Physics for Scientists & Engineers & Modern Physics, 9th Ed

1.4 
Pitfall Prevention 1.3
Always include units When per-
forming calculations with numeri-
cal values, include the units for 
every quantity and carry the units 
through the entire calculation. 
Avoid the temptation to drop the 
units early and then attach the 
expected units once you have an 
answer. By including the units in 
every step, you can detect errors if 
the units for the answer turn out 
to be incorrect.
 
Write an expression for a with a dimensionless constant 
of proportionality k:
a 5 kr
n
v
m
Substitute the dimensions of a, r, and v:
L
T
2
5 L
n
a
L
Tb
m
5
L
n1m
T
m
Equate the exponents of L and T so that the dimen-
sional equation is balanced:
n 1 m 5 1 and m 5 2
Solve the two equations for n:
n 5 21
Write the acceleration expression:
a 5 kr
2
1
v
2
5 k 
v
2
r
In Section 4.4 on uniform circular motion, we show that k 5 1 if a consistent set of units is used. The constant k would 
not equal 1 if, for example, v were in km/h and you wanted a in m/s
2
.
Example 1.2 
Analysis of a Power Law
Suppose we are told that the acceleration a of a particle moving with uniform speed v in a circle of radius r is propor-
tional to some power of r, say r
n
, and some power of v, say v 
m
. Determine the values of n and m and write the simplest 
form of an equation for the acceleration.
S o L u t i o n
Identify the dimensions of a from Table 1.5 and multiply 
by the dimensions of t:
3at4 5
L
T
2
T 5
L
T
Therefore, v 5 at is dimensionally correct because we have the same dimensions on both sides. (If the expression were 
given as v 5 at
2
, it would be dimensionally incorrect. Try it and see!)
▸ 1.1
c o n t i n u e d