Combined Translational and Rotational Motion
3. Gyroscopes and Rotational Stability
Learning outcomes
- I can explain how gyroscopes work.
- I can describe rotational stability using angular momentum.
- I can identify applications of gyroscopes.
- I can explain precession qualitatively.
- I can connect angular momentum to stability.
Gyroscopes and Rotational Stability
A gyroscope is a rapidly rotating object whose spinning motion gives it a strong tendency to maintain the direction of its rotation axis.
A simple gyroscope can be made from a spinning wheel or disk mounted so that its axis is free to change direction. When the wheel is not spinning, gravity can easily make it tip. When it spins rapidly, its behaviour changes dramatically because it possesses angular momentum.
For a rigid object rotating about a fixed principal axis:
L = Iω
where:
- L = angular momentum, measured in kg·m²/s
- I = moment of inertia, measured in kg·m²
- ω = angular velocity, measured in rad/s
The greater the angular momentum, the more difficult it is for an external torque to rapidly change the direction of the rotation axis.
Angular Momentum and Stability
Angular momentum is a vector quantity. It has both magnitude and direction.
For a spinning wheel, the angular momentum vector points along the rotation axis.
Its direction can be determined using the right-hand rule:
- curl the fingers of your right hand in the direction of rotation
- your thumb points in the direction of angular momentum
The spinning wheel tends to maintain this angular momentum unless an external torque acts.
This tendency produces what we commonly call rotational stability.
What Rotational Stability Means
Rotational stability does not mean that a spinning object is impossible to move.
It means that changing the direction of its angular momentum requires a torque acting over time.
The relationship is:
Στ_external = ΔL/Δt
or, in instantaneous form:
Στ_external = dL/dt
Therefore, an external torque changes angular momentum.
If the external torque is small, the direction of a large angular momentum changes relatively slowly.
Why a Fast-Spinning Gyroscope Is Stable
Consider two identical wheels.
One is spinning slowly.
The other is spinning rapidly.
Since:
L = Iω
the faster wheel has greater angular momentum.
For the same external torque, the faster-spinning wheel experiences a smaller rate of directional change relative to the size of its angular momentum.
This gives it greater rotational stability.
Factors Affecting Gyroscopic Stability
For a simple rigid rotor:
L = Iω
Therefore, rotational stability generally increases when:
- moment of inertia I increases
- angular velocity ω increases
Moment of inertia can be increased by placing more mass farther from the rotation axis.
Angular momentum can be increased by spinning the rotor faster.
A large, rapidly rotating wheel can therefore have substantial rotational stability.
Effect of Angular Velocity
Suppose a gyroscope has:
I = 0.20 kg·m²
At:
ω = 20 rad/s
its angular momentum is:
L = Iω
L = (0.20)(20)
L = 4.0 kg·m²/s
If its angular velocity increases to:
ω = 60 rad/s
then:
L = (0.20)(60)
L = 12 kg·m²/s
Tripling the angular velocity triples the angular momentum.
The faster-spinning gyroscope therefore has greater resistance to rapid changes in the direction of its axis under the same applied torque.
Effect of Moment of Inertia
Suppose two rotors spin at the same angular velocity:
ω = 30 rad/s
Rotor A:
I = 0.10 kg·m²
Rotor B:
I = 0.40 kg·m²
For Rotor A:
L = (0.10)(30)
L = 3.0 kg·m²/s
For Rotor B:
L = (0.40)(30)
L = 12 kg·m²/s
Rotor B has four times the angular momentum.
Its larger moment of inertia gives it greater angular momentum at the same angular velocity.
Mass Distribution Matters
Moment of inertia depends not only on mass but also on where the mass is located.
For a point mass:
I = mr²
The r² term means that moving mass farther from the axis can greatly increase moment of inertia.
This is why a rotor with substantial mass near its outer edge can have a large angular momentum.
Torque Changes Angular Momentum
A torque does not necessarily make a spinning gyroscope simply fall over.
Instead, torque changes its angular momentum.
Remember:
Στ = dL/dt
Because angular momentum is a vector, torque can change:
- the magnitude of L
- the direction of L
- or both
In many gyroscope situations, gravity mainly changes the direction of the angular momentum vector.
