A gas bubble, from an explosion under water, oscillates with period T proportional to P a d b E c where P is the static pressure, d is the density of water, and E is the total energy of explosion. Find the values of a,b and c.
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Sol. T ∝ P a d b E c
MºLº[T] ∝ [ML –1 T –2 ] [ML –3 ] b [ML 2 T –2 ] c MºLº[T] ∝ M
a+b+c L –a–3b+2c T –2a –2c Equating dimensions both sides
(i) a + b + c = 0 ⇒ a + c = – b
(ii) –a – 3b + 2c = 0 b = ½
(iii) –2a – 2c = 1 ⇒ a + c = –1/2
from (ii)
a + 3b – 2c = 0
a + c + 3b – 3c = 0
–1/2 + 3(1/2) = 3c
1 = 3c
c = 1/3
a + c = –1/2
a = –1/2 – c = –1/2 –1/3
a = –5/6
Hence; a = –5/6, b = 1/2 , c = 1/3
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