Towards understanding Precessional Motion

Dr. Muhammed Arif M

Assistant professor Chemistry,

Govt. Engineering College Kozhikode,

Kerala, India arifmmm@gmail.com

Key words

Precessional motion, precession, top, earth’s precession, gyroscope, NMR, gyroscopic couple.

Precessional motion can be observed when a spinning top is kept under gravitational field,

or in a rotating ceiling fan (in case of imbalance). It is also underlying principle behind the balancing

of bicycle. Precessional motion is also important for understanding the theory of NMR spectroscopy.

Consider a spinning top in gravitational field. Why the tilted spinning top is not falling down while

spinning?

Falling torque

In the case of a top the centre of mass is above the point of suspension, therefore it is unstable

when it is not spinning, whereas it is stable while spinning. It is because the falling torque is

balanced by rate of change of spinning angular momentum.

The torque is the cross product of field strength vector and dipole moment vector.

In the case of gravitational field the field strength is the force produced on unit mass, it is equal to

acceleration due to gravity. If is the vector joining the centre of mass (C) and point of suspension

(O), then the dipole moment or simply moment is equal to , where is the mass of the

top

Therefore torque

If θ is the angle between the vectors and

Then the magnitude of torque, -------(1)

Expression potential energy

Potential energy is the negative of dot product of field strength vector and dipole moment vector

Potential energy

=

Special cases

Case I Case 2 case 3

Case 1 (centre of mass is above the point of suspension):

Torque is zero and potential energy is at maximum

Case 2 (centre of mass lying in the horizontal plane contain the point of suspension) :

Here torque is at maximum and potential energy is zero

Case 3 ( centre of mass is below the point of suspension eg. Ceiling fan) :

Torque is zero and potential energy is at minimum .

Rate of change of angular momentum

When the top is spinning in anti-clockwise the spin angular momentum vector is directed

out words along the spinning axis as shown in fig 1 . The angular momentum vector can be

resolved into two components and as shown in fig 2 , Here the component ,

remains constant, and the component undergoes change due to change of direction as

shown in fig 1 . The change of angular momentum can be worked out in polar co-ordinates (θ, φ) as

shown in fig 3. In polar co-orates θ varies from 0 to π, whereas φ varies from 0 to 2π.

From the fig 3 the change in angular momentum for an infinitesimal change in angle

occurred in a time interval dt is obtained by subtracting initial direction of form its final

direction

arc length = radius x angle

Therefore the rate of change of angular momentum can be written as

Where is the processional angular velocity vector, which is directed along the precessional axis,

When a spinning top undergo precessional motion the falling torque (eq (1) ) is balanced by rate of

change of angular momentum (eq (2))

Therefore

Where I is the moment of inertia of spin and is spin angular velocity

Therefore it is clear that the precessional angular velocity is independent of the angle with which it

precesses (θ)

According to laws of mechanics the angular momentum conservation principle is valid only when

there is no external torque acting on the system. In the case of a spinning top the total angular

momentum (sum of spin and precession) is conserved or not?, as there are torques acting on the

system, namely frictional torque and gravitational torque. The frictional torque reduces the spin

angular momentum, while gravitational torque changes the spin orientation with respect to the

field. If there is no friction while précising at certain angle or if there is continuous drive from a

motor, the angle of precession (θ) remains constant. In such a situation the torque due to gravity is

balanced by rate of change on angular momentum as shown in eq.(2) and frictional torque is

balanced by electric motor torque, thus net torque acting on the system is zero. Therefore the total

angular momentum (the sum of spin and precessional) remains conserved.

