Friday, October 20, 2023

H C Verma solutions, The Special Theory of Relativity, Chapter-47, Objective-II, Concepts of Physics, Part-II

OBJECTIVE-II


       1. Mark the correct statements: 
(a) Equations of special relativity are not applicable for small speeds.
(b) Equations of special relativity are applicable for all speeds. 
(c) Nonrelativistic equations give exact results for small speeds. 
(d) Nonrelativistic equations never give exact results.    

ANSWER: (b), (d).    

EXPLANATION: The equations of special relativity are applicable for all speeds. When applied to small speeds they give similar results as nonrelativistic equations because the effect of speed is negligible. Hence option (b) is correct not (a). 
      Though at small speeds both types of equations have almost the same results the results of nonrelativistic equations are not exact because the effect of relative speed is neglected. For higher speeds, the results of nonrelativistic equations are quite erroneous. Hence option (d) is correct not (c).   






       2.  If the speed of the rod moving at a relativistic speed parallel to its length is doubled, 
(a) the length will become half of the original value
(b) the mass will become double of the original value 
(c) the length will decrease
(d) the mass will increase.     

ANSWER: (c), (d).    

EXPLANATION: In the given case, the length of the rod becomes, 
L' =L√(1 -v²/c²) and the mass becomes, 
m' =m/√(1 -v²/c²). 

Clearly, the length is not inversely proportional to speed v, hence doubling the speed will not make the length half. The length is proportional to the factor √(1 -v²/c²) which is always less than 1. Hence the length will surely decrease. So option (a) is incorrect but option (c) is correct. 
   Similarly, the mass is not proportional to the speed. Hence doubling the speed, mass will not be doubled. Mass is proportional to a factor 1/√(1 -v²/c²) which is always greater than 1. So surely the mass will increase. Thus option (b) is incorrect but option (d) correct.     





       3. Two events take place simultaneously at points A and B as seen in the lab frame. They also occur simultaneously in a frame moving with respect to the lab in a direction
(a) parallel to AB
(b) perpendicular to AB
(c) making an angle of 45° with AB
(d) making an angle of 135° with AB.    

ANSWER: (b).    

EXPLANATION: Only if the frame moving with respect to the lab is in a direction perpendicular to AB, the separation between A and B will be observed as unchanged. In this case, the two events will again take place simultaneously.    





       4. Which of the following quantities related to an electron has a finite upper limit? 
(a) mass
(b) momentum
(c) speed
(d) kinetic energy.    

ANSWER: (c).    

EXPLANATION: The mass is given as, 
m' =m/√(1 -v²/c²). 
Here if v →c, √(1 -v²/c²) →0 which results in
m' →∞.
So the mass has no finite upper limit.
 
The magnitude of momentum is given as
p ={m/√(1 -c²/v²)}v
   =m'v
As we have just seen m' has no finite upper limit, so p will also have no finite upper limit.

 The kinetic energy of a particle is,
mc² -mₒc² = (m -mₒ)c²
Here rest mass mₒ and the speed of light c is constant but the relative mass m has no finite upper limit. Hence the kinetic energy also has no finite upper limit.
  Thus options (a), (b), and (d) are incorrect.

When we calculate the speed of a particle after time t under a constant force F, we get
Speed v =Ftc/√(mₒ²c²+F²t²)
→v =c/√(1 +mₒ²c²/F²t²)
From this, it is clear that if t is quite large, say t →∞, then v =c. It means that the speed can not be more than c whatever the force and what may be the duration. So the speed of the electron has a finite upper limit. Option (c) is correct. 





       5. A rod of rest length L moves at a relativistic speed. Let L' =L/𝜸. Its length
(a) must be equal to L'
(b) may be equal to L
(c) maybe more than L' but less than L
(d) maybe more than L.      

ANSWER: (b), (c).    

EXPLANATION: The length L' is given as 
L' =L/𝛾.
Where 𝛾 =1/√(1 -v²/c²).

It depends on the direction of the speed. If the movement is parallel to the rod, the length will be equal to L'. But if the movement is perpendicular to the rod the length will be equal to L. So it is not necessarily equal to L'. Option (a) is incorrect and option (b) is correct. 
   
   Since 𝛾 > or = 1, length can not be more than L. Option (d) is incorrect. 

  If the direction of movement is other than parallel or perpendicular to the rod its length may be between L' and L. Option (c) is correct. 

 




       6. When a rod moves with a relativistic speed v, its mass 
(a) must increase
(b) may remain unchanged
(c) may increase by a factor other than 𝛾
(d) may decrease.   

ANSWER: (a).    

EXPLANATION: The mass of the rod is given as, 
m =mₒ/√(1 -v²/c²)

For a relativistic speed v, the factor 
1/√(1 -v²/c²) is always greater than 1. Hence mass m is always greater than mₒ.
So in this problem, the mass of the rod must increase. Option (a) is correct.    
 
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Links to the Chapters


CHAPTER 47- The Special Theory of Relativity





CHAPTER- 34- Magnetic Field

CHAPTER- 29- Electric Field and Potential











CHAPTER- 28- Heat Transfer

OBJECTIVE -I







EXERCISES - Q51 to Q55


CHAPTER- 27-Specific Heat Capacities of Gases

CHAPTER- 26-Laws of Thermodynamics


CHAPTER- 25-CALORIMETRY

Questions for Short Answer

OBJECTIVE-I

OBJECTIVE-II


EXERCISES - Q-11 to Q-18


CHAPTER- 24-Kinetic Theory of Gases







CHAPTER- 23 - Heat and Temperature






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".

---------------------------------------------------------------------------------

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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