PROBLEM #1: ANGULAR SPEED AND LINEAR SPEED
You are working with an engineering group testing equipment that might be used on a
satellite. To equalize the heat load from the sun, the satellite will spin about its center.
Your task is to determine the forces exerted on delicate measuring equipment when the
satellite spins at a constant angular speed. You know that since any object traveling in a
circular path must have exerted on it a non-zero net force, that object must be
accelerating. As a first step in finding the net force, you decide to calculate the linear
speed of any object in the satellite as a function of its distance from the center of the
satellite and the satellite’s angular speed. From the linear speed of the object in circular
motion, you calculate its acceleration. You will test your calculations in a laboratory
before launching the satellite.
Instructions: Before lab, read the laboratory in its entirety as well as the required reading in the
textbook. In your lab notebook, respond to the warm up questions and derive a specific prediction
for the outcome of the lab. During lab, compare your warm up responses and prediction in your
group. Then, work through the exploration, measurement, analysis, and conclusion sections in
sequence, keeping a record of your findings in your lab notebook. It is often useful to use Excel to
perform data analysis, rather than doing it by hand.
Read: Tipler & Mosca Chapter 5, section 5.3 and Chapter 9, section 9.1.
EQUIPMENT
You have an apparatus that spins a horizontal
platform. A top view of the device is shown to
the right. You also have a stopwatch, meterstick
and the video analysis equipment.
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If equipment is missing or broken, submit a problem report by sending an email to
labhelp@physics.umn.edu. Include the room number and brief description of the
problem.
WARM UP
The following questions will help you to reach your prediction and the analysis of your
data.
1. Draw the trajectory of a point on a beam rotating. Choose a coordinate system.
Choose a point on that trajectory that is not on a coordinate axis. Draw vectors
representing the position, velocity, and acceleration of that point.
ANGULAR SPEED AND LINEAR SPEED – 1301Lab6Prob1
2. Write equations for each component of the position vector, as a function of the
distance of the point from the axis of rotation and the angle the vector makes with
an axis of your coordinate system. Next, calculate how that angle depends on time
and the constant angular speed of the beam. Sketch three graphs, (one for each of
these equations) as a function of time. Explain why one of the graphs increases
monotonically with time, but the other two oscillate.
3. Using your equations for components of the position of the point, calculate an
equation for each component of the velocity of the point. Graph these two equations
as a function of time. Compare these graphs to those for the components of the
position of the object (when one component of the position is at a maximum, for
example, is the same component of the velocity at a maximum value?) Draw these
components at the point you have chosen in your drawing; verify that their vector
sum gives the correct direction for the velocity of the point.
4. Use your equations for the point’s velocity components to calculate its speed. Does
the speed change with time? Should it?
5. Use the equations for the point’s velocity components to calculate an equation for
each component of the point’s acceleration. Graph these two equations as functions
of time, and compare to the velocity and position graphs. Verify that the vector sum
of the components gives the correct direction for the acceleration of the point you
have chosen in your drawing. Use the acceleration components to calculate the
magnitude of the acceleration.
6. For comparison, write down the expression for the acceleration of the point as a
function of its speed and its distance from the axis of rotation.
PREDICTION
What are you trying to calculate? Restate the problem to clearly identify your objective.
Illustrate
EXPLORATION
Practice spinning the beam at different angular speeds. How many rotations does the
beam make before it slows down appreciably? Select a range of angular speeds to use
in your measurements.
Move the apparatus to the floor and adjust the camera tripod so that the camera is
directly above the middle of the spinning beam. Make sure the beam is level. Practice
taking some videos. Find the best distance and angle for your video. How will you
make sure that you always measure the same position on the beam?
ANGULAR SPEED AND LINEAR SPEED – 1301Lab6Prob1
Plan how you will measure the perpendicular components of the velocity to calculate
the speed of the point. How will you also use your video to measure the angular speed
of the beam?
MEASUREMENT
Take a video of the spinning beam. Be sure you have more than two complete
revolution of the beam. For best results, use the beam itself when calibrating your
video.
Determine the time it takes for the beam to make two complete revolutions and the
distance between the point of interest and the axis of rotation. Set the scale of your axes
appropriately so you can see the data as it is taken.
Decide how many different points you will measure to test your prediction. How will
you ensure that the angular speed is the same for all of these measurements? How
many times will you repeat these measurements using different angular speeds?
ANALYSIS
Analyze your video by following a single point on the beam for at least two complete
revolutions. Use the velocity components to determine the direction of the velocity
vector. Is it in the expected direction?
Analyze enough different points in the same video to make a graph of speed of a
point as a function of distance from the axis of rotation. What quantity does the slope
of this graph represent?
Calculate the acceleration of each point and graph the acceleration as a function of the
distance from the axis of rotation. What quantity does the slope of this graph
represent?
CONCLUSION
How do your results compare to your predictions and the answers to the warm up
questions? Did the measured acceleration match the acceleration predicted by your
equation from Warm up question 5? Question 6? Explain.
Was the measured linear speed of each point on the beam a constant? Demonstrate
this in terms of your fit equations for velocity.
ANGULAR SPEED AND LINEAR SPEED – 1301Lab6Prob1