Team Quanta gladly presents all possible short questions of BS Physics book Mechanics – II’s Chapter#05: Special Theory of Relativity.
Q.1 The Sweep rate of the tail of a comet exceed the speed of light. Explain this phenomenon and show that there is no contradiction with relativity.
Answer: The tail of comet contains particles which separate from nucleus of comet to become its tail. The sweep rate of each particle leaving the nucleus does not exceed the speed of light because the nucleus supplies new particles to tail continuously. There is no contradiction to relativity.
Q.2 A beam from a laser fall at right angle on a plane mirror and reflects from it. What is the speed of the reflected beam if the mirror is (a) fixed in the laboratory and (b) moving directly toward the laser speed v?
Answer: If mirror is fixed in laboratory then laser beam reflects with same speed and if the mirror is moving then speed of laser light mirror becomes in relative motion. The relative speed is equal to difference of speed of mirror and reflected light.
Q.3 Give an example from the classical physics in which the motion of a clock effects its rate that is, the way it runs. ( The magnitude of the effect may depend on the detailed nature of the clock.)
Answer: Suppose we have two clocks A & B. A is standing still B zooms towards it narrowly missing. As B passes A we observe that they are perfectly synchronized. Clock B continuing to move away from A, its time will be slower than A, will be gradually fall behind time-wise.
Q.4 Although in relativity ( where motion is relative and absolute) we find that “moving clock run slowly” , this effect has nothing to do with the motion altering the way a clock works. With what does it have to do?
Answer: The phrase “ moving clock run slow” indicates that a clock moving relative to a frame containing in an array of synchronized clocks will be found to run slow when timed by those clocks. That is only in sense of comparing a single moving clock with two separated synchronized stationary clocks can we declare that moving clocks run slow. Due to presence of gravitational spin and orbit coupling there would appear a clock effect.
Q.5 How does the concept of simultaneously enter into the measurement of the length of an object?
Answer: While measuring the length of an object relative to some scale or other object there is always disagreement between two observers on the time order of two events for measuring the length of same object simultaneously.
Q.6 In relativity the time and space coordinates are intertwined and treated on a more or less equivalent basis. Are times and space fundamentally of the same nature, or is there some essential difference between them that is preserved even in relativity?
Answer: Yes, time and space coordinated are treated on equivalent basis, but time and space has essential difference which is even preserved in relativity that time dominates the space coordinate in relative motion.
Q.7 How many relativistic expressions can you think of in which Lorentz factor enter as a simple multiplier?
Answer: The Lorentz factor enters as a simple multiplier in following transformations:
(i)- Lorentz transformations (ii)- Inverse Lorentz transformations
(iii)- Time dilation (iv)- Relativity of mass (v)- Relativity of simultaneity
Q.8 Is the mass of stable, composite particle(a gold nucleus, for example) greater than, equal to, or less than sum of the masses of its constituents? Explain.
Answer: When protons and neutrons are combined to form a nucleus, we notice that mass of nucleus is less than sum of masses of nucleons. This is known as mass defect. This can be explained on the basis of mass-energy equivalence E=mc2 , that mass and energy are inter convertible.
Q.9 Two events occur at same place and at same time for one observer. Will they be simultaneous for all other observers?
Answer: Suppose two events occur at same place and at same time for one observer, then these events may not be simultaneous for a moving observer.
Q.10 Has classical mechanics is failed to explain the ordinary process of daily life?
Answer: No, classical mechanics is still essential to explain the ordinary processes in everyday life. Classical mechanics deals with objects of larger mass moving with small velocities.
Q.11 Name the transformations under which Maxwell equations are invariant.
Answer: Maxwell equations are invariant under Lorentz transformations.
Q.12 Some distant galaxies are moving away from us at speed greater than 0.5c. What is speed of light received on earth from these galaxies?
Answer: As speed of light is ‘c’ in vacuum, so speed of light received on earth from these galaxies is ‘c’.
Q.13 If we keep on applying force on a material object, can it gain speed of light?
Answer: No, if we keep on applying force on a material object, it cannot gain speed of light, because mass becomes infinite at .
Q.14 When we view a distant galaxy, we see that the light coming from it has a longer wavelength than the corresponding light here on earth. Is this a violation of postulate of speed of constancy of light?
Answer: Speed of light is constant in vacuum irrespective of wavelength. However the relation shows that smaller frequency has larger wavelength.
Q.15 Name at least two nuclear phenomenon supporting the mass-energy relation.
