InterviewSolution
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InterviewSolution
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Find the value of a and b in the following cases : (a) The velocity `v` of the ball falling freely under gravity is proportional to `g^(a) h^(b)`, where `g` is the acceleration due to gravity, `h` is the height from which the ball is dropped. (b) The kinetic energy `K` of a rotating body is proportional to `I^(a) omega^(b)` where `I` is the moment if inertia and `omega` is the angular speed. (c ) The time-period `T` of spring pendulum is proportiona to `m^(a) k^(b)`, where `m` is the mass of block attached to the spring and `k` is the spring constant. The speed of sound `v` in a gaseous medium is proportional to `P^(a) rho^(b)`, where `P` is the pressure and `rho` is the density of medium. |
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Answer» (a) `v prop g^(a) h^(b)` `[LT^(-1)] prop [LT^(-2)]^(a) [L]^(b)` `L^(1) T^(-1) prop L(a + b) T^(-2a)` Comparing powers of `L` and `T` `a + b = 1` `-2 a = - 1` `a = (1)/(2), b = (1)/(2)` `K prop I^(a) omega^(b)` `[ML^(2) T^(-2)] prop [ML^(2)]^(a) [T^(-1)]^(b)` `M^(1) L^(2) T^(-2) prop M^(a) L^(2a) T^(-b)` Comparing powers of `M` and `T` `a = 1` `-b = - 2` `a = 1, b= 2` (c ) `T prop m^(a) k^(b)` `[T] prop [M]^(a) [MT^(-2)]^(b)` `M^(0) T^(-1) prop M^(a + b) T^(-2b)` Comparing power of `M` and `T` `a + b = 0` `-2 b = 1` `a = (1)/(2), b = - (1)/(2)` (d) `v prop P^(a) rho^(b)` `[LT^(-1)] prop [ML^(-1) T^(-2)]^(a) [ML^(-3)]^(b)` `M^(0) L^(1) T^(-1) prop M^(a + b) L^(-a - 3b) T^(-2a)` Comparing powers of `M` and `T` `a + b = 0` `- 2a = - 1` `a = (1)/(2), b = - (1)/(2)` |
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