Ch13 Rigid Bodies

8/12/2019 Ch13 Rigid Bodies

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Chapter 13

Rigid Body Motion and Rotational

Dynamics

13.1 Rigid Bodies

A rigid body consists of a group of particles whose separations are all fixed in magnitude. Sixindependent coordinates are required to completely specify the position and orientation of arigid body. For example, the location of the first particle is specified by three coordinates. Asecond particle requires only two coordinates since the distance to the first is fixed. Finally,a third particle requires only one coordinate, since its distance to the first two particles

is fixed (think about the intersection of two spheres). The positions of all the remainingparticles are then determined by their distances from the first three. Usually, one takesthese six coordinates to be the center-of-mass position R = (X,Y,Z) and three anglesspecifying the orientation of the body (e.g. the Euler angles).

As derived previously, the equations of motion are

P =i

mi ri , P =F(ext) (13.1)

L=i

mi ri ri , L= N(ext) . (13.2)

These equations determine the motion of a rigid body.

13.1.1 Examples of rigid bodies

Our first example of a rigid body is of a wheel rolling with constant angular velocity = ,and without slipping, This is shown in Fig. 13.1. The no-slip condition is dx = R d, sox= VCM= R. The velocity of a point within the wheel is

v= VCM+ r , (13.3)

1


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2 CHAPTER 13. RIGID BODY MOTION AND ROTATIONAL DYNAMICS

Figure 13.1: A wheel rolling to the right without slipping.

where r is measured from the center of the disk. The velocity of a point on the surface isthen given by v= R

x+ r).

As a second example, consider a bicycle wheel of mass M and radius R affixed to alight, firm rod of length d, as shown in Fig. 13.2. Assuming L lies in the (x, y) plane,one computes the gravitational torque N =r (Mg) =M gd . The angular momentumvector then rotates with angular frequency . Thus,

d=dL

L = = M gd

L . (13.4)

But L = M R2, so the precession frequency is

p= = gd

R2 . (13.5)

For R = d = 30cm and /2 = 200rpm, find p/2 15 rpm. Note that we have hereignored the contribution to L from the precession itself, which lies along z, resulting in thenutationof the wheel. This is justified ifLp/L= (d

2/R2) (p/) 1.

13.2 The Inertia Tensor

Suppose first that a point within the body itself is fixed. This eliminates the translationaldegrees of freedom from consideration. We now have

dr

dt

inertial

= r , (13.6)


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13.2. THE INERTIA TENSOR 3

Figure 13.2: Precession of a spinning bicycle wheel.

since rbody = 0. The kinetic energy is then

T = 12

i

mi

dridt

2inertial

= 12

i

mi( ri) ( ri)

= 12 imi

2 r2i ( ri)2 12I , (13.7)

where is the component of along the body-fixed axis e. The quantity I is theinertia tensor,

I=i

mi

r2i ri, ri,

(13.8)

=

ddr (r)

r2 r r

(continuous media) . (13.9)

The angular momentum is

L=i

mi ri

dridt

inertial

=i

mi ri ( ri) =I . (13.10)

The diagonal elements of I are called the moments of inertia, while the off-diagonalelements are called the products of inertia.


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13.2.1 Coordinate transformations

Consider the basis transformation

e = Re . (13.11)We demand e e=, which meansR O(d) is an orthogonal matrix, i.e.Rt = R1.Thus the inverse transformation is e= Rte. Consider next a general vectorA = Ae.Expressed in terms of the new basis{e}, we have

A= A

e Rt e =

A RAe (13.12)

Thus, the components ofAtransform as A= RA. This is true for any vector.Under a rotation, the density (r) must satisfy(r) =(r). This is the transformation

rule for scalars. The inertia tensor therefore obeys

I=

d3r (r)

r

2 r r

=

d3r (r)

r2

RrRr= R IRt . (13.13)

I.e.I = RIRt, the transformation rule for tensors. The angular frequency is a vector, so =R . The angular momentum L also transforms as a vector. The kinetic energyis T = 12

t

I

, which transforms as a scalar.

