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 ≈ 9.806 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]T = 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]T = [1 + 12/22sin2(θ/2) + 1232/(2242)sin4(θ/2) + 123252/(224262)sin6(θ/2) …]*2π√(L/g)
[3] T = [1 + (1/16)θ2 + (11/3072)θ4 + (173/737280)θ6 + (22931/1321205760)θ8…]*2π√(L/g)
[4]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) / T10 / 1.001907*2π√(L/g)
20 / 1.007669*2π√(L/g)
30 / 1.017409*2π√(L/g)
40 / 1.031341*2π√(L/g)
50 / 1.049783*2π√(L/g)
60 / 1.073182*2π√(L/g)
70 / 1.102145*2π√(L/g)
80 / 1.137493*2π√(L/g)
90 / 1.180341*2π√(L/g)
If a pendulum does not have most of its mass located near a point, then it is referred to as a physical pendulum. One can use the rotational analogue to Newton’s second law to predict the period of such a pendulum.
Στ = Iα = -mgd*sinθ
d = distance from center of mass to pivot point
I = moment of inertia
We again use the approximation sinθ ≈ θ, then solve for α.
α = -(mgd/I)θ
This is a solvable differential equation, so we can predict the period and angular frequency.
ω = √(mgd/I)
T = 2π√[I/(mgd)]
The above equation is a very general formula. Now let’s analyze the specific situation of a rod with an arbitrary pivot point. A table and the parallel axis theorem provide a formula for the moment of inertia which can be substituted into the formula for period.
I = (1/12)mL2 + md2
L = length of rod
[5]T = 2π√{[(1/12)L2 + d2]/(gd)}
Physics is fun!
Pre-Lab Activity
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. Show your results to your instructor, but do not include this table in your report.
θ (°) / θ (rad) / sin(θ) / % difference= 100*[θ – sin(θ)]/sin(θ) / sin(θ) ≈ 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 Simple Pendulum
1)Assemble the H-bar support system. Cut alength of support cord of several meters in length to be usedthroughout the experiment. Suspend the 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 and measure the length of the pendulum from the pivot point to the middle of the bob. Use a protractor to measure the release angle. Check if sinθ ≈ θ 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 a wide variety of lengths. You may go outside or to building 2 for this experiment to test a very large pendulum.
5)Calculate the theoretical period for each length using formula [1]. Consult the general lab instructions regarding graph formats. Graph the experimental and theoretical periodsas a function of lengthon a single graph.Evaluate the formula for period of a simple pendulum.
The Relationship between Period and Mass for a Simple Pendulum
1)Replace the bob with a hooked mass and choose a length of at least 50 cm. Bigger is better. Measure the period using the methoddescribed above.
2)Repeat the above step using several more hooked masses.Since the masses may be different sizes, you may need to adjust the length of the cord in order to maintain a constant distance from the pivot point to the center of mass.
3)Graph the experimental and theoretical period as a function of mass on a single graph. 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.
Period of a Physical Pendulum
1)Use a skinny crossbar to support the physical pendulum at a pivot point location of 5 or 10 cm. Record the location of the pivot point.
2)Use a protractor to measure the release angle. Check if sinθ ≈ θ for your choice of release angle.
3)Measure the amount of time it takes the pendulum to complete 5 cycles. Calculate the experimental period by dividing the measured time by 5.
4)Calculated, the distance from the center of mass to the pivot point. Note that this is not the location of the pivot point.
5)Calculate the theoretical period using equation [5].The length, L, will be a fixed value of approximately 1 m, while the distance from the pivot point to the center of mass, d, will vary.
6)Repeat steps 1 through 5, in increments of 5 or 10 cm until you reach a d of 5 cm.
7)Graph the experimental and theoretical period as a function of d on a single graph. Compare your results and evaluate the efficacy of equation [5].
[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.