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Physics 115: Spring 2005
Rolling Something
Introduction
The purpose of this laboratory is to investigate the dynamics of a wheel and axle
rolling without slipping down an incline from two standpoints: the dynamical equation, τ =
Iα; and conservation of energy.
Equipment
Brass disk on axle, inclined track, meter stick,
stopwatch, scale, and calipers.
Theory
For a system consisting of a wheel mounted on an axis
which is allowed to roll without slipping down an inclined
plane, it is possible to calculate the acceleration of the
wheel/axle system down the plane using either Newton's
laws for translation and rotation or energy conservation.
First, let us consider the analysis using Newton's laws.
Refer to Figure 1. Since, for a rolling body, the axis of
rotation is not fixed, we can consider the motion a
combination of a translation of the center of mass plus a rotation about the center of mass.
It is necessary, therefore, to consider Newton's second law for both the linear and the
rotational motion involved in this system. Let us first consider the linear motion described
by the equation ∑F = ma. If we consider the linear motion of the axis (which remains
parallel to the incline), we see that the sum of the forces in this direction is just mgsinθ - f ,
where f is the (static) friction force between the inclined plane and the axle. Thus, the
∑ F = ma = mg sin θ − f
above equation becomes:
Likewise, in considering the rotational motion of the wheel with moment of inertia I, we
have Στ = Iα. Since the rotational motion of the wheel about its axis is due to the friction
force, this equation can be written:
∑τ = I α = f r
where r is the radius of the axle.
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We also know that the linear and rotational acceleration are coupled by the equation
a = αr. From here, it is possible to combine Equations 1 and 2 to obtain the following
result:
a=
mg sin θ
I
(m + 2 )
r
By determining the moment of inertia of the object, you can calculate a theoretical value
for the acceleration using equation 3. Since the acceleration is constant, you can measure
the time it takes the wheel/axel system to travel a known distance and then use the
equations for constant acceleration to find the acceleration experimentally.
Alternatively, you can equate the total change of potential energy of the wheel/axle
system mgh to the total change in kinetic energy. In this case, there is both kinetic energy
½mv2 associated with the linear motion and kinetic energy associated with the rotational
motion ½Iω2.
Procedure
1) Angle of the incline:
From equation (3), you can see that knowing the angle of the incline that the disk
will roll down is essential in determining the amount of force from gravity the disk will
experience. To determine this angle accurately, carefully measure two heights along the
incline, and the length between those heights. Use your knowledge of trigonometry to
calculate the angle of the incline which the disk will roll down. Show your work below:
Angle of Incline____________
2) Calculate the maximum amount of error in your calculation of the Angle of Incline.
(Hint: Using Propagation of error is quite difficult here. Try to sub in the max error and
find the maximum effect in the results.) Show your work on back of this page.
Maximum error in angle of incline____________
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3) Moment of Inertia:
To find the total moment of inertia, I, of the system, you need to examine both the
disk, and the axle on which it is attached. Each of these elements has a mass, and, as such,
a moment of inertia which you must calculate separately. The formula for the moment of
inertia of a solid disk of radius R is
(4)
I=mR2/2
Take apart the disk and the axle, and use the balance to determine the weight and radius
of each:
Weight of the axle____________
Radius of the axle____________(To measure this value as accurately as possible, use
the vernier calipers provided instead of a typical ruler)
(show work here)
Moment of inertia, I, of the axle_______________
Weight of the disk____________
Radius of the disk____________
(show work here)
Moment of inertia, I, of the disk_______________
Total moment of Inertia (Idisk + Iaxle)____________
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4) Determine a theoretical value for the acceleration of the disk/axle:
Now use equation (3) to find a theoretical value for the acceleration of the
disk/axle. Remember, the “r” in equation (3) refers to the radius of the axle, since the
frictional force f acts on the axle, not the disk. Show your calculations below:
atheoretical__________________
5) Time to experiment!!!:
Place the disk/axle combination at the top of the track and use a stick or ruler to
hold it in place. Use a clean, quick release and allow the disk and axle to roll down the
track, making sure that the axle is not slipping on the track. Make several
measurements of the time it takes to reach the bottom of the track and discard any
wildly inconsistent results. Take the average of your remaining measurements and use it
in the calculations for final velocity and acceleration:
Table: Part IV
Time to reach bottom (seconds)
Trial 1
Trial 2
Trial 3
Extra trials:
Average time
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6) Find the actual acceleration of the disk/axle:
Using the equation
x=vinitialt+.5at2
Calculate the experimental acceleration of the disk/axle:
(5)
aexperimental________________
7) Find the final velocity of the disk/axle:
Using the equation
vfinal=aexperimental t
Find the experimental final velocity of the disk/axle:
(6)
vfinal____________
8) Remember, there is a difference between the rotational velocity, ω, and the
translational velocity, v, which you just calculated. Come up with an expression for v
in terms of ω and whatever other variables you need. Justify that expression.
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9) Determine the rotational velocity, ω final, at the bottom of the incline:
ω final________________________
10) Is energy conserved?
From the initial potential energy, mgh, and the final translational and rotational
velocities, determine whether or not the total energy of the system was conserved as the
disk/axle rolled down the incline (show work below):
Is the total energy conserved_____________
If not, how much energy is lost____________
Is most of the kinetic energy in the form of rotation of the disk/axle about its
center of mass, or in the form of translation___________
Calculate the relative percentages of each:
Percentage of Kinetic Energy made up by Rotational Kinetic Energy_______
by Translational Kinetic Energy___________
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11) One last question:
In the theory section, we included the forces due to friction in the analysis using
Newton’s laws, but we did not include any friction in the analysis using conservation of
energy. Why doesn’t friction have to be included in the energy analysis?
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