HPP A10
Relating Linear and Angular Kinematics
Equipment needed for this lab:
whiteboard w/markers
paper plate
- Rotary Motion Sensor PS-2120
-Pasport Xplorer – PS-2000
- 2 D or 3 D Acceleration Sensor PS-2118/2119
- Tennis ball
Exploration: Relating Angular and Linear Motion
Have each member of your group, in turn, put their elbow in the crook of the corner of the
whiteboard, holding the marker pen perpendicular to the board. Draw an arc subtending a 90
degree angle between the side and bottom of the whiteboard.
Activity Guide
 2010 The Humanized Physics Project
Supported in part by NSF-CCLI Program under grants DUE #00-88712 and DUE #00-88780
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GE 1.
1. Is there a relationship between the length of your arm and the arc length you
made with the pen?
2.Using a tape measure, measure the length of the quarter circle you drew, and
measure the distance from your elbow to the pen in your hand.
GE 2. Radian Measure
1. Draw a line on a paper plate showing the radius of the plate. Measure this
radius with a tape measure.
2. Measure off the length of one radius on the edge of the plate, and make the
plate at that point. How many lengths of one radius fit going around the edge
of the plate? Hint: what is the relationship between the radius of a circle and
its circumference?
From the one radius mark, draw a line to the center of the plate, creating a pie
wedge shape with the other line. Mark and measure with a protractor or
goniometer the angle of this pie wedge in degrees.
This angle is one radian in units of radians.
Invention
One radian is the angle subtended by an arc length of one radius. Using radians we can directly
relate arc length (linear) to angle.
Activity Guide
 2010 The Humanized Physics Project
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The radius fits around the circumference, C, 2 times
2r = C
The angle of one radian fits around the circle 2 times
2(1rad) = 1 rev (360)
By definition the units of radians is the ratio of arc length, s, subtended to radius, r.
rad =s/r
Therefore the arc length is equal to radius times the angle in radians
s = rrad
How fast something is moving along an arc, v = s/t - the linear velocity, is equal to the radius
times the change in radians with time,
v = rad /t
or
v = r
Application
GE 3.
1. Could you determine the linear
speed, v, of the pen if you knew its
angular speed?
Activity Guide
 2010 The Humanized Physics Project
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Attach the rotation sensor to your elbow. Set up the computer to display
angular velocity versus time. Hold a ball, as shown.
2. What is the maximum speed of the ball in your hand as it makes the arc
when you throw, it before you let it go?
3. Keeping your wrist still, take angular velocity measurements of your
forearm as it moves at a constant angular velocity, holding a ball. Using the
graphical results, determine the linear speed of the ball. What other
information do you need?
4. Repeat, this time throw the ball and estimate the maximum linear speed of
the ball, from the maximum angular velocity on the graph.
Exploration: Centripetal and Tangential Accelerations
You are going rotate on your stool with a 2 D or 3 D Acceleration Sensor held straight and steady
in your horizontally outstretched hand. Any hand wobble will make the graph too messy to
interpret. The radial (centripetal) direction is aligned with the forearm.
Activity Guide
 2010 The Humanized Physics Project
HPP A10
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GE 4.
1. On your whiteboard draw your prediction of the centripetal acceleration
and tangential acceleration vs time of your hand for the following movement
sequence on the stool:
a) start rotating quickly by one big push with your feet on the floor.
b) rotate at constant angular speed, with feet off floor, for several rotations
c) come to sharp stop with your feet.
Now you will look at the centripetal and tangential accelerations of your hand
as your forearm rotates. In addition to the Rotary Motion Sensor on your
elbow, now secure on your hand a 2D or 3D Acceleration Sensor, having one
axis aligned with your forearm (the centripetal direction) and one axis aligned
tangential to your hand's rotation.
You will be repeating the previous arm motion - moving your hand and
forearm up and back down, keeping your upper arm vertical.
Activity Guide
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2. On your whiteboard, draw your prediction for the graphs for the radial
(centripetal) acceleration vs. time and tangential acceleration vs. time for the
hand.
Indicate how they relate to the angular velocity and angular acceleration
graphs.
2. Now do the motion. As before, have a prolonged period of stillness at the
start position before moving, to establish a good zero baseline. Do not rotate
your wrist. Paste your graph below with the indicated plots.
Activity Guide
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3. Examine the graphs. Were your predictions correct? How would the
graphs be different if the accelerometer was attached on the arm up closer to
the elbow?
4. Measure and record the distance between the pivot of the Rotary Motion
Sensor and the middle of the Acceleration Sensor.
5. Using this distance from elbow to hand (rotary motion sensor to
accelerometer) and the maximum angular velocity and angular acceleration
graph data, confirm the values of the maximum data for centripetal and
tangential acceleration. What relationships exist between these quantities?
Invention
Recall that the radial (centripetal) acceleration represents the rate at which the direction of the
velocity is changing and the tangential acceleration is the rate at which the magnitude of the
velocity is changing.
aT = r 
Activity Guide
 2010 The Humanized Physics Project
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aC = r 2 = v2/r
Application - True throwing a ball
To see the compound effects of both elbow and wrist action, attach the 3 D Acceleration Sensor
to the hand so that the z axis is tangential. Try to keep the upper arm relatively still and vertical.
GE.
1. Collect data for the motion and paste a graph below.
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2. What is the maximum acceleration of the ball in your hand?
3. What is the net force on the ball at that point? Approximate the force of
your hand on the ball, assuming the arm is nearly horizontal at that point.
Optional Application: Walking and Running
You are going to examine the rotary motions of a group member’s upper leg while walking
and/or running. The rotary motion sensor will be placed approximately at the hip joint, with one
arm attached to the thigh. A reading of zero will be when both legs are straight, before they start.
Depending which leg the Rotary Motion Sensor is placed on, either the leg going forward or back
will register as positive. Play around with rotary motion sensor, deciding which leg to use. You
will also attach an Acceleration Sensor to the group member's leg right above the knee.
Activity Guide
 2010 The Humanized Physics Project
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GE.
1. On the whiteboard Predict what the centripetal and tangential acceleration
graphs of right above the knee would look like as the person starts from a
“both legs straight” position. Make sure these graphs correlate with the thigh
angular motion graphs.
2. Do the motion and collect data. Paster graphs below.
The upper leg motions are similar to the forearm motions, except the upper leg can rotate in both
directions.
Activity Guide
 2010 The Humanized Physics Project