When a spinning top descents down under gravity it follows a spiral path. Here there is a question

whether a part of torque due to gravity balances with frictional torque?. Another part is goes to

precession KE of precession; such that the magnitude of total angular momentum remains

conserved. When the spin speed decreases due to friction, the spin angular momentum decreases

which is compensated by increase in precession angular momentum. From eq. (3) we have seen that

the precessional angular velocity (ωp ) is inversely related to spin angular speed (ωs) and

independent of angle of precession (θ). Therefore there is a decrease in precession angular speed as

top descends down. But the moment of inertia of precession increases as the top descents down; as

a result there is increase in precessional angular momentum. When the top descents down from

case 1 to case 2 , the kinetic energy of precession increases at the expenses of KE of spin and

gravitational potential energy. When the top further descents down from case 2 to case 3 , the

decrease in KE of precession a part of gravitational potential energy are transferred into KE energy

of spin such that the magnitude of total angular momentum remains conserved until it reaches the

lowest potential energy mode case 3. The energy difference between the two extreme orientations

(case1 and case3) is 2mgr which is converted to frictional heat as the top descents down in spiral

path. Though we can apply the law of conservation of angular momentum, the total KE of rotation is

not conserved. Here we can see that the KE of spin decreases as it moves down from case 1 to case 2

and then regain the initial value when reach the minimum PE mode ( case 3 ).

Imagine a top spinning on the top of a cliff and imagine the point of suspension is made of a special

type of bearing such that the centre of mass can decent down below the point of suspension ( i.e

descent down from case 2 to case3 ). If the top spinning in the anti-clock wise direction when

viewed form top, as it starts precessing with a precessional angular momentum which is also

anticlockwise direction, when the top is in upright mode or positive gravitational potential energy

mode (case 1). The vertical component of spin angular is in upward direction when the top is

spinning with positive gravitational potential energy. But when the centre of mass comes below

point of suspension ie ., when potential energy become negative and the spin angular momentum

get directed down wards while keeping the precessional angular momentum in upward direction.

Therefore it can be concluded that in positive gravitational potential energy mode both precessional

angular momentum and vertical component of spin angular momentum are directed along the same

direction. While in negative potential energy mode (case 3) both are in opposite direction.

Celling fan

We can observe precessional motion of ceiling fan (case 3: centre of mass is below the point of

suspension) Here, The precessional angular velocity is inherent as r > 0. As potential energy of the

system is at minimum, we can observe precision in case of slight imbalance in mass distribution

about the spinning axis and the rod connecting the fan and the celling is somewhat flexible. The

depends on r and spin angular velocity as per the equation (3). The increase of potential energy

due to precision can be treated as in conical pendulum. Though the spinning ceiling fan is in negative

potential energy mode, both precession and vertical component of spin vector are along the same

direction, as it is a case of assenting against gravity.

Gyroscope

Gyroscope is a device used as a navigation tool like magnetic compass. With the advancement of

ship building from wood to steel, the magnetic compass becomes obsolete and gyroscopes were

begun to use in ships. The gyroscope was invented in 1852 by the French experimental

physicist Leon Foucault. Gyroscope consists of electrically driven spinning disk, mounded on

gambles. Gumball is a special type of baring system comprising of three rings which can move in

three mutually perpendicular axes. The gyroscope works on the principle of conservation of angular

momentum, which is valid only when no external torque is acting. In the case of gyroscope the

centre of mass and point of suspension are same, ie, r = 0 ie, no torque from gravity. Therefore

there is no tendency to decrease its potential energy, or change of spin orientation. It can maintain

its initial orientation even when the ship is turning. When an external torque is applied (say wind) it

starts précising , when the torque is removed it regains the original orientation. Gyroscopes are

usually kept inside a glass cage such that there are no effects of wind. Even now gyros are widely

used in space navigation, aircrafts and underground tunnelling. Modern gyros like FOG works on the

principle of interference rather than mechanical motion.

FOG

A fibre optic gyroscope (FOG) senses changes in orientation, thus performing the function of a mechanical gyroscope. However its principle of operation is instead based on the interference of light which is passed through a coil of optical fibre which can be as long as 5 km. Two beams from a laser are injected into the same fibre but in opposite directions. Due to the Sagnac effect, the beam travelling against the rotation or turning experiences a slightly shorter path delay than the other beam. The resulting differential phase shift is measured through interferometry, thus translating one component of the angular velocity into a shift of the interference pattern which is measured photometrically. A FOG provides extremely precise rotational rate information, in part because of its lack of cross-axis sensitivity to vibration, acceleration, and shock. Unlike the classic spinning-mass gyroscope, the FOG has absolutely no moving parts and doesn't rely on inertial resistance to movement. Hence this is perhaps the most reliable alternative technology to the mechanical gyroscope. Because of their intrinsic reliability, FOGs are used for high performance space applications. Apple’s latest creation –the iPhone 4 – features a MEMS gyroscope within its pretty shell.