13.2.2 The case of no fixed point

If there is no fixed point, we can let r denote the distance from the center-of-mass (CM),which will serve as the instantaneous origin in the body-fixed frame. We then adopt thenotation whereRis the CM position of the rotating body, as observed in an inertial frame,and is computed from the expression

R= 1

M

i

mi i= 1

M

d3r (r) , (13.14)

where the total mass is of courseM=

i

mi=

d3r (r) . (13.15)

The kinetic energy and angular momentum are then

T = 12MR2 + 12I (13.16)

L= M RR+ I , (13.17)

whereI is given in eqs. 13.8 and 13.9, where the origin is the CM.


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13.3. PARALLEL AXIS THEOREM 5

Figure 13.3: Application of the parallel axis theorem to a cylindrically symmetric massdistribution.

13.3 Parallel Axis Theorem

SupposeIis given in a body-fixed frame. If we displace the origin in the body-fixed frame

by d, then let I(d) be the inertial tensor with respect to the new origin. If, relative tothe origin at 0a mass element lies at position r , then relative to an origin at d it will lie atr

d. We then have

I(d) =i

mi

(r2i 2d ri+ d2) (ri, d)(ri, d)

. (13.18)

Ifri is measured with respect to the CM, theni

mi ri = 0 (13.19)

andI(d) =I(0) + M

d2 dd

, (13.20)

a result known as the parallel axis theorem.

As an example of the theorem, consider the situation depicted in Fig. 13.3, where acylindrically symmetric mass distribution is rotated about is symmetry axis, and about anaxis tangent to its side. The component Izz of the inertia tensor is easily computed whenthe origin lies along the symmetry axis:

Izz =

d3r (r) (r2 z2) =L 2

a0

dr

r3

= 2 La4 = 12M a

2 , (13.21)


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where M = a2L is the total mass. If we compute Izz about a vertical axis which istangent to the cylinder, the parallel axis theorem tells us that

I

zz =Izz + M a2

= 32M a

2

. (13.22)

Doing this calculation by explicit integration of

dm r2

would be tedious!

13.3.1 Example

Problem: Compute the CM and the inertia tensor for the planar right triangle of Fig.13.4, assuming it to be of uniform two-dimensional mass density .

Solution: The total mass isM= 12 ab. The x-coordinate of the CM is then

X= 1

M

a0

dx

b(1x

a

)0

d y x=

M

a0

dx b

1 xa

x

= a2b

M

10

du u(1 u) = a2b

6M = 13a . (13.23)

Clearly we must then have Y = 13b, which may be verified by explicit integration.

We now compute the inertia tensor, with the origin at (0, 0, 0). Since the figure is planar,z = 0 everywhere, hence Ixz = Izx = 0, Iyz = Izy = 0, and also Izz = Ixx+ Iyy . We nowcompute the remaining independent elements:

Ixx=

a0

dx

b(1xa)

0

dy y2 =

a0

dx 13b3

1 xa3

= 13 ab3

10

du (1 u)3 = 112 ab3 = 16M b2 (13.24)

and

Ixy =

a

0

dx

b(1xa)

0

d y x y=

12 b

2

a

0

dx x 1 xa2

= 12 a2b21

0

du u (1 u)2 = 124 a2b2 = 112Mab . (13.25)

Thus,

I=M

6

b2 12ab 012ab a2 0

0 0 a2 + b2

. (13.26)


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13.3. PARALLEL AXIS THEOREM 7

Figure 13.4: A planar mass distribution in the shape of a triangle.

Suppose we wanted the inertia tensor relative in a coordinate system where the CM liesat the origin. What we computed in eqn. 13.26 is I(d), with d=

a

3x

b

3y. Thus,

d2 d d=1

9

b2 ab 0ab a2 0

0 0 a2 + b2

. (13.27)

SinceI(d) =ICM + M

d2 d d

, (13.28)

we have that

ICM =I(d) Md2 d d

(13.29)

=M

18

b2 12ab 012ab a

2 00 0 a2 + b2

. (13.30)

13.3.2 General planar mass distribution

For a general planar mass distribution,

(x,y ,z) =(x, y) (z) , (13.31)

which is confined to the plane z = 0, we have

Ixx=

dx

dy (x, y) y2 (13.32)

Iyy =

dx

dy (x, y) x2 (13.33)

Ixy =

dx

dy (x, y) xy (13.34)

and Izz =Ixx+ Iyy , regardless of the two-dimensional mass distribution (x, y).


