Solution of Dimensional Analysis Problems

1)     Use the method of dimensions to obtain a formula for the force experienced by a sphere of radius r moving at velocity v through a fluid of viscosity h.  (Hint: Set out the force as:           
  F = k h^x v ^y r ^z  )


Solution: Use the table for dimensions to write them out for each factor:


For Force :  F has dimensions M L T ^-2


Velocity v  has dimensions L T ^{- 1}


h has dimensions M L^-1 T ^-1


r has dimension L


Then, equating dimensions on both sides:


 M L T ^– 2  = [M L^-1 T ^-1]^x [LT ^{- 1}] ^y [L]^z

We next equate indices for M, L, and T on both sides:

For M:  1 =   x 

 

For L: 1  = -x  - y   + z 

 

For T:   -2 = -x – y

 

Next, solve for each of the indices:

a)  x =  1

b) y =   1

c) z =   3

 

Finally:  F = k h v  r  ³  

2)  Repeat the exercise above to obtain the equation  for the period (T) of a pendulum’s swing. (Hint: The pendulum's period T should depend on its length L and the acceleration of gravity, g.)

Solution: Write out the provisional eqn.:

t  =    g ^x  M  ^y r ^z

Write out dimensions for each side:

[T]   =  [ L T ^{- 2} ] [M]y  {L] ^z 

Solving for the indices:

1 = - 2x or x = - 1/2

0 = y

0 = x + z or x = - z = - (-1/2) = 1/2

Then:

t  =    k g - 1/2   ℓ  1/2

(k  will be found  by separate analysis to be:  2 p)

So that:

t  =   2 p  Ö ℓ  / Ö g