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A particle with mass m and initial speed `V_(0)` is a subject to a velocity-dependent damping force of the form `bV^(n)`.With dimensional analysis determine how the stopping time depends on m, `V_(0)` and b for begin with writing `Deltat=Am^(alpha)b^(beta)V_(0)^(gamma)`, powers `alpha, beta` and `gamma` will be.A. `alpha=1, beta=-1, gamma=1-n`B. `alpha=2-n, beta=-1, gamma=2`C. `alpha=1, beta=1, gamma=1-n`D. `alpha=-1, beta=-1, gamma=1-n` |
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Answer» Correct Answer - A `F=bv^(n)` `b=F/v^(n)=[(M^(1)L^(1)T^(-2))/((L^(1)T^(-1)))]^(beta)` `Deltat=A m^(alpha) b^(beta) v_(0)^(gamma)` `M^(0) L^(0)T^(1)=A M^(a+b)xxL^(beta-n)T^(-2beta+n)xxL^(gamma)T^(gamma)` By comparison of power of MLT `alpha+beta=0` `beta-n+gamma=0 rArr beta+gamma=n` `-2beta+n-gamma=1 rArr +2beta+gamma=n-1` `beta+n=n-1` `beta=-1` `alpha=1` `gamma=1+n` |
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