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13.4 Principal Axes of Inertia

We found that an orthogonal transformation to a new set of axes e

=Re entailsI =RIRt for the inertia tensor. Since I = It is manifestly a symmetric matrix, it canbe brought to diagonal form by such an orthogonal transformation. To findR, follow thisrecipe:

1. Find the diagonal elements ofI by setting P() = 0, where

P() = det

1 I , (13.35)is the characteristic polynomial for I, and 1 is the unit matrix.

2. For each eigenvalue a, solve the d equations

Ia=a a . (13.36)Here,a is the

th component of the ath eigenvector. Since ( 1 I) is degenerate,these equations are linearly dependent, which means that the first d 1 componentsmay be determined in terms of the dth component.

3. Because I = It, eigenvectors corresponding to different eigenvalues are orthogonal.In cases of degeneracy, the eigenvectors may be chosen to be orthogonal, e.g. via theGram-Schmidt procedure.

4. Due to the underdetermined aspect to step 2, we may choose an arbitrary normaliza-tion for each eigenvector. It is conventional to choose the eigenvectors to be orthonor-mal: ab = ab.

5. The matrixRis explicitly given byRa = a, the matrix whose row vectors are theeigenvectors a. Of courseRt is then the corresponding matrix of column vectors.

6. The eigenvectors form a complete basis. The resolution of unity may be expressed asa

aa= . (13.37)

As an example, consider the inertia tensor for a general planar mass distribution, whichis of the form

I=

Ixx Ixy 0Iyx Iyy 0

0 0 Izz

, (13.38)

whereIyx = Ixy and Izz =Ixx+ Iyy . Define

A= 12

Ixx+ Iyy

(13.39)

B=

14

Ixx Iyy

2+ I2xy (13.40)

= tan1

2IxyIxx Iyy

, (13.41)


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13.5. EULERS EQUATIONS 9

so that

I= A + B cos B sin 0

B sin A

B cos 0

0 0 2A , (13.42)The characteristic polynomial is found to be

P() = ( 2A)

( A)2 B2

, (13.43)

which gives 1= A, 2,3= A B. The corresponding normalized eigenvectors are

1 =

00

1

, 2 =

cos 12sin 12

0

, 3 =

sin 12cos 12

0

(13.44)

and therefore

R = 0 0 1cos 12 sin 12 0

sin 12 cos 12 0

. (13.45)

13.5 Eulers Equations

Let us now choose our coordinate axes to be the principal axes of inertia, with the CM atthe origin. We may then write

=12

3

, I= I1 0 00 I2 00 0 I3

= L= I1 1I2 2I3 3

. (13.46)The equations of motion are

Next =

dL

dt

inertial

=

dL

dt

body

+ L

=I + (I) .

Thus, we arrive at Eulers equations:

I1 1= (I2 I3) 2 3+ Next1 (13.47)I2 2= (I3 I1) 3 1+ Next2 (13.48)I3 3= (I1 I2) 1 2+ Next3 . (13.49)

These are coupled and nonlinear. Also note the fact that the external torque must beevaluated along body-fixed principal axes. We can however make progress in the case


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Figure 13.5: Wobbling of a torque-free symmetric top.

where Next = 0, i.e. when there are no external torques. This is true for a body in freespace, or in a uniform gravitational field. In the latter case,

Next =i

ri (mi g) =

i

miri

g , (13.50)

whereg is the uniform gravitational acceleration. In a body-fixed frame whose origin is theCM, we have

i miri= 0, and the external torque vanishes!

Precession of torque-free symmetric tops: Consider a body which has a symme-

try axis e3. This guaranteesI1 = I2, but in general we still have I1= I3. In the absenceof external torques, the last of Eulers equations says 3 = 0, so 3 is a constant. Theremaining two equations are then

1=

I1 I3

I1

3 2 , 2=

I3 I1

I1

3 1 . (13.51)

I.e.1= 2 and 2= + 1, with

=

I3 I1

I1

3 , (13.52)

which are the equations of a harmonic oscillator. The solution is easily obtained:

1(t) =cos

t +

, 2(t) =sin

t +

, 3(t) =3 , (13.53)

where

andare constants of integration, and where || = (2

+ 23)1/2. This motion is

sketched in Fig. 13.5. Note that the perpendicular components of oscillate harmonically,and that the angle makes with respect to e3 is = tan

1(

/3).

