5BL S20 Lab 5: Pendulum
1. Introduction
When we go at slow speed, we walk. However, if we need to go
faster, we run. Can we not simply walk faster? Is there a
physical limit to how fast we can walk? And if so, does it
depend on the height or weight of a person?
In this lab you will investigate the pendulum and how it relates
to walking. Please see chapter 14.6 in Knight.
Required equipment:
Smartphone (stop-watch), string, tape-measure, small and a large known mass.
2. Experiment
Measurement 1: simple pendulum
A pendulum is a mass suspended from a pivot point by a light string or rod. The mass
oscillates back and forth along a circular arc as shown in figure 1. This oscillator is
driven by the weight of the mass. In this simple experiment you will measure the
frequency of a pendulum and its dependence on the mass m and length L of the string.
Figure 1. Geometry of the pendulum. The mass m oscillates back and forth along the circular
arc of radius L. The angle Θ and x=L٠Θ indicate the displacement from the equilibrium position.
The restoring force -mg sin(Θ) is the component of the weight tangent to the arc.
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Start by constructing a simple string pendulum consisting of a mass suspended by a
string from a pivot point, such as a hook in the ceiling or a table-top. When you displace
the mass horizontally from its equilibrium position and let it go, it will oscillate back and
forth. Using a stop-watch measure the period T of one full oscillation, i.e. the time
required for the mass to swing to the opposite side and back to where it started. Repeat
the measurement for three different lengths, for two different masses (so measure at
least 6 points). Your longest length should be at least three times your shortest length,
and the large mass should be at least twice the small mass. This experiment is
straight-forward. However, in order to produce high fidelity data, a lot of details must be
considered:
(1) The string should be light compared to the mass and should not stretch. Cooking
twine, sewing thread, or fishing line work well for example. You need to measure the
length from the pivot point to the center of mass of the oscillating object.
(2) The masses should be known and as compact as possible to mimic a point mass. In
other words, the size of your mass should be much smaller than the length of your
string. In our experiment we used ten quarters taped together (m = 57 g) as small mass,
and a bottle of detergent as the large mass (1.6 kg as indicated on the label).
(3) The human reaction time to visual stimuli is about 0.25 s and of the same order of
magnitude as the period you want to measure. To reduce the measurement errors you
should therefore measure the time ΔT required for several oscillations. The period can
then be calculated with much higher accuracy as T=ΔT/N, where N is the number of
oscillations per time ΔT. In our experiment we measured over N=10 full oscillations.
(4) In order to assure simple harmonic motion, do not displace the mass by more than
30 degrees (see section 3 as to why). In other words, the horizontal displacement from
the equilibrium should be small compared to the length of your string.
The frequency f = 1/T calculated from the measured period T is plotted in figure 2 as a
function of length L. As you can see, the frequency decreases with length, first fast and
then slower. Furthermore, the frequency appears to be independent of the mass.
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Figure 2. Measured frequency versus length for two different masses. The frequency decreases
with length, first fast and then slower. The mass does not seem to affect the frequency.
Measurement 2: the leg as a pendulum
The motion of a walking animal’s legs can be modeled as pendulum motion. When we
walk, our legs alternately swing forward about the hip joint as a pivot. In this experiment
you will measure the period T of your legs pendulum motion.
Place a low stool or a stack of books a foot or so away from the wall. Step your right
foot on the stool and gently place your right hand against the wall for stability. Then,
swing your left leg back and forth from your hips. Keep the leg straight and the muscles
relaxed, and let the momentum swing the leg. You may know this activity as a yoga
exercise (pendulum leg swings, or hip flexors).You should find that the period is
independent of the amplitude, unless you flex your muscles. If you swing your leg
further it simply moves faster but the period stays the same.
Figure 3. Measuring the period of pendulum leg swings.
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Just like in experiment 1 you should measure the duration of several full oscillations for
higher accuracy, as well as average over several repeated measurements. We
measured a duration of 15.9 seconds for ten full oscillations for a period of T=1.6 s. We
also repeated the measurement over 5 trials with different oscillating amplitude and
found a standard-deviation of 0.2 s between different measurements. Our period is
therefore 1.6 ∓ 0.2 s.
For the analysis below measure the length L of your leg from foot to hip. Next, measure
your relaxed walking speed and step size. The step size is defined as the distance from
the point where your left foot touches the ground to the point where the right foot
touches the ground. Measure the time Δt required to walk a certain distance Δx and
count your number N of steps. Your walking speed is then v=Δx/Δt, and your step size is
then Δs = Δx/N. The larger your distance the more accurate will be your measurements.
In our case it took Δt=11 s to walk a distance of Δx=10 m in N=16 steps. Our velocity
and step size were therefore v=0.91 m/s and Δs=0.63 m, respectively. You may also
use the ‘Location (GPS)’ tool in phyphox to measure your speed while walking at your
relaxed pace in a straight line. However, due to the poor spatial resolution of the satellite
based GPS (> 10 feet), you will need to walk a much larger distance for a high accuracy
measurement. Our GPS indicated a walking speed of 1 ∓ 0.2 m/s.
3. Evaluation and explanation of results
What determines the frequency of the pendulum?
The pendulum is a gravity driven oscillator and the restoring force is
where m is the oscillating mass, g=9.81 m/s2, Θ is the deflection angle from the vertical,
L is the length of the string, and x=L۰Θ is the displacement along the arc (when Θ is
measured in radians). The restoring force is zero at the equilibrium and increases with x
but not linearly. The pendulum is therefore not a SHO. However, for small angles Θ we
can approximate sin(x/L)≅x/L when x/L is expressed in radians. For example, for Θ=20o
= 0.349 rad we find that sin(0.349) = 0.342 differs only by 2% from 0.349 rad. The
discrepancy increases to 10% for angles of 45o. Therefore, as long as the amplitude
does not exceed about 20-30 degrees, the pendulum performs simple harmonic motion.
