H C VERMA Solutions for Class 12-science Physics Chapter 25 - The Special Theory of Relativity

Page / Exercise

Chapter 25 - The Special Theory of Relativity Exercise 458

Question 1

The guru of a yogi lives in a Himalayan cave, 1000 km away from the house of the yogi. The yogi claims that whenever he thinks about his guru, the guru immediately knows about it. Calculate the minimum possible time interval between the yogi thinking about the guru and the guru knowing about it.

Solution 1

We know,    

   

for minimum time velocity is to be maximum.

 s

Question 2

A suitcase kept on a shop's rack is measured 50cm×25cm×10cm by shop's owner. A traveller takes this suitcase in a train moving with a velocity 0.6c. If the suitcase is placed with its length along the train's velocity, find the dimensions measured by (a) the traveler and (b) a ground observer.

Solution 2

(a) When suitcase is observed by traveler it is in same frame as traveler. Thus, dimensions remain same and are:

l=50cm, b=25cm, h=10cm

(b) When suitcase is observed by a person on ground then the frame of reference differ. Thus, length of suitcase will appear contracted to observer while breadth and height appears same. Thus, dimensions are:

  

l = 40cm

b=25cm

h=10cm

Question 3

The length of a rod is exactly 1m when measured at rest. What will be its length when it moves at a speed of (a)  , (b)   and (c)  ?

Solution 3

We know,

  

(a) If   

 m

(b) If   

 m

(c) If   

 m

Question 4

A person standing on a platform finds that a train moving with velocity 0.6c takes one second to pass by him. Find (a) the length of the train as seen by the person and (b) the rest length of the train.

Solution 4

(a) Length observed is given as:

  m

(b) We know,

  

  

Question 5

An aeroplane travels a rectangular field 100m × 50m, parallel to its length. What should be the speed of the plane so that field becomes square in the plane frame?

Solution 5

Field appears square instead of rectangle.

  and   and we know,

  

  

  

Question 6

The rest distance between Patna and Delhi is 1000 km. A nonstop train travels at 360km/h. (a) What is the distance between Patna and Delhi in the time frame? (b) How much time elapses in the time frame between Patna and Delhi?

Solution 6

(a) We know,

   

  

[As l=1000km=106 m and v=360km/h=100m/s]

  

  

(b) Time elapses between Patna and Delhi is:

  

  

  

Question 7

A person travels by a car at a speed of 180km/h. It takes exactly 10hours by wristwatch to go from the station A to the station B. (a) What is the rest distance between the two stations? (b) How much time is taken in the road frame by the car to go from the station A to the station B?

Solution 7

(a) Distance travelled by car is given as:

 km

Now, rest distance is given as,

   

 km

  

(b) Time taken is given as:

   

  

  

  

Question 8

A person travels on a spaceship moving at a speed of 5c/13 (a) Find the time interval calculated by him between the consecutive birthday celebrations of his friend on the earth. (b) Find the time interval calculated by the friend on the earth between the consecutive birthday celebrations of the traveller.

Solution 8

(a) Time interval is given as:

  

  

  

(b) Also, when friend on earth calculates then time interval is:

   

  

Question 9

According to the station clocks, two babies are born at the same instant, one in Howrah and other in Delhi. (a) Who is elder in the frame of 2301 Up Rajdhani Express going from Howrah to Delhi? (b) Who is elder in the frame of 2302 Dn Rajdhani Express going from Delhi to Howrah?

Solution 9

Station clocks are according to ground frame of reference thus, it is proper. While the clocks that are moving shows improper time which is more than proper time.

Time is given as:

ΔT' = v ΔT 

(a) In case of up train, baby born in Delhi is elder according to moving frame.

(b) In this down train, baby born in Howrah is elder.  

Question 10

Two babies are born in a moving train, one in the compartment adjacent to the engine and the other in the compartment adjacent to the guard. According, to the train frame, the babies are born at the same instant of time. Who is elder according to the ground frame? 

Solution 10

As frame is in motion clocks are out of sync. So, if L' is length of train in rest and speed of train is v in moving frame then the clock at one end where guard is standing leads the one near engine by  .

Therefore, baby born in compartment near guard is elder in comparison with baby born near engine.

Question 11

Suppose swarglok (heaven) is in constant motion at a speed of 0.9999c with respect to the earth. According to the earth's frame, how much time passes on the earth before one day passes on swarglok? 