Answer: Mass-energy relation is given by,
![\[E=mc^2\]](https://quantapublisher.com/wp-content/ql-cache/quicklatex.com-3c024f1e436e0ade0ab6b41649b66707_l3.png "Rendered by QuickLaTeX.com")
(i)- Fusion and fission are important examples of conservation of mass into energy.
(ii)-Annihilation of electron-positron pair also supports the mass-energy relation.
Q.16 Two observers are moving relative to one another. Will they measure their relative speed and time between two events to have same value?
Answer: They will measure same relative speed but time interval may be different due to time dilation effect.
Q.17 Show that an object of finite mass cannot be accelerated from rest to a speed greater than speed of light in vacuum?
Answer: From mass variation with velocity we know that,
![\[m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}\]](https://quantapublisher.com/wp-content/ql-cache/quicklatex.com-d120e6fd21dd138265e9dd46e40f8dd8_l3.png "Rendered by QuickLaTeX.com")
When v we have , this is physically impossible. Hence an object of finite mass cannot be accelerated from rest to speed greater than speed of light in vacuum.
Q.18 You are in a spaceship, travelling directly away from moon with a speed of 0.9c. A light signal is sent in your direction from surface of moon. As the signal passes your ship, do you measure its speed equal to 0.1c?
Answer: Yes, since speed of light signal is same regardless of speed of ship.
Q.19 What are measurements on which two observers in relative motion will agree upon?
Answer: The measurements on which two observers in relative motion will agree upon are speed of light in free space, magnitude of force and acceleration of a moving object.
Q.20 An uncharged capacitor is charged by moving some electrons from one plate of capacitor to other plate. Is mass of charged capacitor same as that of uncharged capacitor?
Answer: No, in charging process, the charged capacitor stores energy. Thus charged capacitor has more mass than uncharged capacitor according to relation .
Q.21 Two identical clocks are synchronized. One is then put in orbit directed eastward around the earth while the other remains on the Earth. Which clock runs slower? When the moving clock returns to the Earth, are the two still synchronized?
Answer: The clock in orbit runs slower. The clocks are not synchronized. After return, a time difference has developed between the two clocks although both tick at the same rate
Q.22The speed of light in water is 
. Suppose an electron is moving through water at . Does that violate the principle of relativity?
. Suppose an electron is moving through water at 
. Does that violate the principle of relativity?Answer: No, since no object can move faster than speed of light which is 
.
Q.23 When we compress a spring between our fingers, does its mass stay same?
Answer: No, in corresponding process, energy of spring increases according to relation .
Q.24 How would velocities add if speed of light were infinitely large?
Answer: Lorentz velocity addition law is,
![\[u=\frac{\acute{u}+v}{1+\frac{v}{c^2}\acute{u}}\]](https://quantapublisher.com/wp-content/ql-cache/quicklatex.com-fc7432e9af1e69b8c851224e0166aab0_l3.png "Rendered by QuickLaTeX.com")
In classical limits i.e., 
, we have
![\[u=\acute{u}+v\]](https://quantapublisher.com/wp-content/ql-cache/quicklatex.com-e5a0da62edab5218c25acaf3970f899d_l3.png "Rendered by QuickLaTeX.com")
This classical law of addition of velocities.
Q.25 Two observers are moving relative to one another. Will they measure the length of object and speed of light in vacuum to have some value?
Answer: They will measure same speed of light but length of object may be different due to length contradiction effect.
Q.26 A clock moving with a finite speed v is observed to run slow. If speed of light were twice as large as it actually is, would the factor by which clock runs slow be unchanged?
Answer: No, the factor by which time is dilated increases as speed of clock approaches speed of light according to relation,
![\[t=\ \frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}\]](https://quantapublisher.com/wp-content/ql-cache/quicklatex.com-6c179564b6e83fc02d0ece6589401366_l3.png "Rendered by QuickLaTeX.com")
When speed of light will be doubled, then the factor by which clock runs slow be decreased.
Q.27 What is special theory of relativity? Give its some results.
Answer: Special theory of relativity deals with how physical quantities change between inertial frames of reference. It starts by assuming that;
(i)- There is no “special” inertial frame with respect to which “absolute velocities” can be measured(i.e. there is no such thing as absolute velocity. All velocities are relative) and
(ii)- The speed of light in vacuum is constant in all inertial frames.
From these two postulates , we can drive inserting results such as relativistic length contraction, relativistic mass increase, mass-energy equivalence formula, relativistic time dilation etc. It also says that nothing can travel faster than light. Another result is that time is not an absolute parameter as assumed in Newtonian mechanics, but a 4th coordinate like the 3- space coordinates( with some difference, of course) and together, they can be described as a 4- dimensional space-time.
