ROTATIONAL DYNAMICS
Name: ____________________
Pre-Lab Questions Page
Roster Number: _____________
Class:_____________________
Instructor:__________________
1. Write the symbolic representation and one possible unit for angular
velocity, angular acceleration, torque and rotational inertia.
angular velocity = _____ _____ angular acceleration =_____ ______
torque = ______ ______
rotational inertia = ______ _______
3. Describe a situation in which ω < 0 and ω and α are antiparallel.
4. In the figure 1, a force whose magnitude is 55.0N is applied to a door.
However, the lever arms (moment arms) are different in the three parts of the
drawing: (a) d = 0.80 m, (b) d = 0.060 m and (c) d = 0. Find the magnitude of
the torque in each case.
THIS PAGE LEFT BLANK..........................
Name:_______________________
Date:________________________
Lab Partners:__________________
__________________
________________________________________________
ROTATIONAL DYNAMICS
OBJECTIVE: To determine the rotational inertia (I) of a disc and cylindrical axle by
measuring its angular acceleration and applying the rotational form of Newton's laws of
motion to the rotating disc. This value is compared to the geometrically computed value
for a disc. (Optional: compared to computed value for disc and cylindrical axle)
APPARATUS:
Rotational inertia apparatus
1-meter string
50g mass holder
Assorted masses (1g - 100g)
PC and ULI timer
Meter Stick
Vernier caliper
Photogate
Flag
ULI
INTRODUCTION:
The amount of torque(τ)
applied to the axle of the rotational
Figure 1
dynamics apparatus is varied by
adding additional mass to the string
attached to that axle. The resulting angular accelerations (α) are measured by timing the
flag as it passes through the photo gate. From the plots of angular velocity(ω) vs time the
angular accelerations are found. The values of the applied torque are computed from
τ = rF sin θ = Fd and a graph of applied torque vs angular acceleration is prepared. From
this graph an experimental dynamic value of the rotational inertia and the frictional
torque are determined. In the last equation, d, is known as the lever arm(moment arm).
PROCEDURE:
1.
Check to be sure the apparatus is connected as shown in Figure 1.
2.
Make sure the computer is on. Double click on the “Physics Lab” folder and
select <Rotational Dynamics> icon.
3.
Rotate the disc so that the string is wrapped around the axle as shown in figure 1.
Hang the 50g mass holder from the end of the string.
4.
5.
Click on the
Start
button, then give the rim of the disc a small push to start
Stop
the rotation. After 10 rotations click on the
button. If the angular velocity
was nearly constant the mass attached provides a torque just sufficient to
overcome the friction in the axle. Observe the angular speed column of the table.
If the angular velocity increases, lower the amount of mass attached to the string
and repeat the timing run. If the angular velocity decreases, add additional mass
and repeat the timing run. Repeat this step until the mass required to overcome
friction is determined to within 1g and record this value.
Start
Add 50g, rewind the string, click on
and release the mass from rest .
Stop
Click
after 10 rotations. Check the angular velocity in the table to see if
the results are reasonable.
6.
If your data is reasonable then go to the <View> menu and select “Auto Scale
Once.”
7.
The computer can find the best straight line fit for a set of data points but it calls it
a “linear fit” line. Do this by clicking and holding on the first data point and then
drag to the last data points. Let go and the box will remain there.
8.
Next, go up to the <Analyze > menu and select “Linear Fit.”
9.
Go to the <View> menu, select Graph options and make sure the following are
selected “Point Protector Every”, “Graph Title” and “Grid.” At the bottom of the
page in the box below “graph title” type “Velocity vs. Time”.
Click “ok”
10.
Save this data on your diskette and record the name you assigned to the saved file
on your data sheet. To do this select Save As from the <File> menu. Save the file
as a MBL file. For example, “50 grams added.mbl”
6.
Repeat steps 5 through 10 for at least 4 additional times with attached masses in
increments of 50 grams.
7.
