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  5. Precession of a gyroscope

11.4 Precession of a gyroscope

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By the end of this section, you will be able to:
  • Describe the physical processes underlying the phenomenon of precession
  • Calculate the precessional angular velocity of a gyroscope

[link] shows a gyroscope , defined as a spinning disk in which the axis of rotation is free to assume any orientation. When spinning, the orientation of the spin axis is unaffected by the orientation of the body that encloses it. The body or vehicle enclosing the gyroscope can be moved from place to place and the orientation of the spin axis will remain the same. This makes gyroscopes very useful in navigation, especially where magnetic compasses can’t be used, such as in manned and unmanned spacecraft, intercontinental ballistic missiles, unmanned aerial vehicles, and satellites like the Hubble Space Telescope.

A drawing of a gyroscope, consisting of a disk that can spin on an shaft, perpendicular to the plane of the disk and through its center. Two rings surround the gyroscope. One is attached to the shaft above and below the disk, and the other is attached to the first ring and is in the plane of the disk so that this second ring is concentric with the disk.
A gyroscope consists of a spinning disk about an axis that is free to assume any orientation.

We illustrate the precession    of a gyroscope with an example of a top in the next two figures. If the top is placed on a flat surface near the surface of Earth at an angle to the vertical and is not spinning, it will fall over, due to the force of gravity producing a torque acting on its center of mass. This is shown in [link] (a). However, if the top is spinning on its axis, rather than topple over due to this torque, it precesses about the vertical, shown in part (b) of the figure. This is due to the torque on the center of mass, which provides the change in angular momentum.

Figure a: An x y z coordinate system is show, with x out of the page, y to the right, and z up. The origin is point O. A top is shown with its point at the origin and its axis tilted away from the vertical z axis. The axis of the top is the line O O prime. The vector r extends from the origin to the center of the mass, labeled as C M, of the top. The force M g acts downward at the center of mass. The torque about the origin is equal to vector r crossed with M vector g. This torque is a vector in the x y plane, perpendicular to the r vector. Figure b: The x y z coordinate and the top are shown. The top is again tilted away from the z axis and is spinning rapidly counterclockwise about the O O prime axis as viewed from above. The precession of the top traces a counterclockwise circle as viewed from above, centered on the z axis. The cone swept by the precession of the top is indicated using dashed lines.
(a) If the top is not spinning, there is a torque r → × M g → about the origin, and the top falls over. (b) If the top is spinning about its axis O O ′ , it doesn’t fall over but precesses about the z- axis.

[link] shows the forces acting on a spinning top. The torque produced is perpendicular to the angular momentum vector. This changes the direction of the angular momentum vector L → according to d L → = τ → d t , but not its magnitude. The top precesses around a vertical axis, since the torque is always horizontal and perpendicular to L → . If the top is not spinning, it acquires angular momentum in the direction of the torque, and it rotates around a horizontal axis, falling over just as we would expect.

An x y z coordinate system is show, with x out of the page, y to the right, and z up. The origin is point O. A top is shown with its point at the origin and its axis tilted by an angle theta away from the vertical z axis, clockwise as we view it. The vector r extends from the origin to the center of the mass, labeled as C M, of the top. The force M g acts downward at the center of mass. The torque, tau, about the origin is equal to vector r crossed with M vector g. This torque is a vector in the x y plane, perpendicular to the r vector, into the page. The angular velocity, omega, of the top is counterclockwise as viewed from above. The angular momentum, L, is in the same direction as the r vector, tilted up along the axis of the top. The circle traced by the precession of the top is shown as a horizontal circle above the top. The precession angular velocity omega sub p is counterclockwise as viewed from above. The radius of the precession circle is L sine theta. The vector d L is tangent to the circle, pointing into the page, and is equal to vector tau d t. The triangle formed L sine theta and d L is shown, and the angle across from d L is labeled as d phi.
The force of gravity acting on the center of mass produces a torque τ → in the direction perpendicular to L → . The magnitude of L → doesn’t change but its direction does, and the top precesses about the z -axis.

We can experience this phenomenon first hand by holding a spinning bicycle wheel and trying to rotate it about an axis perpendicular to the spin axis. As shown in [link] , the person applies forces perpendicular to the spin axis in an attempt to rotate the wheel, but instead, the wheel axis starts to change direction to her left due to the applied torque.

In figure a, a woman, facing the viewer, is holding a spinning bike wheel of radius r by the axle. The wheel is so that the angular velocity omega and angular momentum L are along the axis of rotation of the wheel, to her left (the viewer’s right.) That is, the motion of the wheel is such that the bottom of the wheel is moving toward her (into the page.) The direction of the force F applied by her left hand is shown downward and that by her right hand in upward direction. The torque tau is toward her (into the page.) In figure b, addition of two vectors L and delta-L, which is parallel to torque tau, is shown. The resultant of the two vectors is labeled as L plus delta L. The direction of rotation, omega sub p, is counterclockwise as viewed from above.
(a) A person holding the spinning bike wheel lifts it with her right hand and pushes down with her left hand in an attempt to rotate the wheel. This action creates a torque directly toward her. This torque causes a change in angular momentum Δ L → in exactly the same direction. (b) A vector diagram depicting how Δ L → and L → add, producing a new angular momentum pointing more toward the person. The wheel moves toward the person, perpendicular to the forces she exerts on it.

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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