1J11.

50 Double Cone on an Incline

Abstract
As a ball rolls down an inclined plane, its center of mass lowers. A double cone can however, roll up two rails
on an incline to lower its center of mass. In this case the altitude of the cone decreases as it rolls from the bottom
to the top of the incline.

Picture

Setup
Setup is 5 minutes.

Safety Concerns
Catch the double cone if it falls through the ramp.

Equipment
Double cone Wooden cylinder
T-shaped base Hinged V-shaped ramp

Procedure
Place the T-shaped base on a level surface, with the flat part of the base on the bottom. Open the V-shaped
ramp and place it on the T-shaped base with the hinged side in the groove. If the angle of opening between the

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two rails of the ramp is greater than 20.4 degrees, the double cone rolls up the ramp. When the angle between
the two rails is adjusted to 20.4 degrees, the double cone does not roll on the ramp, and the cone rolls down the
hill when the opening angle is less than 20.4 degrees. The cylinder, in contrast, rolls down the hill at any angle.

Theory
Consider a ball atop an inclined plane. When the ball rolls down the plane its gravitational potential energy
is converted to kinetic energy. Ignoring any work done by non-conservative forces, the decrease in gravitational
potential energy can be equated to the increase in kinetic energy by the conservation of energy.
The gravitational potential energy, Ug , of an object is defined to be

Ug = mghcm , (1)

where m is the mass of the object, g is the acceleration due to gravity, and hcm is the altitude of the objects
center of mass from the reference point chosen as zero. In this demonstration, it is useful to choose the surface
of the table on which the inclined plane rests as the point of zero potential energy.
It is possible to set up the V-shaped rails so that a cylinder will roll down the incline while a double cone rolls
up. The motion of the double cone does not violate the conservation of energy because its center of mass moves
downward as the cone rolls up the rails. The cones gravitational potential energy is still converted to kinetic
energy in the process.
Using symmetry, it is easy to locate the center of mass on the
double cone. Consider a coordinate system as shown in Figure 1.
The double cone exhibits symmetry about the x-y, y-z, and x-z y
coordinate planes. As a result, the center of mass of the cone
must lie at the intersection of all three planes i.e. the origin of
the coordinate system. If the cone is viewed from the side as it
rolls up the hill, the motion of the center of mass can be traced z
by the motion of the metal tip on the cone. The metal tip moves
downward as the cone rolls uphill.
The motion of the double cone depends primarily on the three x
parameters shown in Figures 2 and 3. These parameters are the
vertex angle of the cone, , the opening angle between the rails Figure 1: Diagram of a double cone with an
of the ramp, , and the angle between the ramp and the table, . x-y-z co-ordinate system.
Note that in this demonstration, the angles and are fixed.

y xcm

Pp
Cp
O x

2
S 2 Qp
A
x

xcm z

Figure 2: Cross-sectional diagram of the ap- Figure 3: Diagram of the overhead view or x-z
paratus in the x-y plane. plane of the apparatus.

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 If the angle between the two rails in the V-shaped ramp is adjusted, it is possible to find a critical angle at
which the cone remains at rest on the ramp. At an angle smaller than the critical angle the cone rolls down the
ramp, while at an angle larger than the critical angle the cone rolls up. Defining a coordinate system like the one
shown in Figure 2, it is possible to find the equation of motion of the center of mass. From this, the critical angle
can be deduced.
y

r
C

P
S Q

2 R
ycm

A
Table z

Figure 4: Cross-sectional diagram of the apparatus in the y-z plane.

Consider a projection of the apparatus on the table. A view from above of this projection is illustrated in
Figure 4. The double cone depicted can be thought of as the shadow of the double cone on the table. Pp and
Qp are the projections of the points of contact, O is the origin, Cp is the projection of the center of mass of the
cone, and xcm is the x-coordinate of the center of mass.
Using trigonometry, it is seen that

Pp Cp = xcm tan
2

Pp Qp = 2xcm tan , (2)
2
where Pp Cp is the distance between projections Pp and Cp and Pp Qp is the distance between the projections Pp
and Qp .
Next, consider a view of the demonstration from the front, as if the cone were rolling up the rails towards you.
This is shown in Figure 4. Here, ycm is the y-coordinate of the center of mass, P and Q are the points of contact,
S is the point where the lines P Q and CA intersect, R is the lowest point on the cone, and r is the radius of
the cross-sectional circle at the center of the cone. Note that the distance between the points of contact, P Q, is
equal to the distance between projections of the points of contact, Pp Qp , calculated in Equation 2, that is,

P Q = Pp Qp = 2xcm tan . (3)
2
From Figure 4, the y-coordinate of the center of mass is
ycm = CS + SA. (4)
Looking at the triangle RP S in Figure 4, it is seen that,
RS RS
tan = =  (5)
2 PS PQ
2

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Substituting P Q from Equation 3 into Equation 5 yields,

RS
tan = 
2 (2 tan 2 )(xcm )
2

RS = xcm tan tan (6)
2 2
it can also be seen from Figure 4 that
CS = r RS. (7)
Substituting RS from Equation 6 into Equation 7 yields,

CS = r xcm tan tan . (8)
2 2
Lastly, consider a view of the apparatus from the side, as shown in Figure 2. The x-coordinate of the point S
is equal to the x-coordinate of the center of mass because point S lies directly below the center of mass, as seen
in Figure 4. From Figure 2,

SA
tan =
xcm
SA = xcm tan . (9)

Substituting Equations 8 and 9 into Equation 4 yields

which is an equation for the center of mass in the form y = mx + b. The critical angle is the angle at which the
cone does not roll on the rails. This occurs when the slope in Equation 10 is 0, that is,

tan tan tan = 0
2  2 
tan
= 2 tan1 (11)
tan 2

In the demonstration, is approximately 3.3 degrees and is approximately 38.8 degrees so the theoretical
critical angle between the two rails is 18.6 degrees. Experimentally, the critical angle between the two rails is
approximately 20.4 degrees.

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References

[1] G. D. Freier and F. J. Anderson. A Demonstration Handbook for Physics, Mr-1. Rolling Up Hill. American
Association of Physics Teachers One Physics Ellipse, College Park MD, 1996. pg M-41.
[2] Wallace A. Hilton. Physics Demonstration Experiments at William Jewell College,M-18a.3 Center of Mass
Apparatus. American Association of Physics Teachers, June 1982. pg 20.
[3] http://plus.maths.org/issue40/features/uphill/index-gifd.html.

[4] Richard Manliffe Sutton. Demonstration Experiments in Physics, M-37.. Mcgraw-Hill Book Company Inc,
New York and London, 1938, pg 26.