This produces precession.
What Is Precession?
Precession is the gradual turning of the rotation axis of a spinning object.
Instead of simply falling downward under gravity, a spinning gyroscope can have its axis move sideways around a vertical direction.
The tip of the rotation axis traces approximately a circular path.
This behaviour can seem surprising because the gravitational force acts downward, yet the gyroscope's axis moves mainly sideways.
Angular momentum explains why.
A Gyroscope Supported at One End
Imagine a spinning wheel with one end of its axle supported.
Gravity acts downward on the wheel's centre of mass.
The support point acts as a pivot.
Because the line of action of gravity does not pass through the pivot, gravity produces a torque.
The magnitude of this torque is approximately:
τ = rmg
when the axle is horizontal, where r is the horizontal distance from the pivot to the centre of mass.
This torque changes the direction of the angular momentum.
The result is precession.
Why the Gyroscope Does Not Immediately Fall
If the wheel were not spinning, gravity would cause it to rotate downward about the support.
When it is spinning rapidly, however, it already has a large angular momentum vector along its axle.
The gravitational torque continuously changes the direction of this vector.
Instead of immediately rotating downward, the axis moves around the pivot.
This sideways turning is precession.
A Vector View of Precession
Suppose the angular momentum initially points horizontally.
During a short time Δt, gravity produces a torque.
The resulting change in angular momentum is:
ΔL = τΔt
This small ΔL points in the direction of the torque.
The new angular momentum is the vector sum of:
L_new = L_old + ΔL
If ΔL is mostly perpendicular to L, the magnitude of L changes little, but its direction changes.
Repeated continuously, this produces precession.
Precession Rate
For a simple rapidly spinning gyroscope under gravity, the precession angular speed can be approximated by:
Ω = τ/L
Since:
L = Iω
then:
Ω = τ/(Iω)
For a gyroscope supported at one end with its axle approximately horizontal:
τ = mgr
so:
Ω = mgr/(Iω)
where:
- Ω = precession angular speed
- m = mass
- g = gravitational field strength
- r = distance from pivot to centre of mass
- I = moment of inertia of the spinning rotor
- ω = rotor angular velocity
This equation is an approximation for steady precession under appropriate conditions, but it reveals the important relationships clearly.
Faster Spin Means Slower Precession
Since:
Ω = τ/L
a larger angular momentum produces a smaller precession rate for the same torque.
Therefore:
greater ω → greater L → slower precession
This explains why a rapidly spinning gyroscope can appear very stable.
As it slows, its precession generally becomes more noticeable.
Worked Precession Example
A gyroscope has:
m = 1.5 kg
r = 0.20 m
I = 0.060 kg·m²
ω = 100 rad/s
First calculate angular momentum:
L = Iω
L = (0.060)(100)
L = 6.0 kg·m²/s
Gravitational torque:
τ = mgr
τ = (1.5)(9.8)(0.20)
τ = 2.94 N·m
Precession rate:
Ω = τ/L
Ω = 2.94/6.0
Ω = 0.49 rad/s
What Happens as the Gyroscope Slows?
Real gyroscopes experience:
- bearing friction
- air resistance
- internal deformation
These processes reduce the rotor's angular speed.
As ω decreases:
L = Iω decreases
For a similar gravitational torque:
Ω = τ/L
therefore increases.
The gyroscope often precesses faster and becomes less stable as it slows.
Eventually, it may wobble strongly and fall.
Precession Is Not the Same as Spin
A gyroscope can have several motions occurring simultaneously.
Spin — rapid rotation of the rotor around its own axis.
Precession — slower rotation of the spin axis around another direction.
Nutation — small oscillations or wobbling of the axis that can occur in addition to precession.
These motions should not be confused.
Nutation
A gyroscope does not always settle immediately into smooth precession.
Its axis may bob or wobble as it moves.
This motion is called nutation.
Nutation can occur when:
- the gyroscope is released suddenly
- its initial motion does not match steady precession
- external disturbances act on it
Friction may gradually reduce the wobbling.
Spinning Tops
A spinning top is a familiar example of gyroscopic behaviour.