For the earth, (I3 I1)/I1 1305 , so3 , and /305, yielding a precession periodof 305 days, or roughly 10 months. Astronomical observations reveal such a precession,


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13.5. EULERS EQUATIONS 11

known as the Chandler wobble. For the earth, the precession angle is Chandler 6 107rad, which means that the North Pole moves by about 4 meters during the wobble. TheChandler wobble has a period of about 14 months, so the nave prediction of 305 days is

off by a substantial amount. This discrepancy is attributed to the mechanical properties ofthe earth: elasticity and fluidity. The earth is not solid!1

Asymmetric tops: Next, consider the torque-free motion of an asymmetric top, whereI1=I2=I3=I1. Unlike the symmetric case, there is no conserved component of . True,we can invoke conservation of energy and angular momentum,

E= 12I1 21+

12I2

22+

12I3

23 (13.54)

L2 =I2121+ I

22

22+ I

23

23 , (13.55)

and, in principle, solve for 1

and 2

in terms of 3

, and then invoke Eulers equations(which must honor these conservation laws). However, the nonlinearity greatly complicatesmatters and in general this approach is a dead end.

We can, however, find a particular solution quite easily one in which the rotation isabout a single axis. Thus, 1 = 2 = 0 and 3 = 0 is indeed a solution for all time,according to Eulers equations. Let us now perturb about this solution, to explore itsstability. We write

= 0e3+ , (13.56)

and we invoke Eulers equations, linearizing by dropping terms quadratic in . This yield

I1 1= (I2

I3) 0 2+

O(2 3) (13.57)

I2 2= (I3 I1) 0 1+ O(3 1) (13.58)I3 3= 0 + O(1 2) . (13.59)

Taking the time derivative of the first equation and invoking the second, and vice versa,yields

1= 2 1 , 2= 2 2 , (13.60)with

2 =(I3 I2)(I3 I1)

I1 I2 20 . (13.61)

The solution is then 1(t) =C cos(t + ).

If2 >0, thenis real, and the deviation results in a harmonic precession. This occursifI3 is either the largest or the smallest of the moments of inertia. If, however, I3 is themiddle moment, then 2


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13.5.1 Example

PROBLEM: A unsuspecting solid spherical planet of mass M0 rotates with angular velocity0. Suddenly, a giant asteroid of mass M0 smashes into and sticks to the planet at alocation which is at polar angle relative to the initial rotational axis. The new massdistribution is no longer spherically symmetric, and the rotational axis will precess. RecallEulers equation

dL

dt + L= Next (13.62)

for rotations in a body-fixed frame.

(a) What is the new inertia tensor Ialong principle center-of-mass frame axes? Dont

forget that the CM is no longer at the center of the sphere! RecallI= 25MR2 for a solid

sphere.

(b) What is the period of precession of the rotational axis in terms of the original lengthof the day 2/0?

SOLUTION: Lets choose body-fixed axes with z pointing from the center of the planet tothe smoldering asteroid. The CM lies a distance

d=M0 R+ M0 0

(1 + )M0=

1 + R (13.63)

from the center of the sphere. Thus, relative to the center of the sphere, we have

I= 25M0R2

1 0 00 1 0

0 0 1

+ M0R2

1 0 00 1 0

0 0 0

. (13.64)

Now we shift to a frame with the CM at the origin, using the parallel axis theorem,

I(d) =ICM + M

d2 dd

. (13.65)

Thus, withd= dz,

ICM = 25M0R

2

1 0 00 1 0

0 0 1

+ M0R2

1 0 00 1 0

0 0 0

(1 + )M0d2

1 0 00 1 0

0 0 0

(13.66)

=M0R2

25 + 1+ 0 00 25+ 1+ 0

0 0 25

. (13.67)


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13.6. EULERS ANGLES 13

In the absence of external torques, Eulers equations along principal axes read

I1d1

dt

= (I2

I3) 2 3

I2d2dt

= (I3 I1) 3 1I3

d3dt

= (I1 I2) 1 2(13.68)

Since I1 = I2, 3(t) = 3(0) = 0cos is a constant. We then obtain 1 = 2, and

2= 1, with=

I2 I3I1

3= 5

7 + 23 . (13.69)

The period of precession in units of the pre-cataclysmic day is

T =

=

7 + 2

5 cos . (13.70)

13.6 Eulers Angles

In d dimensions, an orthogonal matrixR O(d) has 12d(d 1) independent parameters.To see this, consider the constraintRtR = 1. The matrixRtR is manifestly symmetric,so it has 12d(d+ 1) independent entries (e.g. on the diagonal and above the diagonal).This amounts to 12d(d+ 1) constraints on the d

2 components ofR, resulting in 12d(d 1)freedoms. Thus, in d = 3 rotations are specified by three parameters. The Euler angles{,,} provide one such convenient parameterization.

