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NEET-XII-Physics

47: The Special Theory of Relativity

with Solutions - page 2

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  • #
    Section : iii
  • Qstn #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 result for small speeds.
    (d) Nonrelativistic equations never give exact result.
    digAnsr:   b,d
    Ans :
    (b) Equations of special relativity are applicable for all speeds.
    (d) Nonrelativistic equations never give exact result.
    According to special relativity, if a particle is moving at a very high speed v, its mass
    `` m=\gamma {m}_{o},\,\mathrm{\,length\,}l=\frac{{l}_{o}}{\gamma },\,\mathrm{\,change\,}\,\mathrm{\,in\,}\,\mathrm{\,time\,}\Delta t=\gamma \Delta {t}_{o}``
    `` \,\mathrm{\,where\,}\gamma =\frac{1}{\sqrt{1-{\displaystyle \frac{{v}^{2}}{{c}^{2}}}}}ifv<<c\Rightarrow \gamma \cong 1``
    `` ``
    `` ``
    `` ``
    that is at non-relativistic speed (small speed), `` m\cong {m}_{o,}l\cong {l}_{o},\Delta t\cong \Delta {t}_{o}``     where `` {m}_{o},{l}_{o}\,\mathrm{\,and\,}\Delta {t}_{o}``     are the rest mass, length and time interval respectively. Therefore, relativistic equations are applicable for all speeds. But
    `` \gamma ={\left(1-\frac{{v}^{2}}{{c}^{2}}\right)}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}``
    `` \Rightarrow \gamma =1+\frac{{v}^{2}}{2{c}^{2}}+...(\,\mathrm{\,expanding\,}\,\mathrm{\,binomially\,})``
    `` \frac{{v}^{2}}{2{c}^{2}}+...=k<<1\,\mathrm{\,if\,}v<<c\,\mathrm{\,but\,}\,\mathrm{\,still\,}k>0``
    Hence, non relativistic equations in which `` \gamma `` factor is taken to be exactly 1 never give exact results.

    Page No 457:
  • Qstn #2
    If the speed of a 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
    digAnsr:   c,d
    Ans :
    (c) the length will decrease


