A particle is performing SHM along the axis of a fixed ring. Due to gravitational force, its displacement at time t is given by x = a sin ω t.
In this equation ω is found to depend on radius of the ring (r), mass of the ring (m) and gravitational constant (G).Using dimensional analysis, find the expression of ω in terms of m, r and G
Text Solution
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ω = (some number) 
.
Sol. Let, ω = KM a r b G c where K is a dimensionless constant.
Writing the dimension of both the sides and equating then we have,
T –1 = M a L b (M –1 L 3 T –2 ) c
= M a–c L b+3c T –2c
Equating the exponents
– 2c = – 1 or c = 
,
b + 3c = 0 or – 3 c = b = – 
a – c = 0 . c = a = + 
Thus the required equation is ω = K 
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