A general rotationR(,,) is built up in three steps. We start with an orthonormaltriad e0 of body-fixed axes. The first step is a rotation by an angle about e

03:

e = Re03,

e0 , R

e03,

=

cos sin 0 sin cos 0

0 0 1

(13.71)

This step is shown in panel (a) of Fig. 13.6. The second step is a rotation by about thenew axis e1:

e = Re1,

e , R

e1,

=1 0 00 cos sin

0 sin cos

(13.72)This step is shown in panel (b) of Fig. 13.6. The third and final step is a rotation by about the new axis e3:

e = Re3,

e , R

e3,

=

cos sin 0 sin cos 0

0 0 1

(13.73)


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Figure 13.6: A general rotation, defined in terms of the Euler angles{,,}. Threesuccessive steps of the transformation are shown.

This step is shown in panel (c) of Fig. 13.6. Putting this all together,

R(,,) = Re

3, Re

1, Re03, (13.74)

=

cos sin 0 sin cos 0

0 0 1

1 0 00 cos sin

0 sin cos

cos sin 0 sin cos 0

0 0 1

=

cos cos sin cos sin cos sin + sin cos cos sin sin sin cos cos cos sin sin sin + cos cos cos cos sin

sin sin sin cos cos

.

Next, wed like to relate the components = e (with e e) of the rotation in

the body-fixed frame to the derivatives ,

, and

. To do this, we write

= e+e+

e , (13.75)

where

e03= e= sin sin e1+ sin cos e2+ cos e3 (13.76)

e = cos e1 sin e2 (line of nodes) (13.77)e =e3 . (13.78)


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This gives

1=

e1=

sin sin + cos (13.79)

2= e2= sin cos sin (13.80)3= e3= cos + . (13.81)

Note that

precession , nutation , axial rotation . (13.82)

The general form of the kinetic energy is then

T = 12I1

sin sin + cos 2

+ 12I2 sin cos sin

2+ 12I3 cos +

2

. (13.83)

Note that

L= pe+p e+ p e , (13.84)

which may be verified by explicit computation.

13.6.1 Torque-free symmetric top

A body falling in a gravitational field experiences no net torque about its CM:

Next = i ri (

mi g) =g

imi ri= 0 . (13.85)

For a symmetric top with I1 = I2, we have

T = 12I1

2 + 2 sin2

+ 12I3

cos + 2

. (13.86)

The potential is cyclic in the Euler angles, hence the equations of motion are

d

dt

T

(, , )=

T

(,,) . (13.87)

Since and are cyclic in T, their conjugate momenta are conserved:

p= L

=I1 sin2+ I3( cos + ) cos (13.88)

p =L

=I3(

cos + ) . (13.89)

Note thatp =I3 3, hence 3 is constant, as we have already seen.

To solve for the motion, we first note that L is conserved in the inertial frame. We aretherefore permitted to define L=e03 = e. Thus,p = L. Since e e = cos , we have


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Figure 13.7: A dreidlis a symmetric top. The four-fold symmetry axis guarantees I1= I2.The blue diamond represents the center-of-mass.

that p = L e = L cos . Finally, e e = 0, which means p = L e = 0. From theequations of motion,

p = I1=

I1 cos p

sin , (13.90)

hence we must have= 0 , =

pI1cos

. (13.91)

Note that = 0 follows from conservation ofp = L cos . From the equation forp, wemay now conclude

=pI3

pI1

=

I3 I1

I3

3 , (13.92)

which recapitulates (13.52), with = .

13.6.2 Symmetric top with one point fixed

Consider the case of a symmetric top with one point fixed, as depicted in Fig. 13.7. TheLagrangian is

L= 12I1

2 + 2 sin2

+ 12I3

cos + 2 M g cos . (13.93)

Here, is the distance from the fixed point to the CM, and the inertia tensor is defined alongprincipal axes whose origin lies at the fixed point (not the CM!). Gravity now supplies atorque, but as in the torque-free case, the Lagrangian is still cyclic in and , so

p= (I1sin2+ I3cos

2) + I3cos (13.94)

p =I3cos + I3 (13.95)


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Figure 13.8: The effective potential of eq. 13.102.

are each conserved. We can invert these relations to obtain and in terms of{p, p, }:

=p p cos

I1 sin2

, =pI3

(p p cos ) cos I1 sin

2 . (13.96)

In addition, since L/t= 0, the total energy is conserved:

E=T+ U= 12I12+

Ueff()

(p p cos )22I1 sin

2 + p

2

2I3+ Mg cos , (13.97)

where the term under the brace is the effective potential Ueff().