Note and record the measured mass of the disc with the axle. This value is
marked on the apparatus along with its implied uncertainty.
8.
Use the most precise tool available to measure (1) the diameter of the axle,
(2) the distance that the axle protrudes on each side of the disc, (3) the thickness of the disc and (4) the diameter of the disc. Record these values with
their uncertainties. Record this information in a data chart.
(Check out the Vernier Caliper from the instructor. Return it when you are finished.)
CALCULATIONS:
Combine Newton's second law for rotation applied to the disc and Newton's
second law for translation applied to the attached mass to derive Equation 1. Refer to
figure 2.
2
Iα = mgr − mr α − τ f
Equation 1
Where m is the suspended mass, r is the radius of the axle, g is the acceleration due to
gravity,α is the angular acceleration of the disc and τf is the frictional torque. (Refer to
figure 2)
Equation 1 can be rearranged as: mgr = (I + mr ) α + τf
2
Equation 2
Because applied masses and axle radius are so small using this apparatus, mr2 is
negligibly small compared to ( I ) this equation can be written as:
Fig. 2
mgr = Iα + τ f
Equation 3
M
1.
Print out your graphs from your experiment using the
computers in SM252 by selecting Logger Pro from
the“Physics Lab” folder. Open your first graph of data.
r
R
FT
2.
Print graphs of your angular velocity in rad/s vs time as
FT
usual for each trial. Be sure to include the regression
and have the vertical intercept shown on the graph. Your
name should be printed in the header or footer. To do this
Go to the <File> menu and select “printing options” then
FW = mg
type your name and roster#. In the comment section type the
amount of mass that was suspended(be sure to include the mass holder).
Next, click “page setup” and select “landscape”.
3.
From these graphs determine the angular acceleration for each trial.
4.
Compute mgr (the torque applied by the weight force) for each trial. Be sure to
calculate mgr for the case with no angular acceleration when the torque was
applied only to overcome the friction in the axle.
5.
Use Graphical Analysis to make a graph of mgr vs angular acceleration.
6.
From the regression box of the graph from step 5 determine dynamic values for
the moment of inertia (I) and the frictional torque ( τ f ). Hint: the equation for a
linear situation is y = mx + b and also see equation 3.
7.
Compute a geometric value for the moment of inertia of the rotating system
according to:
(a) Use the mass (total of large disc and axle) marked on the apparatus and
the equation for the moment of inertia of a disc (I =
1
MR 2 ).
2
(b)* Compute the moment of inertia of the large disc and axle by adding
the separate moments of inertia (Itotal = Idisc + Iaxle ).
*First find the volume of the large disc and the axle separately using the formula
2
for the volume of a cylinder ( V = πr h ), where h is the thickness and r is the
radius. Find the combined volume. Because the density is uniform and the same
for both the disc and its axle the density is:
ρ=
M total M disc M axle
=
=
Vaxle
Vdisc
Vtotal
to find the mass of the disc alone:
⎛M
⎞
Mdisc = ⎜ total ⎟ Vdisc
⎝ Vtotal ⎠
to find the mass of the axle alone:
⎛M ⎞
Maxle = ⎜ total ⎟ Vaxle
⎝ Vtotal ⎠
Treat each as a cylinder and find the combined moment of inertia:
Itotal = Idisc + Iaxle
8.
Compare the dynamic values of I and τ f , from the graph, to each
computed(geometric) value.
A E
x100
Recall: %error =
, where E = experimental value and A = calculated
A
value.
QUESTIONS:
1. Was your experimental value for rotational moment of inertia (I) greater than or less
than the calculated value? Is this what you expected? What affect did friction play, if
any?
2. Was your experimental value for the frictional torque (τf) greater than or less than the
calculated value? Is this what you expected?
3. If torque is the product of the moment of inertia and rotational acceleration, calculate
the torque supplied by a motor that rotates a circular blade from rest to an angular
rad
speed of 660
in 306 rev. Suppose that the moment of inertia for the blade is 1.61 x
s
10-3 kg m2.