**(New tips are continually
added to these pages. Check back in a few months' time for more)**

**TOPIC
1: Units, Dimensions, Errors and Uncertainties**

__Tip 1:__

What is the difference between Errors and Uncertainties? Errors are uni-directional whereas Uncertainties are bi-directional. Therefore, Errors can be added or subtracted, but Uncertainties can only be added. (Note: this statement may sound paradoxical but the following example shall illustrate).

For example, let's use the formula for the period of pendulum:

*T* = 2p
Ö(*l*/*g*)

__Error__

The length of the
pendulum, *l*, which is supposed to be 0.80 m, is measured as 0.86 m (with an error of +0.06m).
The acceleration due to gravity, *g*, which is supposed to be 9.81 m s^{-2},
is measured as 9.84 m s^{-2} (with
an error of +0.03 m s^{-2})

To find the error that
results in the measurement of the period *T*, we have to first find the fractional
error of *l* and *g*.

D*l*
/ *l* = 0.06 / 0.80 = 0.075

D*g*
/ *g* = 0.03 / 9.81 = 0.003

Since *l* is in the
numerator, its positive error will result in an increase in the period *T*.
Since *g* is in the denominator, its positive error will result in a decrease in the
period *T*. Therefore, the
fractional error of *l/g* will be 0.075 - 0.003 = 0.072

Since *l/g* is square-rooted
(ie. to the power of 1/2), the fractional error of of D*l/g*
will be 1/2 x 0.072 = 0.036.

Next, evaluate the period
*T* by substituting *l* = 0.80m and *g* = 9.81 m s^{-2}
, we find that *T* = 1.794s

The error of *T* will be its
value (1.794 s) multiplied by the fractional error (0.036), the answer is 0.065,
or 0.07 after rounding off to 1 significant figure. The erroneous value is
therefore 1.79 + 0.07 = 1.86 s.

Let's check the answer by
substituting *l* = 0.86m and *g* = 9.84 m s^{-2}
into the equation, the answer is 1.86 s.

__Uncertainty__

Using the same example,
instead of a positive error of +0.06 m for the length *l*, we have an uncertainty
of __+__0.06 m; and instead of a positive error of +0.03 m s^{-2}
for the acceleration due to gravity *g*, we have an uncertainty of __+__0.03 m s^{-2}.
Then the total fractional uncertainty is 0.075 + 0.003 = 0.078.

The uncertainty of *T* will
be its value (1.794) multiplied by the fractional uncertainty (0.078), the
answer is 0.14.

Therefore, period *T* =
1.794 __+__ 0.14, but since uncertainties have to be rounded upward to only have 1 significant figure,
the proper presentation is *T* = (1.8 __+__ 0.2) s.

© **Copyright Physics.com.sg (Registration No. 52890077C). All rights reserved.**

® *First Class in Physics Tuition* is the
Registered Trademark (TM No. T02/02149B) of Physics.com.sg

.