3. Assuming that the frequency gamma of a
vibrating string may depend upon i)
applied force (F) ii) length (C) iii) mass
per unit length (m), prove that you gamma, alpha
1
F
IVM
using dimensional analysis.
(related to JIPMER 2001)
Answers
Force =1 /time
f = T ^-1
F unit is kgms ^-2
dimension formula is MlT ^-2
length unit M dim formula is L
mass length is Kgm ^-1
dimensions formula is Ml ^-1
(T ^-1 ) = (MlT ^-2 )^a (l ) ^b (ml -1)c
(T^-1 ) = (M ) ^a+c (L ) ^a+b-c (T )^-2 a
Now a =0
b=0
c=1
a+c =0
1/2 +c =0
c =-1/2
a+b-c =0
1/2 +b -1/2 =0
b =-1
-2a =1
a = 1/2
K F ^1/2 l ^-1 m ^1/2
V = K /L root of f / m is proved
Answer:
v=La Tb mc ________________ EQ. 1
v=T-1 , L=L , T=MLT-2 , m=ML-1
Putting values of v,L,T and m in eq 1
T-1=[L]a [MLT-2] b [ML-1]C
T-1=La+b-c Mb+c T-2b
Comparing powers on both sides,
a+b-c=0 => a=-1
b+c=0 => C=-2
-2b=-1 => b=1
Putting values of a,b and c in eq. 1 , we get
v=L-1 T-2 M-1/2
v=k 1/L √T/ √M where k is constant
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