    (d) the mass will increase
    If the speed of a rod moving at a relativistic speed v parallel to its length, its mass
    `` m=\gamma {m}_{o}=\frac{{m}_{o}}{\sqrt{1-{\displaystyle \frac{{v}^{2}}{{c}^{2}}}}}``
    and its length
    `` l=\frac{{l}_{o}}{\gamma }={l}_{o}\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}``
    `` \,\mathrm{\,where\,}\gamma =\frac{1}{\sqrt{1-{\displaystyle \frac{{v}^{2}}{{c}^{2}}}}}={\left(1-\frac{{v}^{2}}{{c}^{2}}\right)}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=1+\frac{{v}^{2}}{2{c}^{2}}+...>1\,\mathrm{\,as\,}v<c``
    If the speed is doubled, its multiplying factor
    `` \gamma \text{'}=\frac{1}{\sqrt{1-{\displaystyle \frac{4{v}^{2}}{{c}^{2}}}}}={\left(1-\frac{4{v}^{2}}{{c}^{2}}\right)}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=1+\frac{2{v}^{2}}{{c}^{2}}+...>2\gamma ``
    `` \,\mathrm{\,and\,}m=\gamma \text{'}{m}_{o}>2\gamma {m}_{o},l=\frac{{l}_{o}}{\gamma \text{'}}<\frac{{l}_{o}}{2\gamma }``
    Hence, the mass will increase but more than double and length will decrease but not exactly half of the original values.
    Page No 457:
  • Qstn #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
    digAnsr:   b
    Ans : (b) perpendicular to AB
    Page No 457:
  • Qstn #4
    Which of the following quantities related to an electron has a finite upper limit?
    (a) Mass
    (b) Momentum
    (c) Speed
    (d) Kinetic energy
    digAnsr:   c
    Ans : (c) speed
    If an electron is given a very high speed v, its mass
    `` m=\gamma {m}_{o}=\frac{{m}_{o}}{\sqrt{1-{\displaystyle \frac{{v}^{2}}{{c}^{2}}}}}``
    `` \,\mathrm{\,momentum\,},p=mv=\gamma {m}_{o}v=\frac{{m}_{o}v}{\sqrt{1-{\displaystyle \frac{{v}^{2}}{{c}^{2}}}}}``
    `` \,\mathrm{\,kinetic\,}\,\mathrm{\,energy\,},k=\frac{1}{2}m{v}^{2}=\frac{1}{2}\gamma {m}_{o}{v}^{2}=\frac{1}{2}\frac{{m}_{o}{v}^{2}}{\sqrt{1-{\displaystyle \frac{{v}^{2}}{{c}^{2}}}}}``
    `` \gamma =\frac{1}{\sqrt{1-{\displaystyle \frac{{v}^{2}}{{c}^{2}}}}}``
    `` \,\mathrm{\,at\,}v=c,\gamma =\frac{1}{\sqrt{1-{\displaystyle \frac{{c}^{2}}{{c}^{2}}}}}=\infty ``
    `` \Rightarrow \,\mathrm{\,at\,}v=c,m=p=k=\infty ``
    Therefore, there's an upper bound for v to be always less than c but no upper limits for mass, momentum and kinetic energy of the electron.
    Page No 457:
  • Qstn #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) may be more than L‘ but less than L
    (d) may be more than L
    digAnsr:   b,c
    Ans : (b) may be equal to L
    (c) may be more than L' but less than L
    If a rod of rest length L is moving at a relativistic speed v and its length is contracted to L', then
    `` L\text{'}=\frac{L}{\gamma }=L\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}``
    `` \,\mathrm{\,If\,}\gamma =\frac{1}{\sqrt{1-{\displaystyle \frac{{v}^{2}}{{c}^{2}}}}},\,\mathrm{\,then\,}v<<c,\gamma \cong 1.``
    `` ``
    `` \Rightarrow L\text{'}\cong L``
    But the length of the rod may be more than L' depending on the frame of the observer. However, it cannot be more than L because as the speed of the rod increases, its length contracts more and more due to increasing value of `` \gamma ``.
    Page No 457:
  • Qstn #6
    When a rod moves at a relativistic speed v, its mass
    (a) must increase by a factor of γ
    (b) may remain unchanged
    (c) may increase by a factor other than γ
    (d) may decrease
    digAnsr:   a
    Ans : (a) must increase by a factor of γ
    If a rod is moving at a relativistic speed v, its mass is given by
    `` m=\gamma {m}_{o}=\frac{{m}_{o}}{\sqrt{1-{\displaystyle \frac{{v}^{2}}{{c}^{2}}}}}``
    Here,
    `` \gamma =\frac{1}{\sqrt{1-{\displaystyle \frac{{v}^{2}}{{c}^{2}}}}}``
    Thus, its mass must increase by a factor of `` \gamma ``.
    Page No 458:
  • #
    Section : iv
  • Qstn #1
    The guru of a yogi lives in a Himalyan 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.
    Ans : Given: Distance between the house of the yogi and his guru, s = 1000 km = 106 m
    So, for the minimum possible time interval, the velocity should be maximum. We know that maximum velocity can be that of light, i.e. v = 3 × 108 m/s.
    We know,
    `` \,\mathrm{\,Time\,},t=\frac{\,\mathrm{\,Distance\,}}{\,\mathrm{\,Speed\,}}``
    `` =\frac{{10}^{6}}{3\times {10}^{8}}=\frac{1}{300}\,\mathrm{\,s\,}``
    Page No 458:
  • Qstn #2
    A suitcase kept on a shop’s rack is measured 50 cm × 25 cm × 10 cm by the shop’s owner. A traveller takes this suitcase in train moving with velocity 0.6c. If the suitcase is placed with its length along the train’s velocity, find the dimensions measured by
    Ans : Given:
    Length of suitcase, l = 50 cm
    Breadth of suitcase, b = 25 cm
    Height of suitcase, h = 10 cm
    Velocity of train, v = 0.6c
  • #2-a
    the traveller and
    Ans : The observer in the train notices the same values of l, b and h because the suitcase is in rest w.r.t. the traveller.
  • #2-b
    a ground observer.
    Ans : Since the suitcase is moving with a speed of 0.6c w.r.t. the ground observer, the component of length parallel to the velocity undergoes contraction, but the perpendicular components (breadth and height) remain the same.
    So, the length that is parallel to the velocity of the train changes, while the breadth and height remain the same.
    `` l\text{'}=l\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}``
    `` =50\sqrt{1-\frac{{\left(0.6c\right)}^{2}}{{c}^{2}}}``
    `` =50\sqrt{1-0.36}``
    `` =50\sqrt{0.64}``
    `` =50\times 0.8=40\,\mathrm{\,cm\,}``
    Thus, the dimensions measured by the ground observer are 40 cm × 25 cm × 10 cm.
    Page No 458:
  • Qstn #3
    The length of a rod is exactly 1 m when measured at rest. What will be its length when it moves at a speed of
    Ans : Given:
    Proper length of the rod, L = 1 m
    If v is the velocity of the rod, then the moving length of the rod is given by
    `` L\text{'}=L\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}``
  • #3-a
    3 × 105 m s-1,
    Ans : Here,
    v = 3 × 105 m/s
    `` L\text{'}=1\times \sqrt{1-\frac{9\times {10}^{10}}{9\times {10}^{16}}}``
    `` =\sqrt{1-{10}^{-6}}=0.9999995\,\mathrm{\,m\,}``
  • #3-b
    3 × 106 m s-1 and
    Ans : Here,
    v = 3 × 106 m/s
    `` L\text{'}=1\times \sqrt{1-\frac{9\times {10}^{12}}{9\times {10}^{16}}}``
    `` =\sqrt{1-{10}^{-4}}``
    `` =0.99995\,\mathrm{\,m\,}``
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