F = k hx 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
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