Space-Time

Practice

practice problem 1

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solution

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practice problem 2

Gravity Probe B is an earth-orbiting relativity experiment developed by NASA and Stanford University to test Albert Einstein's general theory of relativity. The media have reported that Gravity Probe B has gone in search of "the missing inch". What the hell are they talking about? Do the following calculations for a satellite in a circular, low earth orbit (LEO) at an altitude of 642 km. (Ignore the rules of significant digits when calculating distances. Record these answers with as much precision as your calculator will allow.)
  1. Using classical euclidean geometry, determine the circumference of this orbit.
  2. Using classical Newtonian mechanics, determine the speed of a satellite in such an orbit.
  3. Using special relativity, determine the amount by which the circumference of the orbit has been shortened due to the satellite's motion. State your answer in …
    1. centimeters
    2. English inches

solution

  1. Circumference …
     
    C = 2πr
    C = 2π(6,378,100 m + 642.000 m)
    C = 44108589.174932 m
     
  2. The centripetal force is supplied by gravity for a satellite in circular orbit. Set the centripetal force formula equal to Newton's law of universal gravitation.
             
    Fc = Fg  ⇒  mv2  =  Gm1m2
    r r2
             
    Solve for speed (v), plug in numbers, and calculate an answer.
             
    v =  √  Gm  = √  (6.67 × 10−11 Nm2/kg2)(5.98 × 1024 kg)
    r (6,378,100 m + 642,000 m)
    v =  7538 m/s
             
  3. Apply the length contraction formula to the circumference of the orbit moving at the speed of the satellite.
     
    ℓ =  ℓ0 √ ⎛
    ⎝    		 1 −  v2 ⎞
    ⎠    		  
    c2
    ℓ =  44108589.174932 m √ ⎛
    ⎝    		 1 −  (7538 m/s)2 ⎞
    ⎠
    (3.00 × 108 m/s)2
    ℓ =  44108589.161009 m
     
    Subtract the contracted length from the rest length.
     
    Δℓ = ℓ0 − ℓ
    Δℓ = 44108589.174932 m − 44108589.172716 m
    Δℓ = 0.013923 m
     
    1. Convert to centimeters with a simple shift of the decimal point.
       
      Δℓ = 1.4 cm
       
    2. Convert to English inches using the definition of an inch. (An inch is 0.0254 m by definition.)
       
      Δℓ = 0.013923 m ÷ 0.0254 m/in. = 0.55 inches
       
      OK, so it's closer to half an inch than an inch, but its of the right order. The more important thing to note is that Gravity Probe B is not looking for space-time distortion caused by motion, it's looking for space-time distortion caused by gravity. It's not trying to confirm the predictions of special relativity, it's trying to confirm the predictions of general relativity. The two effects are of about the same size, however, and together they make an inch.

practice problem 3

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solution

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practice problem 4

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solution

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The Physics Hypertextbook

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