ch13-p072

# ch13-p072 - M by 2 10 30 kg. Thus, admitting some tolerance...

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72. (a) The gravitational acceleration a g is defined in Eq. 13-11. The problem is concerned with the difference between a g evaluated at r = 50 R h and a g evaluated at r = 50 R h + h (where h is the estimate of your height). Assuming h is much smaller than 50 R h then we can approximate h as the dr which is present when we consider the differential of Eq. 13-11: | da g | = 2 GM r 3 dr 2 GM 50 3 R h 3 h = 2 GM 50 3 (2 GM/c 2 ) 3 h . If we approximate | da g | = 10 m/s 2 and h 1.5 m, we can solve this for M . Giving our results in terms of the Sun’s mass means dividing our result for
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Unformatted text preview: M by 2 10 30 kg. Thus, admitting some tolerance into our estimate of h we find the critical black hole mass should in the range of 105 to 125 solar masses. (b) Interestingly, this turns out to be lower limit (which will surprise many students) since the above expression shows | da g | is inversely proportional to M . It should perhaps be emphasized that a distance of 50 R h from a small black hole is much smaller than a distance of 50 R h from a large black hole....
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