If we then approximate F = - (mg/L)۰x and compare with the restoring force for the SHO
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(F = - k۰x) we see that the effective spring constant of a pendulum is k = mg/L.
Substituting this effective k into the frequency formula for the SHO (equation 6 in the
week 3 lab manual) we find that the frequency of the pendulum is
The mass has cancelled out and the frequency of the pendulum only depends on the
length and the gravitational acceleration. This is consistent with our measurements
(figure 4).
Figure 4. Comparison of measured frequency versus length with the theoretical prediction
(equation 2). The frequency decreases with length but is independent of mass. See
https://trinket.io/glowscript/0f9cbe2430 for the python code used to create this plot.
The fact that the frequency of the pendulum is independent of its amplitude makes it a
powerful timing device. In fact, for more than 400 years and until the invention of the
quartz-oscillator in the 1970s, all clocks were pendulum based.
What is your maximum walking speed?
Animals evolve towards maximum energy efficiency. The most efficient way for
land-animals to move is walking. Walking is essentially a controlled fall forward, while
the legs perform synchronized pendulum motion to keep you off the ground. Energy is
exerted mostly to lift one foot barely off the ground, while the forward momentum swings
the legs. The leg that touches the ground acts like an inverted pendulum, while the
airborne leg performs regular pendulum motion (figure 5).
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Figure 5. The leg that touches the ground (shown in red) acts like an inverted pendulum. The
airborne leg swings like a regular pendulum. The step size is Δs and Fc is the centrifugal force.
Unlike a simple point-mass pendulum, the leg serves both as mass and rod. Different
parts of the leg swing at a different length. This is a so-called “physical pendulum”. The
oscillating frequency of a physical pendulum depends on the solid object's moment of
inertia. Treating the leg as a uniform rod of length L, its pendulum frequency is
where the additional 3/2 term stems from the correction for the elongated shape of the
mass (see example 14.10 in Knight). The frequency of the swinging legs is fixed by their
length and their spatial distribution of mass. It does not depend on the amplitude, or the
total mass of the animal.
For our measured length of L=0.97 m we calculate a frequency of f=0.62 Hz. This
agrees remarkably well with our measured frequency of f=1/T = 1/(1.6 s) = 0.63 Hz.
Knowing the step-size we can calculate our relaxed walking speed. One full pendulum
oscillation is actually two steps, i.e. left foot forward and then the right foot forward. The
relaxed walking speed is then
For our measured step size of Δs=0.63 m and period of T=1.6 s we calculate a relaxed
walking speed of v = 0.79 m/s, which is quite close to our measured velocity of 0.91
m/s. The small difference is due to the fact that when you walk you slightly bend your
knees, which effectively reduces the length of your legs and the period, without
changing the step-size. As you can see in figure 5, the step size Δs = 2۰L۰sin(Θ) is
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proportional to the length L of the leg, where Θ is the maximum angle a leg will make
relative to the vertical. The maximum leg angle we calculate for our measurements is
Θ=arcsin(Δs/2L) = 19o. You may also measure this angle from a photograph of the leg
at its largest angle using imageJ.
The relaxed walking speed is proportional to Δs/T, while Δs is proportional to L, and T is
proportional to the square root of L. The speed is therefore proportional to the square
root of L:
This means that an animal with long legs walks at a faster speed. Giraffes with their 2
meter long legs have a cruising speed of 10 mph. You can somewhat increase your
walking speed with additional muscle power at the expense of energy efficiency. For
example, step size may be increased for a given leg length by increasing the maximum
leg angle. Changing the oscillating frequency is also possible but difficult. An oscillator
that is driven by an external force (such as your muscles) always oscillates at the drive
frequency. However, the efficiency of a driven oscillator peaks when it is driven at its
natural frequency, and decreases quickly at higher or lower frequencies. You may force
your legs to swing at a higher frequency to increase the walking velocity past its relaxed
pace but not for long.
Is there an upper limit to how fast we can walk? The path that the body takes during a
stride is the arc of a circle. The radius of this circular motion is the length L of the leg
(see figure 5). The body tries to lift off as it pivots over the foot but is held in place by its
weight. The upward centrifugal force Fc = mv2/L increases with the square of the walking
speed. The maximum possible walking speed vmax occurs when this upward pointing
centrifugal force equals the weight mg:
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For faster speeds the magnitude of the centrifugal force exceeds the magnitude of the
weight and the net-force points up. The maximum possible walking speed also
increases with the square root of the length of the leg. The mass has cancelled out so it
does not depend on the weight of the animal. For our leg we calculate a maximum
speed of 3.0 m/s or 6.7 mph and three-times that of our relaxed walking speed. If we
want to move any faster we must lift off the ground and run.
4. Deliverables
For full credit, please include the following in your lab report. Follow the template
provided on the Weebly Lab 5 page and include one deliverable per Google slide in the
order that they are presented below. Always use proper units and label your plots.
1.
2.
3.
4.
5.
6.
7.
8.
Photograph of your pendulum setup.
Table of your T vs L and m data from experiment 1.
f-versus-L graph of all your data and comparison to theory (similar to figure 4).
Measurement of your leg pendulum period and comparison to equation 3.
Measurement of your walking speed and comparison to theory (equation 4).
Calculation of your maximum walking speed (equation 6).
Calculation of your maximum leg deflection angle Θ for your measured Δs.
Enter one of your pendulum T vs L measurements in the crowdsourced
spreadsheet. What is that data point?
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