Find the value of #a, b and c#?
A gas bubble from an explosion under water, oscillates with a time period T proportional to #p^a d^b E^c# , where #p,d, and E# are pressure, density and total energy respectively.
Also pls explain how to get the values of #a, b and c#
A gas bubble from an explosion under water, oscillates with a time period T proportional to
Also pls explain how to get the values of
1 Answer
Explanation:
Given time period
#T prop p^a d^b E^c#
#T=K p^a d^b E^c# ......(1)
where#K# is a constant of proportionality and dimensionless quantity.
Inserting the dimensions of Time, pressure, density and Energy in equation (1) we get
#[T] = [ML^-1T^-2]^ a [ML^-3]^ b [ML^2T^ -2]^ c#
Equating powers of
# 0= a + b + c # .....(2)
# 0= –a –3b + 2c # .....(3)
#1= –2a – 2c # ......(4)
Solving these equations
From (4)
#a+c=-1/2# .....(5)
Inserting this value in (2)
# 0= b-1/2 #
#=>b=1/2#
From (5)
#a=-1/2-c#
Inserting values of
# 0= -(-1/2-c)-3xx1/2 + 2c #
# => 3c=1#
# => c=1/3#
Inserting value of
#a+1/3=-1/2#
#=>a=-1/2-1/3#
#=>a=-5/6#
#:.a = -5/6, b = 1/2 and c = 1/3#