Scientists sometimes will use a process called dimensional
analysis to predict the form of the relationship between
variables in a system . While this does NOT constitute a theory base, it does
give scientists a better understanding of
what to expect experimentally, and helps them to de sign scientific experiments.
Carefully observe the fol lowing
process of dimensional analysis for a cart accelerating down a fixed inc lined
plane .
Let’s say that a scientist observes the motion of the cart
on the fixed incline, and after conducting a few qualitative
experiments concludes that the distance of the cart is a function of its
accele ration and the time . As sume that the
cart accelerates from rest when the stopwatch starts. What is the expected form
of the relationship between
distance, acceleration, and time?
The form of this relationship can be predicted using an
approach known as dimensional analysis. This process is
based on the knowledge that the distance (d) is a function of the acceleration
(a) and the time (t). That is,
Now, if d (expressed in meters, m) is related to both a
(expressed in meters per second squared , m/s2) and t
(expressed in seconds, s) in some form, then the only thing missing is a
proportionality constant and the powers of
the variable terms. Write a proportionality applying power terms x and y to a
and t respectively Replace variables
by units and solve for the power terms as, in this case, two simultaneous
equations with two unknowns . Working
backward, then find the form of the equation and insert proportionality
constant. (Note that if the unit of time, s, is
present on the right side of the equation, it must also be present on the left
side of the equation. Place s0 on the left
side of the equation in step two below as shown; s0 is equal to 1 and d one not
affect the equality.)
replacing variables with units
simplifying
equating exponents on m and s
1= x
0 = y −2x
and solving simultaneous equations
y = 2x
hence, y = 2
thus,
or, d = kat2
Experiments can be conducted or theoretical work performed
to find the value of the constant , k. In reality , k equals
1/2 giving the familiar kinematic equation