When spinning rapidly:
- angular momentum is large
- the top remains relatively upright
- its axis precesses slowly
As friction reduces its spin:
- angular momentum decreases
- precession becomes more noticeable
- wobbling increases
- the top eventually falls
Why a Non-Spinning Top Falls
A stationary top has no spin angular momentum:
L = 0
Gravity produces a torque about the contact point.
With no large angular momentum already present, the top simply begins rotating in the direction produced by the gravitational torque and falls.
A rapidly spinning top responds differently because its existing angular momentum must be redirected.
Bicycle Wheels and Rotational Stability
A spinning bicycle wheel has angular momentum.
As wheel speed increases:
ω increases
and therefore:
L increases
This can contribute to the behaviour and stability of a moving bicycle.
However, bicycle stability is more complicated than a simple gyroscope.
Real bicycle stability also involves:
- steering geometry
- rider control
- centre of mass
- tire-ground forces
- forward speed
The gyroscopic effect of the wheels is one contributor, not the complete explanation.
Motorcycles
Motorcycle wheels also possess substantial angular momentum.
At higher speeds, changing the orientation of the rotating wheels requires torque.
This contributes to the characteristic handling of motorcycles.
Riders intentionally use steering and body movement to change the direction of the vehicle and its rotating wheels.
Aircraft Gyroscopes
Gyroscopes have historically been important in aircraft instruments.
They can provide stable references for measuring orientation.
Applications include instruments that indicate:
- aircraft attitude
- heading
- turning motion
Modern aircraft increasingly use electronic and inertial sensors rather than relying only on large mechanical spinning gyroscopes.
Ships
Gyroscopes can be used for navigation and stabilization on ships.
A gyrocompass uses rotational principles and Earth's rotation to provide a directional reference.
Gyroscopic stabilization systems can also produce torques that help oppose unwanted rolling motion.
Spacecraft
Spacecraft cannot always rely on aerodynamic surfaces for orientation because there may be essentially no surrounding air.
Rotational systems can therefore be used to control spacecraft orientation.
Examples include:
- reaction wheels
- momentum wheels
- control moment gyroscopes
Reaction Wheels
A reaction wheel is a motor-driven flywheel inside a spacecraft.
Suppose the spacecraft and wheel initially have zero total angular momentum.
If the motor accelerates the wheel in one direction, conservation of angular momentum causes the spacecraft body to rotate in the opposite direction.
Symbolically:
L_spacecraft + L_wheel = constant
If the initial total angular momentum is zero:
L_spacecraft = −L_wheel
This allows spacecraft orientation to be changed without pushing against the surrounding environment.
Worked Reaction-Wheel Example
A spacecraft body has:
I_s = 800 kg·m²
A reaction wheel changes its angular momentum by:
+40 kg·m²/s
Ignoring external torques, the spacecraft must gain:
−40 kg·m²/s
of angular momentum.
Therefore:
L_s = I_sω_s
−40 = (800)ω_s
ω_s = −0.050 rad/s
The negative sign indicates that the spacecraft rotates opposite to the reaction wheel's angular momentum change.
Control Moment Gyroscopes
A control moment gyroscope (CMG) uses a rapidly spinning rotor.
Instead of mainly changing the rotor's spin speed, the system changes the direction of its angular momentum.
Changing that direction requires torque.
By controlling this process, the spacecraft experiences an equal and opposite torque.
CMGs can produce substantial control torques and are useful for orienting large spacecraft.
Gyroscopes in Phones
Modern smartphones commonly use tiny MEMS gyroscopes rather than visible spinning wheels.
MEMS stands for microelectromechanical systems.
These sensors detect rotational motion using tiny vibrating structures.
They help devices determine:
- angular velocity
- orientation changes
- screen rotation
- motion for games
- camera stabilization
The operating mechanism differs from a traditional spinning-wheel gyroscope, but the device performs the same general job of sensing rotational motion.
Navigation Systems
Gyroscopes are important components of inertial navigation systems.
These systems may combine:
- gyroscopes
- accelerometers
- computers
Gyroscopes measure rotational motion while accelerometers measure linear acceleration.