Q.28 What is general theory of relativity? Give its some results.
Answer: The general theory of relativity is the generalization of the special theory where non-inertial frame of references are taken into account. One of the postulate of the general theory is the equivalence principle, which implies that one cannot distinguish between a non-inertial frame of reference and a frame of reference where gravity is present only by observing motion of particles. Due to this , the general theory of relativity becomes a theory of gravitation.
In the general theory, space-time is no longer flat. Gravitation is described as curvature in space- time caused by the presence of mass. The more massive an object is, the more curvature it causes. The curbed trajectories traced by a particle moving in a gravitational field can be expressed as the particle moving in a straight line in a curved space-time. The general theory predicts phenomenon such as black holes and gravitational deflection of light.
Q.29 Give a comparison of special theory of relativity and general theory of relativity.
Answer: A comparison of special theory of relativity and general theory of relativity is given below:
| Special theory of relativity In special theory, the laws of physics are the same for all inertial coordinate system that is those in which Newton’s first law of motion is true.Special theory of relativity explains the time dilation principle at high speed i.e. it does not include gravity. | General theory of relativity In general theory of relativity, the laws of physics are the same in all coordinate systems, whether they are inertial or not. So general theory is a special case of general theory.General theory of relativity explains the relationship between space and time and its properties. This includes gravity. |
Q.30 Newtonian mechanics correctly describes objects moving at ordinary speeds, and relativistic mechanics correctly describes objects moving very fast. Argue for or against statement.
Answer: For acceptance of any theory, it must agree with experiments. Keeping this concept in mind, Newtonian mechanics agrees with experiment for objects moving slow as compared with speed of light whereas Relativistic mechanics agrees with experiment for objects moving at high speeds. Hence the given statement is correct.
Q.31 Two identical clocks are in the same house, one upstairs in a bedroom and the other downstairs in the courtyard. Which clock runs faster?
Answer: Because of gravitational time dilation, the upstairs clock runs faster because it is farther to the earth and hence in a weaker gravitational field than the downstairs clock.
Q.32 A spacecraft zooms past the Earth with a constant velocity. An observer on the earth measures that an undamaged clock on the spacecraft is ticking at one third the rate of an identical clock on the Earth. What does an observer on the space craft measure about the Earth-based clock’s ticking rate?
Answer: Since the relativistic time dilation effect is symmetric between the observers, so it runs at one-third the rate of his own.
Q.33 An astronaut is travelling in a spacecraft in outer space in a straight line at a constant speed of 0.5c Which of the following effects would experience?
Answer: According to theory of relativity, the laws of physics are same in all inertial frames. The frame of reference of astronaut being inertial reference frame, he should experience no effects due to his motion through space.
Q.34 In many cases, a nearby star has been found to have large planet orbiting about it, although light from the planet could not be seen separately from the starlight. Using the ideas of a system rotating about its center of mass and of the Doppler shift for light, explain how an astronomer could determine the presence of the invisible planet.
Answer: According to Kepler’s first law , a star planet moves in elliptical orbit. The time period of planet can be calculated from Kepler’s third law and from time period, size of orbit can be calculated.
Q.35 With regard to reference frames, how does general relativity differ from special relativity?
Answer: General theory of relativity deals with the relationship between physical quantities and laws in all frames of reference where as special relativity describes the relationship between physical quantities and laws only in inertial frames of reference.
Q.36 Explain how the Doppler Effect with microwaves is used to determine the speed of an automobile.
Answer: When microwaves are reflected from moving object, receiver in the radar gun detects the reflected wave and compares its frequency to that of emitted wave. Speed can be calculated by using frequency shift.
Q.37 A spacecraft built in the shape of sphere moves past an observer on the earth with speed of 0.5c. What shape does the observer measure for the spacecraft as it goes by?
Answer: Since the dimension in the direction of motion would be contracted but the dimension perpendicular to the motion would be unchanged, so shape will be like a round pillow shape, flattened along the direction of motion.
Q.38 A distant astronomical object (a quasar) is moving away from us at half the speed of light .What is the speed of the light we receive from this quasar?
Answer: According to second postulate of special theory of relativity, speed of light is c in vacuum, so in the present case, light from source travels in vacuum at speed c.
Q.39 Explain why, when defining the length of a rod, it is necessary to specify that the positions of the ends of the rod are to be measured simultaneously.
Answer: Due to length contraction, it is necessary to specify that the positions of the ends of the rod are to be measured simultaneously.