Precessional motion of earth.

We know the earth’s spinning time period is one day and orbiting time period is one year. Like top

precessing in the gravitational field of earth, The earth also undergoes precessional motion in the

combined gravitational fields of sun and moon. In fact the spin vector orients an angle of 23.5o from

the normal of orbiting plane which is responsible for the occurrence seasons on earth. The angle

23.5o is in fact is the angle of precession θ, and the time period of earths precision is about 26000

years. During the month of December there is winter in northern hemisphere summer in southern

hemisphere. At present winter solstice happens on December 22 in northern hemisphere. But due to

precision of earth there will be shift in winter solstice. It is estimated that winter solstice shifts a day

once in every 72 years (26000/365). It is believed that the birth of Jesus was on the day of winter

solstice in northern hemisphere. December 25th is fixed for Xmas day just before 200 to 300 years

(72 x (25-22) = 216 years). Due to precision of earth there will be slow shift in climate of a place. The

earth’s precession is due to bulging of equator that affects the combined fields of the sun and the

moon.

Bicycle

The basic principle of balancing of bicycle is the precision of wheels. We can balance the bicycle very

well while in motion than when the wheels are at rest. The torque created by the imbalance is

balanced by the rate of change of angular momentum of the wheel. The spin orientation of bicycle

wheel is dissimilar to that of the top, for stable position we have to maintain the spin orientation

angle at 90o with respect to field vector. Any deviation from 90o produces torque that must be

balanced by rate of change of angular momentum. Stability of bikes increases as becomes

smaller. Therefore the stability of bikes (low ) increases as the spin angular velocity ( ) and

moment of inertia of wheels becomes greater, which is evident from eq . (3).

Rocket propulsion

There is a high angular momentum disk at the nose of the rocket to keep the orientation of the

rocket along the direction of propulsion. Therefore even in the presence of small torques from

outside (like wind) it maintains the initial orientation.

Effect of the Gyroscopic Couple on a ship or an aeroplane

Ships and aeroplanes consist of large moment of inertia propeller and fly wheels. When aeroplane or

ship takes right or left turn there occur a rate of change of angular momentum because of change of

direction of angular momentum vector. Thus a counter couple is produced when they terns.

1. When the engine or propeller of aeroplane or fly wheel of ship rotate such that the spin

angular momentum vector is along the sailing direction and when the aeroplane or ship

takes a left turn, then the effect of reactive gyroscopic couple will be to dip the nose (stem:

for ship) and raise the tail of the aeroplane or ship.

2. When the aeroplane or ship takes a right turn under similar conditions above, the effect of

reactive gyroscopic couple will be to raise the nose and dip the tail of the aeroplane or ship.

In the case of ships the fly wheel must be mounded such that the spin axis is along the

length so as to maintain dynamic stability.

NMR spectroscopy

Nuclear magnetic resonance spectroscopy depends on the absorption of energy when the nucleus of an atom is excited from its lowest energy nuclear spin state to the next higher one. The nuclear energy levels are produced by keeping the nuclei in a magnetic field. The energy required for transitions falls in radio frequency region (60 - 500 MHz)

Nuclear spin and the splitting of energy levels in a magnetic field

Subatomic particles (electrons, protons and neutrons) can be imagined as spinning on their axes. In

many atoms (such as 12C) these spins are paired against each other, such that the nucleus of the

atom has no overall spin. However, in some atoms (such as 1H and 13C) the nucleus does possess an overall spin. The rules for determining the net spin of a nucleus are as follows;

1. If the number of neutrons and the number of protons are both even, then the nucleus has no

spin (I=0). eg

2. If the number of neutrons plus the number of protons (mass no.) is odd, then the nucleus has a

half-integral spin (i.e .I = 1/2, 3/2, 5/2….) eg.

