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Physical Pendulum – Period EX- 5518 Page 1 of 4
Physical Pendulum - Period
Equipment
INCLUDED:
1 Large Rod Stand ME- 8735
1 45 cm Long Steel Rod ME- 8736
1 Physical Pendulum Set ME- 9833
1 Rotary Motion Sensor PS- 2120
NOT INCLUDED, BUT REQUIRED:
1 850 Universal Interface UI- 5000
1 PASCO Capstone UI- 5400
Introduction
This experiment explores the dependence of the period of a physical
pendulum (a uniform bar) on the distance between the pivot point and the
center of mass of the physical pendulum.
Think about the two extremes:
- When the pendulum is pivoting about the end, do you expect that the
period is longer or shorter than if it is pivoting about the first hole
down?
- When the pendulum is pivoting about the center of the rod, what will
the period be?
Since the period starts to get less as the pivot is moved toward the center,
but the period is infinitely long at the center, there should be a place where
the period is a minimum. You will apply simple calculus to find the minimum
period.
You will also develop a mathematical model for the system and compare to
theory. By designing a free parameter (the length of the bar) into the model,
the length of the bar can be accurately inferred, showing some of the power
of computer models.
Theory
The period of a physical pendulum is given by
for small amplitude (the error is less than 1% at 20
o
). I is the rotational
inertia of the pendulum about the pivot point, m is the total mass of the
pendulum, and x is the distance from the pivot to the center of mass. A
Physical Pendulum – Period EX- 5518 Page 2 of 4
x
Pivot Point
Center of
Mass
uniform rectangular bar has a rotational inertia about its center of mass
given by
𝑐𝑚
1
12
2
2
where m is the mass, L is the length and w is the width of the bar. For the
28 - cm Pendulum Bar (w/L)
2
< 0.003 and we can simplify this expression to
𝑐𝑚
1
12
2
with an error of only 0.3%.
The parallel axis theorem enables us to
write the rotational inertia of the bar
about a pivot point a distance x from the
center of gravity as
𝑐𝑚
2
and Equation (1) becomes
𝑚(𝐿
2
/ 12 )+𝑚𝑥
2
𝑚𝑔𝑥
1
12
𝐿
2
+𝑥
2
𝑔𝑥
Use calculus to find the derivative of the
period, T, with respect to x, and set it
equal to 0 to find the value of x that will
produce the minimum period.
Physical Pendulum – Period EX- 5518 Page 4 of 4
Analysis
- Find the period of oscillation for each position of the pivot.
a. Select Run #1 on the graph.
b. Click the Coordinates Tool button in the toolbar. Move the
Coordinates Tool to one of the first peaks of Angular Position.
c. Right-click on the Coordinates Tool and turn on the Delta Tool.
Measure the period by measuring the time for 10 periods and
divide by 10.
d. In the table, enter the period you measured in the T column
beside the zero in the distance column.
e. Repeat for each of the seven runs.
- Determine which distance gives the minimum period of oscillation of
the pendulum bar and compare to what you calculated in the theory
section.
Theoretical Curve Fit
- Apply a User-Defined fit to the data on a Period vs. Distance graph. Use
Equation (5) and lock in the appropriate numbers in the Curve Fit
Editor.
- How well does the theoretical curve fit your data? Does the curve fit
better for some of the points than others? Does making a selection of
only some of the points make the curve fit better?
- Unlock the parameter which depends on the length of the pendulum.
Update the fit and calculate the length of the pendulum from the new
parameter value. What length does this give? How close is it to the
actual length of the pendulum?
Questions
- What is the percent difference between the calculated value for the
length that gives minimum period of oscillation and the measured
value for the length?
- Would a pendulum bar with different mass but with the same
dimensions have a different value for the length that gives minimum
period of oscillation? Why or why not?
Physical Pendulum – Inertia EX- 5518 Page 1 of 6
Rotational Inertia Based on Period of Oscillation
Equipment
INCLUDED:
1 Large Rod Stand ME- 8735
1 45 cm Long Steel Rod ME- 8736
1 Physical Pendulum Set ME- 9833
1 Clamp-on Pulley ME-9448B
1 Rotary Motion Sensor PS- 2120
NOT INCLUDED, BUT REQUIRED:
1 850 Universal Interface UI- 5000
1 PASCO Capstone UI- 5400
Introduction
The period of oscillation of a physical pendulum will be measured and used
to calculate the rotational inertia of the pendulum. The rotational inertia is
also determined by measuring the mass and the dimensions of the
pendulum.