Solution 11

We know,

  

v= 70.71

If Δt0 is one day in earth then one day in heaven that is   is given as:

  

  

 days

Question 12

If a person lives on the average 100 years in his rest frame, how long does he live in the earth frame if he spends all his life on a spaceship going at 60% of the speed of light?

Solution 12

Time interval is given as:

   

  

Question 13

An electric bulb, connected to a make and break power supply, switches off and on every second in its rest frame. What is the frequency of its switching off and on as seen from a spaceship travelling at a speed of 0.8c?

Solution 13

We know,

  

  

   

Question 14

A person travelling by a car moving at a speed of 100km/h finds that his wristwatch agrees with the clock on a tower A. By what amount will his wristwatch lag or lead the clock on another tower B, 1000km (in the earth's frame) from the tower A when the car reaches there?

Solution 14

Time interval is given as:

  

  

Now,

   

  

 s

  ns 

Question 15

At what speed the volume of an object shrinks to half its rest value?

Solution 15

As given,   

And we know,

  

  

  

   

Question 16

A particular particle created in a nuclear reactor leaves a 1cm track before decaying. Assuming that the particle moved at 0.995c, calculate the life of the particle (a) in the lab frame and (b) in the frame of the particle.

Solution 16

(a) In laboratory frame time is given as:

  

  

  

 ps

(b) In particle's frame:

  

  

 s

 ps

Question 17

By what fraction does the mass of a spring change when it is compressed by 1cm? The mass of the spring is 200g at its natural length and the spring constant is 500N/m.

Solution 17

Energy stored is given as:

  

  

=0.025J

Increase in mass  

  

  

Frictional change is:

  

  

Question 18

Find the increase in mass when 1kg of water is heated from 0°c to 1000c. Specific heat capacity of water=4200 J/kg-K.

Solution 18

We know,

  

  

=420000J

Also,

  

  

 kg

Question 19

Find the loss in the mass of 1 mole of an ideal monatomic gas kept in a rigid container as it cools down by 10°c. The gas constant R=8.3 J/mol-K.

Solution 19

Energy by monatomic gas is given as:

  

 J

Loss of   

  

 kg 

Question 20

By what fraction does the mass of a boy increase when he starts running at a speed of 12km/h?

Solution 20

We know,

  

  

and   

  

  

  

Question 21

A 100W bulb together with its power supply is suspended from a sensitive balance. Find the change in the mass recorded after the bulb remains on for 1year.

Solution 21

Energy in bulb is:

  

In 1sec energy spend is 100J.

Total energy spend in 1 year is given as:

  

  

Also,

  

 kg

Question 22

The energy from the sun reaches just outside the earth's atmosphere at a rate of 1400 w/m2. The distance between the sun and the earth is  m. (a) Calculate the rate at which the sun is losing its mass. (b) How long will the sun last assuming a constant decay at this rate? The present mass of the sun is   kg.

Solution 22

Power is given as:

P=IA

  

(a) We know

  

  

   

  

 kg/s

(b)  kg/s of disintegration=1sec

 kg/s of disintegration  

  

  

  

Question 23

An electron and a positron moving at small speeds collide and annihilate each other. Find the energy of the resulting gamma photon.

Solution 23

Both electron and positron have opposite charge and equal mass.

 kg

Energy of gamma particle is

  

  

 J

  

Question 24

Find the mass, kinetic energy and the momentum of an electron moving at 0.8c.

Solution 24

(a) We know,

  

 kg

(b) kinetic energy is given as:

  

  

 J

(c) We know,

  

  

 kg m/s

Question 25

Through what potential difference should an electron be accelerated to give it a speed of (a) 0.6c, (b) 0.9c and (c) 0.99c?

Solution 25

(a) We know,

  

  

Solving we get,

  

 V=831kV

(b) Similarly,

   

  

 kV

(c) Again similarly,

  

  

 MeV 

Question 26

Find the speed of an electron with kinetic energy (a) 1eV, (b) 10 keV and (c) 10 MeV. 

Solution 26

 We know,   

 (a)   

  

  

  

 m/s

(b) Similarly,

  

  

 m/s

(c)   

 m/s

Question 27

What is the kinetic energy of an electron in electron volts with mass equal to double its rest mass?

Solution 27

We know,

  

  

And

  

  

 J

Also,

 eV

 keV 

Question 28

Find the speed at which the kinetic energy of a particle will differ by 1% from its nonrelativistic value .

Solution 28

We know,

  

  

After solving we get,

  

  [we can neglect terms with power 4]

  

 m/s