3. If the number of neutrons and the number of protons are both odd, then the nucleus has an

integer spin (i.e. 1, 2, 3) eg.

Quantum mechanics tells us that a nucleus of spin I will have ‘2I + 1’ possible orientations. A

nucleus with spin ½ will have 2 possible orientations.

are the examples for such nuclei. In the absence of an external magnetic field, these orientations are of equal energy. If a magnetic field is applied, the energy levels split. Each level is given a magnetic quantum number, I.

Fig. Energy levels for a nucleus with spin quatum number 1/2

When the nucleus is in a magnetic field, the initial populations of the energy levels are determined by thermodynamics, as described by the Boltzmann distribution. This is very important, and it means that the lower energy level will contain slightly more nuclei than the higher level. It is possible to excite these nuclei into the higher level with electromagnetic radiation. The frequency of radiation needed is determined by the difference in energy between the energy levels.

The precession of nucleus in a magnetic field

Consider the spinning of a top in gravitational field, the spinning axis get tilted by an angle ‘’ and starts precessing with a frequency. In this discussion, we will be taking a “classical” view of the behaviour of the nucleus i.e. the behaviour of a nuclear magnet in a magnetic field.

Imagine a nucleus (of spin 1/2) in a magnetic field. This nucleus is in the lower energy level (i.e. its

magnetic moment align with the applied field). The nucleus is spinning on its axis. In the presence of

a magnetic field, this axis of spin will precess around the magnetic field, just like the Precessional

motion of a spinning top in gravitational field.

Fig. precessional motion of a spinning nucleus

The frequency of precession is termed as Larmor frequency, when this frequency becomes identical to the radiofrequency, resonance occurs and flipping of proton takes place with the absorption of radiation. The potential energy (E) of the précising nucleus is given by;

Where is the angle between the direction of the applied field and the axis of nuclear spin and μ is

the dipole moment of nuclear magnet.

If energy is absorbed by the nucleus, then the angle of precession, , will change to opposite

direction. For a nucleus of spin I=1/2 or cos 60o absorption of radiation “flips” the magnetic moment

to I=-1/2 or -cos 60o so that it opposes the applied field (the higher energy state). Here unlike

classical systems like top the spin orientation is quantised, it can take only two orientations either

60o with the field direction or 60o opposite of field direction and also there is no change in spin

angular velocity. It is in accordance with Heisenberg’s uncertainty principle that the spin orientation

must be uncertain along a circle.

Fig. fliping of spin

Where Lp and Ls represents precession and spin angular momentum respectively, in clock wise precession Lp is in downward direction. It is also clear from our discussion of spinning top on a cliff thought experiment that in negative magnetic potential energy mode (case a), both precessional angular momentum and vertical component of spin angular momentum are directed in opposite directions. While in positive potential energy mode (case b) both are in same direction. Thus when a proton in low potential energy mode (I = +½ ) transit to high energy (I = -½) mode there occurs a change in the spin orientation (θ) for m +cos 60 o to –cos 60 o , the torque required for the change of angular momentum is provided by the photon of angular momentum hυ ( angular momentum and energy are in same dimensions).

Calculating transition energy The nucleus has a positive charge and is spinning. This generates a small magnetic field. The nucleus therefore possesses a magnetic moment, µ, which is proportional to its spin quantum number I.

The constant, , is called the gyromagnetic ratio and is a fundamental nuclear constant which has a different value for every nucleus. h is the Planck’s constant. The energy of nucleus

Energy difference

Where

When an electromagnetic radiation of frequency () comes in resonance,

Fig. 7.16 splitting of energy levels in magnetic field

The energy difference between the two states is directly proportional to the strength of the applied

field. A magnetic field of 4.7 T (which is about 100,000 times stronger than earth’s magnetic field)

separates the two spin states of 1H only by an energy difference corresponding to electromagnetic

radiation having a frequency of 2 x108 Hz (200 MHz) which lies in the radio frequency (RF) region of

the electromagnetic spectrum

2

h I

No energy difference

between spin states in

the absence of magnetic

field

ΔE

At field

strength B

Fig. NMR spectrometer