Together, these measurements can help estimate:
- orientation
- velocity
- position
Such systems are useful when external navigation signals are unavailable.
Camera Stabilization
Gyroscopic sensors can detect unwanted camera rotation.
A stabilization system can then compensate using:
- moving lens elements
- sensor movement
- motorized gimbals
- electronic correction
This helps produce smoother images and video.
Drones
Drones use gyroscopic sensors as part of their flight-control systems.
The sensors detect changes in orientation.
The control system can then adjust motor speeds to correct:
- roll
- pitch
- yaw
This allows rapid automatic stabilization.
Rotational Stability in Sports
Angular momentum also helps explain stability in many sports.
Examples include:
- spinning balls
- bicycle and motorcycle racing
- gymnastics
- diving
- figure skating
- discus throwing
A rotating athlete or object can maintain a relatively stable rotation axis when its angular momentum is large and external torques are small.
Spinning Balls
A spinning ball has angular momentum along its rotation axis.
External torques can change the direction of this angular momentum.
However, a rapidly spinning ball may maintain its spin orientation more effectively than a slowly rotating one.
This contributes to the stable rotation seen in objects such as:
- footballs
- rugby balls
- thrown discs
- spinning projectiles
Aerodynamic forces may also strongly influence their motion.
Stability Does Not Mean No Change
A common misconception is that angular momentum prevents the axis from changing direction.
It does not.
Instead:
torque causes angular momentum to change
A large angular momentum means that a particular torque produces a relatively small directional change over a short time.
Therefore, the axis changes more slowly.
Angular Impulse
If a torque acts for time Δt:
ΔL = τΔt
This is the rotational equivalent of linear impulse:
Δp = FΔt
Suppose a gyroscope has:
L = 20 kg·m²/s
and a perpendicular torque of:
2.0 N·m
acts for:
0.10 s
Then:
ΔL = (2.0)(0.10)
ΔL = 0.20 kg·m²/s
This change is small compared with the original angular momentum.
The axis therefore changes direction only slightly.
Comparing Two Gyroscopes
Gyroscope A:
L = 5 kg·m²/s
Gyroscope B:
L = 50 kg·m²/s
Suppose the same perpendicular torque acts on each for the same time.
They receive the same ΔL.
However, ΔL is a much larger fraction of Gyroscope A's original angular momentum.
Therefore, Gyroscope A experiences the larger change in orientation.
Gyroscope B is more rotationally stable.
Conservation of Angular Momentum
If:
Στ_external = 0
then:
L_total = constant
This is the principle of conservation of angular momentum.
A freely rotating object in space with negligible external torque tends to maintain its angular momentum vector.
That means both the magnitude and direction of total angular momentum remain constant.
This is the fundamental reason angular momentum is so important for rotational stability.
Gyroscopic Stability in Space
In space, external torques can be extremely small.
A spinning spacecraft can therefore maintain a relatively fixed angular momentum direction for long periods.
This principle has historically been used for spin stabilization.
A spacecraft may be deliberately rotated so that its angular momentum helps maintain its orientation.
Spin-Stabilized Spacecraft
Suppose an entire spacecraft rotates.
Its angular momentum is:
L = Iω
If external torque is very small:
L ≈ constant
The spacecraft's spin axis therefore tends to remain pointed in approximately the same inertial direction.
This can provide passive stability without continuous active control.
Earth as a Rotating System
Earth also has enormous angular momentum because:
- it has enormous moment of inertia
- it rotates continuously
Its rotation axis is therefore relatively stable.
However, external gravitational torques from the Moon and Sun cause Earth's axis to undergo very slow precession.
This is a large-scale example of the same basic physics seen in a gyroscope.
Precession of Earth's Axis
Earth is not a perfect sphere. It has a slight equatorial bulge.
The gravitational effects of the Moon and Sun on this bulge produce torque.
This causes Earth's rotation axis to slowly change direction.
The axis traces out a very slow precessional motion over thousands of years.
The physical principle is the same:
external torque → change in direction of angular momentum
Energy and Gyroscopes
A spinning gyroscope also possesses rotational kinetic energy:
E_rot = 1/2 Iω²
Angular momentum is:
L = Iω
These quantities are related but are not the same.
