Physics Online
Pendulum
Introduction
The purpose of this lab is to see if we can predict the period of a pendulum under a variety of
conditions.
We can use Newton’s second law to predict the period of a simple pendulum. If we use a polar
coordinate system, then the tangential acceleration can be calculated as follows:
ΣFtan = ma
-mg*sinθ = ma
m = mass of bob
g = magnitude of free fall acceleration ≈ 9.8 m/s2
θ = angle (measured with vertical)
a = -g*sinθ
This equation is extremely difficult to solve for the period, but we can make an assumption to simplify
the analysis. If θ in radians is “small”, then sinθ ≈ θ. Using the definition of the radian, θ = x/L where
x = displacement and L = length of the pendulum. Putting this all together, the expression for
acceleration of the pendulum bob when the angle is small is
a = -(g/L)x.
The mathematical relationship between the acceleration and the displacement looks very similar to that
of the mass on a spring. This is a differential equation whose solution for the position as a function of
time is the following:
x(t) = A*cos(ωt + φ)
ω = angular frequency = √(g/L)
A = amplitude
φ = phase angle = 0 if released from rest
Since ω = √(g/L) and T = 2π/ω, then we finally have the following:
[1]
T1 = 2π√(L/g)
Note that this formula only works when the assumptions are true. This includes the approximation
sinθ ≈ θ. In reality, sinθ < θ on the relevant domain of 0 < θ < π/2 rad, and the difference between the
two gets larger with increasing angles. To compensate for this difference, one can use a more refined
formula such as one of the following:
[2]
[3]
[4]
T = [1 + 12/22sin2(θ/2) + 1232/(2242)sin4(θ/2) + 123252/(224262)sin6(θ/2) …]*2π√(L/g)
T = [1 + (1/16)θ2 + (11/3072)θ4 + (173/737280) θ6 + (22931/1321205760)θ8…]*2π√(L/g)
T = [cos(θ/2)]^-{0.5*[cos(θ/2)]0.125}*2π√(L/g)
Equation [2] is a theoretically exact solution, but needs a large number of terms to converge for large
release angles. Equation [3] requires fewer terms to converge, but the coefficients do not follow a
simple pattern. Use radians in this formula. Equation [4] has no theoretical justification, but gives
reasonable approximations without the use of an infinite series. Equation [2] with lots of terms gives
the following:
θ (degrees)
10
20
30
40
50
60
70
80
90
T
1.001907*2π√(L/g)
1.007669*2π√(L/g)
1.017409*2π√(L/g)
1.031341*2π√(L/g)
1.049783*2π√(L/g)
1.073182*2π√(L/g)
1.102145*2π√(L/g)
1.137493*2π√(L/g)
1.180341*2π√(L/g)
Physics is fun!
Equipment You Procure
 digital camera
 string
 scissors
 metric tape measure
 protractor (optional)
 large pendulum bobs (optional)
Equipment from Kits
 pendulum bobs
 C clamp
 scale
 stopwatch
Pre-Lab Calculations
Complete the following table using Excel. Use columns 2 and 3 to calculate column 4, the percent difference. If
this difference is small, according to your somewhat subjective standard, then you should answer “yes” for
column 5. Include this table in the calculations section of your report.
θ (°)
θ (rad)
sin(θ)
% difference
sin(θ) ≈ θ ?
= 100*[θ – sin(θ)]/sin(θ)
(yes/no)
10
20
30
40
50
60
70
80
90
Experimental Procedures
Note: for all of the experiments you are certainly welcome and encouraged to try very large lengths. The lengths
are only suggested values.
The Relationship between Period and Length for a Small Angle Simple Pendulum
1) Attach the C clamp to a door jam or other stable location with at least 1 meter of clearance. Cut a length of
string approximately 1 meter in length to be used throughout the experiment and attach it to the C clamp.
Suspend a pendulum bob from the end of the cord. Don’t tie a knot at the pivot point because you will be
adjusting the length.
2) Choose a length of approximately 15 cm and measure the length of the pendulum from the pivot point to
the middle of the bob. Use a protractor (or trigonometry) to measure the release angle. Check if sinθ ≈ θ
(in radians) for your choice of release angle. If this is not true, then reduce your release angle.
3) Measure the amount of time it takes the pendulum to complete 10 cycles. Calculate the experimental period
by dividing the measured time by 10.
4) Repeat steps 2 and 3 for lengths of approximately 25 cm, 35 cm, 45 cm, 55 cm, 65 cm, and 75 cm. These are
only suggested values and you are certainly encouraged to super-size this experiment.
5) Calculate the theoretical period for each length using formula [1]. Read the general lab instructions regarding
graph formats. Make a single graph of the experimental and theoretical periods as a function of length. Test
the claim that formula [1] can be used to predict the period of a simple pendulum.
The Relationship between Period and Mass for a Simple Pendulum
1) Replace the bob and adjust the cord so that the length of the pendulum is the same as it was in the last test
(approximately 75.0 cm). Measure the period using the method described above.
2) Attach both bobs to the end of the cord and measure the period using the method described above.
3) Measure the masses of each of the pendulum bobs.
4) Graph the experimental and theoretical period as a function of mass on a single graph. You should have
three different masses (two individuals and combined) for a single length. Test the claim that period is
independent of mass.
The Relationship between Period and Angle for a Simple Pendulum
1) Attach a single bob and measure the length of the pendulum. Measure the amount of time it takes the
pendulum to complete 5 cycles from a release angle of 10°. Measure the angle at the end of the cycle.
2) Calculate the experimental period by dividing the measured time by 5.
3) Calculate the average of the release and ending angles.
4) Calculate the theoretical period using the average angle and equation [2], [3], [4], or the table (with
interpolation if needed).
5) Repeat the above steps using release angles of 30°, 50°, 70°, and 90°.
6) Graph the experimental and theoretical period as a function of average angle on a single graph. Test the
theory that a significant increase in the angle will increase the period in a detectable and predictable manner.
[2] Young and Freedman, University Physics, 11th edition, 2004, page 496.
[3] Nelson, Robert; M. G. Olsson "The pendulum - Rich physics from a simple system". American Journal of Physics
[4] Hite, Gerald E., The Physics Teacher, Vol. 43, May 2005, page 291.