Theory
The period of oscillation, T, of a physical
pendulum depends on the rotational
inertia about a pivot point, I pivot
, the mass,
M, and the distance from the pivot point to
the center of gravity, d.
𝑝𝑖𝑣𝑜𝑡
The Parallel Axis Theorem states that the
rotational inertia about a pivot point, I pivot
, is equal to the sum of the
rotational inertia about the center of mass, I cm
, and the rotational inertia of
the object as if all its mass was at the center of mass:
𝑝𝑖𝑣𝑜𝑡
𝑐𝑚
2
Conversely, the rotational inertia about the center of gravity could be found
as follows:
𝑐𝑚
𝑝𝑖𝑣𝑜𝑡
2
Figure 1. Pivot and center of gravity of the
disk pendulum
Physical Pendulum – Inertia EX- 5518 Page 3 of 6
- Repeat the process for several more trials. Determine the average of
the values of the period of oscillation and record the average value in
the table.
Analysis
- Using Equation (5), calculate the rotational inertia about the center of
gravity using the period, T, the mass, M, and the distance from the
pivot point to the center of gravity, d.
- Record the calculated value for the rotational inertia in a table.
Part II: Finding Rotational Inertia by Measuring the
Dimensions and Mass
Theory
The theoretical rotational inertia of a disk (radius R and mass M) about its
center of mass given by
1
2
2
Procedure
- Measure the diameter and the mass of the disk.
- Calculate the rotational inertia of the disk about its center of mass.
Questions
- Determine the percent difference between the rotational inertia
calculated from the period and the rotational inertia calculated using
the dimensions.
- Do your results support or disprove the idea that the rotational inertia
of a physical pendulum can be determined from its period of
oscillation. Why or why not?
Extensions
Repeat the procedure for the thin ring, thick ring, and offset hole, each
pivoting about the outside edge. For the offset hole, solve for the rotational
inertia about the pivot, rather than about the center of mass. To find d for
this object, balance the disk flat on the edge of a table and make a small
Physical Pendulum – Inertia EX- 5518 Page 4 of 6
pencil mark at the center of mass (where the edge of the table is when the
disk is balanced).
Irregular Shape
For the irregular shape, it is not possible to find the rotational inertia from
the dimensions. However, it can be found by accelerating it with a known
torque.
To find the rotational inertia of the ring and disk experimentally, a known
torque is applied to the ring and disk, and the resulting angular acceleration,
α, is measured. Since τ = Iα,
𝜏
𝛼
where τ is the torque caused by the weight hanging from the string which is
wrapped around the large step of the 3 - step pulley of the apparatus.
τ=rF (4)
where r is the radius of the pulley about
which the string is wound and F is the
tension in the string when the apparatus
is rotating. Also, a=rα, where "a" is the
linear acceleration of the string.
Applying Newton's Second Law for the
hanging mass, m, gives (See Figure 2)
Solving for the tension in the string gives
a
F
mg
Figure 3. Free-body Diagram
Physical Pendulum – Inertia EX- 5518 Page 6 of 6
Appendix: Rotational Inertia Equations
For a disk about its center of mass: 𝐼 =
1
2
2
For a thick ring about its center of mass: 𝐼 =
1
2
1
2
2
2
For a thin ring about its center of mass: 𝐼 = 𝑀𝑅
2
For a disk of radius R with a hole of radius r, pivoting at the outer edge of the
hole:
𝐷𝑖𝑠𝑘
𝐻𝑜𝑙𝑒
1
2
𝐷𝑖𝑠𝑘
2
𝐷𝑖𝑠𝑘
2
1
2
𝐻𝑜𝑙𝑒
2
𝐻𝑜𝑙𝑒
2
𝐷𝑖𝑠𝑘
𝐻𝑜𝑙𝑒
1
2
𝐷𝑖𝑠𝑘
2
𝐷𝑖𝑠𝑘
2
𝐻𝑜𝑙𝑒
2
𝐷𝑖𝑠𝑘
𝐷𝑖𝑠𝑘
𝑀
( 𝜋𝑅
2
−𝜋𝑟
2
)
2
𝐷𝑖𝑠𝑘
𝑅
2
𝑅
2
−𝑟
2
𝐻𝑜𝑙𝑒
𝐻𝑜𝑙𝑒
𝑀
( 𝜋𝑅
2
−𝜋𝑟
2
)
2
𝐻𝑜𝑙𝑒
𝑟
2
𝑅
2
−𝑟
2
2
2
2
2
2
2
2
2
2
2
2
2
1
2
𝑅
4
− 3 𝑟
4
2
𝑥
2
𝑅
2
−𝑟
2
) x=distance from pivot to center of disk (9)
