Question

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. Find the value of a, b and c.

Answer 1

Hy mate

There is 5 ways to do this question

Given time period T

T∝padbEc
T=KpadbEc ......(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[ML2T−2]c

Equating powers of M,L,andT 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=−12 .....(5)

Inserting this value in (2)

0=b−12
⇒b=12

From (5)

a=−12−c

Inserting values of aandb in (3)

0=−(−12−c)−3×12+2c
⇒3c=1
⇒c=13

Inserting value of c in (5)

a+13=−12
⇒a=−12−13
⇒a=−56

∴a=−56,b=12andc=13

Answer 2

Given T = k p^a d^b E^c, where k is a proportionally constant. Substituting the dimensions of T, p, d and E, we have

(T) = (ML^-1T^-2) ^a (ML^-3)^ b (ML^2T ^-2) ^c

Equating power of M, L, and T on both sides, we have

a + b + c = 0
–a –3b + 2c = 0
–2a – 2c = 1
Solving these equations, we get a = -5/6, b = 1/2 and c = 1/3