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#

1 Answer
Aug 9, 2017

#a = -5/6, b = 1/2 and c = 1/3#

Explanation:

Given time period #T#

#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 #M, L, and T# on both sides we get

# 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 #aand b # in (3)

# 0= -(-1/2-c)-3xx1/2 + 2c #
# => 3c=1#
# => c=1/3#

Inserting value of # c# in (5)

#a+1/3=-1/2#
#=>a=-1/2-1/3#
#=>a=-5/6#

#:.a = -5/6, b = 1/2 and c = 1/3#