Find using dimensional analysis the relationship between the frequency of the vibration of a stretched string and it's m/l , it's tension and it's length
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Answer:
Suppose, that the frequency f depends on the tension raised to the power a, length raised to the power b and mass per unit length raised to the power c.
Then, f∝[F]
a
[l]
b
[μ]
c
or, f=k[F]
a
[l]
b
[μ]
c
...(i)
Here, k is a dimensionless constant.
Thus, [f]=[F]
a
[l]
b
[μ]
c
or, [M
0
L
0
T
−1
]=[MLT
−2
]
e
[L]
b
[ML
−1
]
c
or, [M
0
L
0
T
−1
]=[M
a+c
L
a+b−c
T
−2a
]
For dimensional balance, the dimensions on both sides should be same.
Thus, a+c=0 ...(ii)
a+b−c=0 ...(iii)
−2a=−1 ...(iv)
Solving these three equations, we get
a=
2
1
,c=−
2
1
andb=−1
Substituting these values in Eq. (i), we get
f=k(F)
1/2
(l)
−1
(μ)
−1/2
orf=
l
k
μ
F
Experimentally, the value of k is found to be
2
1
.
Hence, f=
2l
1
μ
F
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