The problem thus reduces to the one-dimensional dynamics of(t), i.e.

I1= Ueff

, (13.98)

with

Ueff() =(p p cos )2

2I1 sin2

+p22I3

+ M g cos . (13.99)

Using energy conservation, we may write

dt=

I12

dE Ueff()

. (13.100)

and thus the problem is reduced to quadratures:

t() =t(0)

I12

0

d 1E Ueff()

. (13.101)


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Figure 13.9: Precession and nutation of the symmetry axis of a symmetric top.

We can gain physical insight into the motion by examining the shape of the effectivepotential,

Ueff() =(p p cos )2

2I1 sin2 + Mg cos +

p2

2I3, (13.102)

over the interval [0, ]. ClearlyUeff(0) =Ueff() = , so the motion must be bounded.What is not yet clear, but what is nonetheless revealed by some additional analysis, is thatUeff() has a single minimum on this interval, at = 0. The turning points for the motionare at = a and = b, whereUeff(a) =Ueff(b) =E. Clearly if we expand about 0 andwrite = 0+ , the motion will be harmonic, with

(t) =0 cos(t + ) , =

Ueff(0)

I1. (13.103)

To prove that Ueff() has these features, let us define u cos . Then u= sin , andfromE= 12I1

2 + Ueff() we derive

u2 =

2E

I1 p

2

I1I3

(1 u2) 2Mg

I1(1 u2) u

p pu

I1

2 f(u) . (13.104)

The turning points occur at f(u) = 0. The functionf(u) is cubic, and the coefficient of the

cubic term is 2Mg/I1, which is positive. Clearly f(u= 1) = (p p)2/I21 is negative,so there must be at least one solution to f(u) = 0 on the interval u (1, ). Clearly therecan be at most three real roots for f(u), since the function is cubic in u, hence there are at

most two turning points on the interval u [1, 1]. Thus, Ueff() has the form depicted infig. 13.8.

To apprehend the full motion of the top in an inertial frame, let us follow the symmetryaxis e3:

e3= sin sin e01 sin cos e02+ cos e03 . (13.105)

Once we know (t) and (t) were done. The motion(t) is described above: oscillatesbetween turning points at a and b. As for (t), we have already derived the result

=p p cos

I1 sin2

. (13.106)


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13.7. ROLLING AND SKIDDING MOTION OF REAL TOPS 19

Figure 13.10: A top with a peg end. The frictional forces f andfskid are shown. When the

top rolls without skidding, fskid = 0.

Thus, ifp> pcos a, thenwill remain positive throughout the motion. If, on the other

hand, we have

pcos b < p< pcos a , (13.107)

then changes sign at an angle = cos1p/p

. The motion is depicted in Fig. 13.9.

An extensive discussion of this problem is given in H. Goldstein, Classical Mechanics.

13.7 Rolling and Skidding Motion of Real Tops

The material in this section is based on the corresponding sections from V. Barger andM. Olsson, Classical Mechanics: A Modern Perspective. This is an excellent book whichcontains many interesting applications and examples.

13.7.1 Rolling tops

In most tops, the point of contact rolls or skids along the surface. Consider the peg end topof Fig. 13.10, executing a circular rolling motion, as sketched in Fig. 13.11. There are three

components to the force acting on the top: gravity, the normal force from the surface, andfriction. The frictional force is perpendicular to the CM velocity, and results in centripetalacceleration of the top:

f=M 2 Mg , (13.108)whereis the frequency of the CM motion and is the coefficient of friction. If the aboveinequality is violated, the top starts to slip.

