory was recorded to be 75 cm. Calculate this pressure in SI and CGS units using the following data: Specific gravity of mercury = 13.6, Density of water = 10³ kg/m³, g = 9.8 m/s² at Calcutta. Pressure = hρg in usual symbols.       


ANSWER: Here h = 75 cm =0.75 m

Density of mercury, ρ = specific gravity*Density of water =13.6*10³ kg/m³

g = 9.8 m/s²

Hence the pressure at the base of the mercury column  in SI units =hρg

=0.75*13.6x10³*9.8 N/m²

=99960 N/m²                  

≈100000 N/m²

=10x10⁴ N/m²


In CGS units, units of mass = g, length = cm and time = s. We know that 1 kg = 1000 g, 1 m = 100 cm.

Hence, 1 N =1 kg-m/s² = (1000 g)(100 cm)/s²
=10⁵ g-cm/s² =10⁵ dyne
So, 1 N/m² =(10⁵ dyne)/(100 cm)²
                 =10 dyne/cm²

So, in CGS units, pressure = 10x10⁴ N/m²

=10x10⁴*(10 dyne/cm²)
=10x10⁵ dyne/cm²

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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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