A gyroscope can have both:
- substantial rotational energy
- substantial angular momentum
Angular momentum is the quantity most directly connected to directional rotational stability.
Angular Momentum vs Rotational Energy
| Quantity | Equation | Main Significance |
|---|---|---|
| Angular momentum | L = Iω | rotational motion and stability |
| Rotational kinetic energy | E_rot = 1/2 Iω² | energy stored in rotation |
| Torque | τ = ΔL/Δt | changes angular momentum |
Increasing ω increases both L and E_rot.
However:
L ∝ ω
while:
E_rot ∝ ω²
Doubling angular velocity doubles angular momentum but quadruples rotational kinetic energy.
Friction and Gyroscope Motion
Real gyroscopes eventually slow because of friction and air resistance.
Energy is transferred from rotational kinetic energy into:
- thermal energy
- sound
- air motion
As ω decreases:
- rotational energy decreases
- angular momentum decreases
- rotational stability decreases
- precession can become more rapid
- wobbling may increase
Eventually the gyroscope can no longer maintain its characteristic stable motion.
Complete Gyroscope Problem
A gyroscope rotor has:
I = 0.080 kg·m²
ω = 150 rad/s
Its mass is:
m = 2.0 kg
and its centre of mass is:
r = 0.15 m
from the pivot.
Angular Momentum
L = Iω
L = (0.080)(150)
L = 12 kg·m²/s
Gravitational Torque
With the axle horizontal:
τ = mgr
τ = (2.0)(9.8)(0.15)
τ = 2.94 N·m
Approximate Precession Rate
Ω = τ/L
Ω = 2.94/12
Ω ≈ 0.245 rad/s
Precession Period
The time for one complete precession is:
T = 2π/Ω
T = 2π/0.245
T ≈ 25.6 s
What If the Spin Speed Halves?
Suppose friction reduces:
ω = 150 rad/s
to:
ω = 75 rad/s
Angular momentum becomes:
L = (0.080)(75)
L = 6.0 kg·m²/s
For approximately the same gravitational torque:
Ω = 2.94/6.0
Ω = 0.49 rad/s
The angular momentum has halved and the approximate precession rate has doubled.
This demonstrates why a slowing gyroscope tends to precess more rapidly.
Analyzing Gyroscopic Systems
When examining a gyroscope, first identify the rotation axis and direction of angular momentum.
Then determine whether an external torque acts.
Ask:
- What force creates the torque?
- Where is the pivot?
- In which direction is L?
- In which direction is τ?
- Is the torque changing the magnitude of L, its direction, or both?
- Is the rotor speeding up or slowing down?
- How will changing I or ω affect stability?
This approach is often more useful than trying to think only in terms of forces.
Common Mistakes
Thinking a gyroscope has no tendency to fall because gravity disappears
Gravity still acts. Its torque changes the angular momentum and produces precession.
Thinking angular momentum is a scalar
Angular momentum has both magnitude and direction.
Confusing spin with precession
Spin is rotation about the rotor's own axis. Precession is the slower motion of that axis.
Thinking torque always increases rotational speed
Torque can change the direction of angular momentum without greatly changing its magnitude.
Assuming a gyroscope is perfectly stable
External torques can always change its angular momentum.
Thinking faster spin causes faster gravitational precession
For a simple rapidly spinning gyroscope under the same torque, greater angular momentum generally means slower precession.
Confusing angular momentum and rotational kinetic energy
L = Iω, while E_rot = 1/2 Iω².
Assuming all modern gyroscopes contain spinning wheels
Many electronic devices use MEMS gyroscopes with microscopic vibrating structures.
Claiming gyroscopic effects completely explain bicycle stability
Bicycle stability also involves steering geometry, rider control, tire forces, and other effects.
Key Terms
Gyroscope — device or rotating system that uses rotational motion to maintain or measure orientation.
Rotational stability — tendency of a rotating object with angular momentum to resist rapid changes in the direction of its rotation axis.
Angular momentum (L) — vector quantity describing rotational motion; for a simple rigid rotor about a fixed principal axis, L = Iω.