The frictional and normal forces combine to produce a torque N = M g sin f cos


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Figure 13.11: Circular rolling motion of the peg top.

about the CM2. This torque is tangent to the circular path of the CM, and causes L toprecess. We assume that the top is spinning rapidly, so that L very nearly points alongthe symmetry axis of the top itself. (As well see, this is true for slow precession but not

for fast precession, where the precession frequency is proportional to 3.) The precession isthen governed by the equation

N=M g sin f cos = L =

L I3 3sin , (13.109)

wheree3 is the instantaneous symmetry axis of the top. Substituting f=M 2,

M g

I3 3

1

2

g ctn

= , (13.110)

which is a quadratic equation for . We supplement this with the no slip condition,

3 =

+ sin

, (13.111)

resulting in two equations for the two unknowns and .

Substituting for () and solving for , we obtain

= I3 3

2M2 cos

1 +

Mg

I3ctn

1 +

Mg

I3ctn

24M

2

I3 M g

I3 23

. (13.112)

This in order to have a real solution we must have

3 2M 2 sin

I3sin + Mgcos

g

. (13.113)

2Gravity of course produces no net torque about the CM.


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If the inequality is satisfied, there are two possible solutions for , corresponding to fastand slow precession. Usually one observes slow precession. Note that it is possible that < 0, in which case the CM and the peg end lie on opposite sides of a circle from each

other.

13.7.2 Skidding tops

A skidding top experiences a frictional force which opposes the skidding velocity, untilvskid = 0 and a pure rolling motion sets in. This force provides a torque which makes thetop rise:

= NskidL

= MgI3 3

. (13.114)

Suppose

0, in which case + sin = 0, from eqn. 13.111, and the point of contact

remains fixed. Now recall the effective potential for a symmetric top with one point fixed:

Ueff() =(p p cos )2

2I1 sin2

+p22I3

+ Mg cos . (13.115)

We demand Ueff(0) = 0, which yields

cos 0 2 psin20 + MgI1sin40= 0 , (13.116)where

p pcos 0= I1sin20 . (13.117)

Solving the quadratic equation for , we find

= I3 3

2I1cos 0

1

1 4MgI1cos 0

I2323

. (13.118)

This is simply a recapitulation of eqn. 13.112, with = 0 and with M 2 replaced by I1.

NoteI1= M 2 by the parallel axis theorem ifICM1 = 0. But to the extent thatI

CM1 = 0, our

treatment of the peg top was incorrect. It turns out to be OK, however, if the precession isslow, i.e. if/3 1.

On a level surface, cos 0> 0, and therefore we must have

3 2I3

MgI1cos 0 . (13.119)

Thus, if the top spins too slowly, it cannot maintain precession. Eqn. 13.118 says that thereare two possible precession frequencies. When 3 is large, we have

slow = M g

I3 3+ O(13 ) , fast=

I3 3I1cos 0

+ O(33 ) . (13.120)

Again, one usually observes slow precession.


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Figure 13.12: The tippie-top behaves in a counterintuitive way. Once started spinning withthe peg end up, the peg axis rotates downward. Eventually the peg scrapes the surface andthe top rises to the vertical in an inverted orientation.

A top with 3 > 2I3

MgI1 may sleep in the vertical position with 0 = 0. Due to the

constant action of frictional forces, 3 will eventually drop below this value, at which timethe vertical position is no longer stable. The top continues to slow down and eventuallyfalls.

13.7.3 Tippie-top

A particularly nice example from the Barger and Olsson book is that of the tippie-top, atruncated sphere with a peg end, sketched in Fig. 13.12 The CM is close to the centerof curvature, which means that there is almost no gravitational torque acting on the top.The frictional force f opposes slipping, but as the top spins f rotates with it, and hencethe time-averaged frictional forcef 0 has almost no effect on the motion of the CM.A similar argument shows that the frictional torque, which is nearly horizontal, also timeaverages to zero:

dLdt inertial 0 . (13.121)In the body-fixed frame, however, N is roughly constant, with magnitude N MgR,

whereRis the radius of curvature and the coefficient of sliding friction. Now we invoke

N=dL

dt

body

+ L . (13.122)

The second term on the RHS is very small, because the tippie-top is almost spherical, hence


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inertia tensor is very nearly diagonal, and this means

L

I= 0 . (13.123)

Thus, Lbody N, and taking the dot product of this equation with the unit vector k, weobtain

Nsin = k N= ddt

k Lbody

= L sin . (13.124)

Thus,

=N

L MgR

I . (13.125)

Once the stem scrapes the table, the tippie-top rises to the vertical just like any other risingtop.

Documents

Ch13 Rigid Bodies