HPP A16.v2
Rotating Your Arms and Legs
Rotating Your Arms and Legs
Exploration: Thinking about different kinds of motion as you swing your arm…
Have one person in your group stand up and make a fist. This person should then
gently swing their arm in a vertical circle, pivoting at the shoulder and keeping the rest of the
arm straight and unbent. Observe the motion of the arm, paying particular attention to the
motion of the elbow and of the hand.
Using a stopwatch, time how long it takes for this person to rotate their arm five
consecutive times without stopping. Again, this should be a gentle motion. You should not
risk pulling or straining a muscle in your shoulder!
Record the following information: time of five rotations, time of one rotation, distance
from the shoulder to the elbow, and distance from the shoulder to the middle of the hand. Use
this information to answer the following questions about the arm rotation. Be sure to show any
work and/or calculations and identify the information that you record in your logbook by
writing short labels in words!
Data Analysis of the Five Arm Rotations
1.
Distance and angular displacement of the hand
(a) How far in degrees did the hand travel during the five rotations?
(b) How far in radians did the hand travel during the five rotations? (Hint: 180 =  rad)
(c) How far in meters did the hand travel during the five rotations? (Hint: the
circumference of a circle = 2  r )
2.
Velocity and angular velocity of the hand
(a) What was the average angular speed (degrees/s and rad/s) of the hand?
(b) What was the average linear speed (m/s) of the hand?
3.
Acceleration and angular acceleration of the hand
(a) What was the average angular acceleration (degrees/s2 and rad/s2) of the hand?
How do you know?
(b) What was the average centripetal acceleration (m/s2) of the hand? (Hint: remember,
it is going in a circle and acp = v2/r.)
4.
Displacement and angular displacement of the elbow
(a) How far (degrees and rad) did the elbow travel during the five rotations?
(b) How far (m) did the elbow travel during the five rotations?
(c) How do these compare to the hand? Why are they the same and/or different?
5.
Velocity and angular velocity of the elbow
(a) What was the average angular speed (degrees/s and rad/s) of the elbow?
(b) What was the average linear speed (m/s) of the elbow?
(c) How do these compare to the hand? Why are they the same and/or different?
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Rotating Your Arms and Legs
Acceleration and angular acceleration of the elbow
(a) What was the average angular acceleration (degrees/s2 and rad/s2) of the elbow?
(b) What was the average centripetal acceleration (m/s2) of the elbow?
(c) How do these compare to the hand? Why are they the same and/or different?
Invention Discussion #1
Be prepared to share some of your results with the class during the class discussion.
Feel free to record notes and definitions in your logbook.
Application #1: Considering a Leg Extension
A. Thinking about energy as you extend your lower leg…
Have a person in your group do this activity:
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7.
Sit on a table with your lower leg hanging freely, i.e. with your knee bent at about
90 and the corresponding foot off of the floor. Swing your leg rapidly up to an
extended position.
Discuss with your partners the energy of your lower leg and foot during this
process.
If you consider the lowest location of your lower leg as its zero energy point, you can fill
in the table of the rotational kinetic energy, potential energy, work and total energy
present in this system. Describe how the magnitude of these kinds of energy increase,
decrease, remain constant, or equal zero as your lower leg/foot is at rest, moving upward
with increasing speed, moving upward with decreasing speed, and is extended.
Leg position
Work
Rotational
Kinetic
Energy
Potential
Energy
Total energy
of the lower
leg and foot
Hanging down at rest
Moving upward with
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increasing speed
Moving upward with
decreasing speed
Extended
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Suppose you had attached a sensor to the person's leg that was able to measure the
angular position of the tibia as a function of time. From the resulting data you would also be
able to plot the angular velocity and angular acceleration as functions of time during this leg
extension.
8.
(a)
(b)
Discuss with your partners and sketch in your logbooks how you would expect
these three curves of data to look, i.e. angle versus time, angular velocity versus time and
angular acceleration versus time. Take the start time as the position of the leg as it
hangs freely and consider that to be 0.
Record a brief description as to why you think the graphs will look this way.
Be prepared to discuss and explain your predictions with other members of the class.
Invention Discussion #2
Your instructor will lead a discussion of important concepts related to this lesson.
B. Trying out a new sensor
Now you are ready to make a more detailed analysis of the motion of your lower leg
and foot. Open the file: MBL – Rotary motion sensor. Select a member of your team and attach
the rotary motion sensor to that person's right leg at the knee using the elastic straps (like
pictured below). The computer should display the angular position of this person's tibia as a
function of time. From these data you will also plot the angular velocity and angular
acceleration as functions of time.
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Test that the rotational motion sensor is able to measure the
angular position of the tibia. Start collecting data with your leg
hanging straight down. Carefully extend the lower leg up by
45 and then to 90 (or other amounts).
Repeat this starting from another position.
Do your data make sense? If you think the sensor is not
working, then have your instructor check the equipment.
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Rotating Your Arms and Legs
Once you are convinced that the rotational motion sensor is working and gives
meaningful angle measurements, then explain your evidence for this conclusion.
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C. Analyzing the motion of your lower leg…
Now you are ready to make a more detailed analysis of the motion of your lower leg
and foot. Select a member of your team to have the rotary motion sensor attached to that
person's right leg at the knee. The person should swing his/her leg rapidly up to the extended
position and then hold it there for at least 2 seconds while data is collected. Repeat this for at
least three trials to be sure you have a good data set.
Save your data and give the file an appropriate name. Record the name of this file.
Select the best looking set of data. Adjust the graphs as necessary to show one complete leg
extension and print out copies for each member of your team's lab book. Label this graph along
with important features (such as when you started to move, when the leg was not moving, etc.).
Data Analysis
10.
11.
12.
(a)
Make and record the necessary measurements of the length of the person's lower leg.
Draw a careful sketch and label the following locations and distances.

The center of mass of the lower leg can be assumed to be located 50% of the
distance from the pivot point to the bottom of the person's heel.

The radius of gyration, k, of a person's lower leg is about 63% of the distance
from the pivot point to the bottom of the person's heel.

The mass of a person's lower leg and foot is typically 6% of the person's mass.
(b)
Using these values calculate the moment of inertia {I (kg m2) = m k2 } of the person's
lower leg and foot.
(a)
Identify and label an example of each of the following events on your graphs: lower
leg hanging down at rest, lower leg moving upward with maximum speed, lower
leg moving upward with decreasing speed, and lower leg extended and at rest.
(b)
Create a data table and record the angular position and angular velocity values at
these 4 points in time.
(c)
Calculate the potential energy of the lower leg for each of these instances.
Hint: you will need to use the angle and trigonometry to determine the height of the
center of mass of the leg.
(d)
Calculate the rotational kinetic energy of the lower leg for each of these instances.
Hint: To calculate these values you will need to first convert your angular velocities
from degrees/s to radians/s.
(e)
Calculate the total energy of the lower leg for each of these four instances.
(a)
Where is the potential energy of the lower leg the least and the most? Mark on the
graphs and describe in words what the leg was doing at those times.
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(b)
Where is the rotational kinetic energy of the lower leg the least and the most? Mark
on the graphs and describe in words what the leg was doing at those times.
(c)
Where is the total mechanical energy of the lower leg the least and the most?
Describe in words what the leg was doing at those times.
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End of Lab Procedures
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If you are the last lab of the day, then "shut down" the computer and turn off the
interface box.
Want More Information?
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Angular displacement, speed, and acceleration (see Walker, Section 10-1)
Center of mass (see Walker, Section 9-7)
Rotational motion (see Walker, Sections 10-1 – 10-3)
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