Moment of inertia (I) — measure of how mass is distributed relative to an axis of rotation.
Angular velocity (ω) — rate and direction of rotation.
Torque (τ) — turning effect that changes angular momentum.
Precession — gradual turning of the rotation axis of a spinning object due to an external torque.
Precession angular speed (Ω) — rate at which the rotation axis precesses.
Nutation — oscillation or wobbling of a rotating object's axis superimposed on precession.
Right-hand rule — method for determining the direction of angular velocity and angular momentum.
Angular impulse — product of torque and time, equal to the change in angular momentum.
Reaction wheel — rotating wheel used to control spacecraft orientation by exchanging angular momentum with the spacecraft.
Control moment gyroscope — device that produces control torque by changing the direction of a spinning rotor's angular momentum.
MEMS gyroscope — microscopic electromechanical sensor used to detect rotational motion.
Spin stabilization — use of angular momentum from rotation to help maintain an object's orientation.
Key Takeaways
- Gyroscopes demonstrate the effects of angular momentum and torque.
- For a simple rigid rotor, angular momentum is L = Iω.
- Angular momentum is a vector directed along the rotation axis.
- A rapidly spinning object can have substantial rotational stability.
- Increasing moment of inertia increases angular momentum when angular velocity is unchanged.
- Increasing angular velocity increases angular momentum when moment of inertia is unchanged.
- External torque changes angular momentum.
- The fundamental relationship is Στ_external = ΔL/Δt.
- Torque can change the direction of angular momentum without greatly changing its magnitude.
- This directional change produces gyroscopic precession.
- Precession is different from the rotor's spin.
- For simple steady gravitational precession, Ω ≈ τ/L.
- Greater angular momentum generally produces slower precession for the same torque.
- As a gyroscope slows because of friction, its angular momentum decreases and its behaviour becomes less stable.
- Nutation is the additional wobbling motion that can accompany precession.
- Angular momentum conservation explains why freely rotating objects tend to maintain their orientation when external torque is negligible.
- Gyroscopic principles are used in spacecraft, navigation, aircraft systems, ships, vehicles and stabilization systems.
- Reaction wheels control spacecraft by exchanging angular momentum internally.
- Modern phones and drones commonly use MEMS gyroscopes rather than large spinning rotors.
- Earth's slow axial precession is a large-scale example of torque changing the direction of angular momentum.
Check Your Understanding
- What is a gyroscope, and what property of rotational motion gives it stability?
- A rotor has I = 0.15 kg·m² and rotates at 80 rad/s. Calculate its angular momentum.
- The angular velocity of the rotor in Question 2 doubles. What happens to its angular momentum?
- Two gyroscopes rotate at the same angular velocity. One has twice the moment of inertia of the other. Compare their angular momenta and rotational stability.
- Explain how the right-hand rule can be used to determine the direction of a gyroscope's angular momentum.
- What does an external torque do to angular momentum?
- Explain qualitatively why a spinning gyroscope supported at one end precesses rather than simply falling straight downward.
- Distinguish between spin, precession, and nutation.
- A gyroscope has angular momentum L = 15 kg·m²/s and experiences a perpendicular torque of 3.0 N·m. Estimate its precession angular speed using Ω = τ/L.
- A gyroscope has I = 0.050 kg·m² and initially rotates at 200 rad/s. Friction reduces its angular velocity to 100 rad/s. Calculate its initial and final angular momentum and predict how this change affects its rotational stability.
- Explain why increasing a rotor's angular velocity generally increases its gyroscopic stability.
- Explain how a reaction wheel can rotate a spacecraft even though the spacecraft has nothing external to push against.
- Compare a traditional mechanical gyroscope with a MEMS gyroscope used in a smartphone.
- A gyroscope rotor has I = 0.10 kg·m², ω = 120 rad/s, mass 2.5 kg, and its centre of mass is 0.20 m from the pivot. With the axle horizontal, calculate its angular momentum, gravitational torque, and approximate precession angular speed.
- A spinning gyroscope gradually slows because of friction. Explain the complete sequence of changes involving rotational kinetic energy, angular momentum, precession, stability, and eventual motion of the gyroscope.