A gas bubble from an explosion under water oscillates with a period T proportional to pap "E where P is the statie pressure, p is the density of water and E is the total energy of the explosion. So, values of a, b and care ..... ... and ... respectively.

A gas bubble from an explosion under water oscillates with a period T

1.	A gas bubble from an explosion under water oscillates with a period T proportional to pap "E where P is the statie pressure, p is the density of water and E is the total energy of the explosion. So, values of a, b and care ..... ... and ... respectively.
Answer» <P> SOLUTION :Here `T prop p^(a) rho^(b) E^(c)` is GIVEN. `:. T=kP^(a) rho^(b)E^(c)` where k is a dimensionless constant. `[T]=T^(1),[K]=M^(0)L^(0)T^(-1)` `:.[T]=[k][P]^(a)[rho]^(b) [E]^(c)` But `[P]=M^(1)L^(-1)T^(-2)` `:.T^(1)=(M^(1)L^(-1)T^(-2))^(a)(M^(1)+L^(-3))^(b)(M^(1)L^(2)T^(-2))^(c)` `( :.[rho]=M^(1)L^(-3)T^(0)[E]=M^(1)L^(2)T^(-2))` `:. T^(1)=M^(a+b+c)L^(-a-3b+2c)T^(-2a-2c)` COMPARING powers of T, we get `-2a-2c=1:.a+c=-(1)/(2)...(i)` Comparing powers of M, we get `-a-3b+2c= :.2c=a+3b=a+(3)/(2)` [ From eq. (ii) `:.2c-a=(3)/(2)...(iii)` On addition of eq. (i) and (ii) `a+c=-(1)/(2)` `(+-a+2c=(3)/(2))/(3c=1)` `:.c=(1)/(3)...(iv)` Putting value of c in eq (i) we get `a+(1)/(3)=-(1)/(3)` `:. a=-(1)/(2)-(1)/(3)` `:.a=-(5)/(6)...(V)` From eq. (ii), (iv) and (v), `a=-(5)/(6),b=(1)/(2) and c=(1)/(3)`