Understanding the contractile response of an animal skeletal system is important

Question

Understanding the contractile response of an animal skeletal system is important in bio-mechanical studies. Towards this end, the studies have measured velocity-force data and modeled it using the following equation. v = Fa/F + b (1 + 1/exp(-k(F-CF_degree)) Where v is velocity, F is applied force, and F_degree is measured force. The model has four constant parameters a, b, C, and k. If M is used to represent dimension of mass, t for dimension of time, and L for dimension of length, choose the correct option below by applying the principle of dimensional homogeneity. a has dimensions of t^2/(ML) b has dimensions of t^2/(ML) k has dimensions of t^2/(ML) Two choices, A and B, are correct None of the A-B-C choices are correct

Explanation / Answer

Dimensional homoginety says that an equation should have balanced units on each side.

force = mass*acceleration
   acceleration = meter/s2 or Lt-2

force = MLt-2

velocity = m/s = Lt-1

Lt-1 = (MLt-2 * a) / (MLt-2 +b) {1+1/exp(-k(F-CFo))}

Lt-1 = MLt-2 * a

a = Lt-1/ MLt-2

a = M-1t+1

similarly b should have same dimensions as force F as it is in addition

so b = MLt-2

similarly for exponential to be unitless ,the term k(F-CFo) has to be unitless

it contains only force and k while c cant effect as k*F(1-C Fo/F) both applied and measured force should have same units so in bracket only terms left is 1 and c so 1 doesn't have any units so as c should have no units.

now k*F should be unit less

k*MLt-2 = 1

k = M-1L-1t2

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