Physical basis for measuring ‘g’ using a simple pendulum:
Before beginning, read these instructions and discuss with your team members or partner:
Is it best to have a long, slow pendulum or a short, fast pendulum? Which would probably result in lower uncertainties?
Is one oscillation enough to time, or is there an advantage in timing a set number of oscillations and then dividing total time by the number of cycles?
How confident are you in measuring length of your pendulum?
Take care that your support system is rigid and not oscillating. Coupled oscillations involve more complex theories and you don’t want to go there as part of this class.
If one value differs significantly from the others when it comes time to calculate average period T, do you ignore the outlier, include it in calculating an average, or ignore it and repeat the measurement, replacing the suspect value with a new value?
Is it best to time oscillations as the pendulum comes to rest at the top of the swing, or when it is traveling most rapidly at the bottom of its swing? Why?
My assistant will supply you with materials useful in building your pendulum and measuring its length.
Any debate need (should) not be recorded and made part of your report. You should, however, report how you made the measurements.
l (Length of rod = l, not 1 <one>).
Consider the rotational motion of a simple pendulum. When the pendulum of
mass m is displaced an angle θ
from its equilibrium position, the force of gravity mg exerts a torque
about the axis O that tends to restore the pendulum to equilibrium. The
magnitude of this torque is 
. For small angles, sin(θ)
» θ (when the angle is expressed in radians)
so we can write (1) 
. The negative sign indicates that
torque is opposite in direction from the direction of displacement. This torque
imparts on the mass an angular acceleration (2)
where (3) 
.
Because 
we can write (4)
.
I is the moment of inertia which, for a point mass at the end of a rigid zero-mass rod of length l, is ml2.
Rearranging terms, we write the equation as (5) 
. This
is called a differential equation because it includes not only a
variable (angular displacement, θ)
but also derivatives of that variable. Equations may contain
derivatives of time, space, or both.
Solving a differential equation involves (1) guessing a solution and (2) testing that guess. If the guess is successful, we have solved the equation; if not, we have to try again. Classes in differential equations teach you how to set up (that is, write) the equation and then how to make a good first guess, based on mathematicians’ research. In this case, let’s guess that the solution is
(6) 
where ω=2πf
where f = frequency in cycles/second (or Hertz).
Take the first derivative: (7) 
and then the second
derivative
. Substituting the hypothetical solution and its second derivative into
(9) 
(and dividing both sides by ml2)
we write
(10)
and examine the result. Note that the solution is independent of mass m!
Note also that this equation is true only if (11)
or (12) 
.
Rearranging, (13) 
.
l can be measured, as can f. So we solve for g
and write (14) 
.
The period T of one oscillation is the time required for one
oscillation. T is the reciprocal of frequency f.
That is, 
and
.
The equation for g written in terms of period T is (15)
.
LAB 1: Construct a simple pendulum and measure g. Work in teams of 2 or 3. Make 10 or more independent measurements and calculate an average and standard deviation of your g values. Measure the time required for a number of oscillations and then divide the total time by the number of oscillations to obtain period T. Your report (due at 9:00 AM on August 29 – my watch) should include the length of your pendulum, how you measured l, and a table summarizing results. Please follow the standard rules regarding the use of significant figures.
Excel has built-in functions for calculating average and standard deviation, as do most scientific calculators. Please use built-in functions instead of calculating required quantities the long way.