Question
Question: The period of a body SHM is represented by \(T \propto {P^a}{D^b}{S^c}\), where \(P\) is the pressur...
The period of a body SHM is represented by T∝PaDbSc, where P is the pressure, D the density and S is the surface tension then the values of a, b and c is
A) 1,3,31
B) −23,21,1
C) −1,−2,3
D) −21,−23,−21
Solution
Hint
Pressure, density and surface tension are related as T∝PaDbSc. The dimensions of these quantities will be substituted and equated with T. Apply principle of homogeneity and solve the equations of powers to find the values of a, b and c.
Complete step-by-step answer
Use the dimensional analysis to solve this problem. Pressure is given by force per unit area,
P=AF
Dimensional formula:
[P]=[ML−1T−2]
Density is the ratio of mass to volume,
D=VM
Dimensional formula:
[D]=[ML−3T0]
Surface tension is force per unit length,
S=LF
Dimensional formula:
[S]=[ML0T−2]
Let T be a quantity with dimensions,
[T]=[M0L0T1].
Given that,
T∝PaDbSc
[M0L0T1]=[ML−1T−2]a[ML−3T0]b[ML0T−2]c
[M0L0T1]=[Ma+b+cL−a−3bT−2a−2c]
Equate the corresponding powers of mass, length and time on both sides. k is the constant of proportionality.
a+b+c=0,
−a−3b=0,
−2a−2c=0,
Solve the equations simultaneously.
a=−23,b=21,c=1
Hence, the values of a, b and c is a=−23,b=21,c=1
The correct option is (B).
Note
Applications of dimensional analysis:
(i) To find the unit of physical quantity.
(ii) To find dimensions of physical constant or coefficients.
(iii) To convert physical quantities to other systems.
(iv) To check the correctness of dimensions.