Understanding Gyroscopes

A deep dive into the intuition and mathematics of why Gyroscopes seemingly defy gravity.

Image for post
Image for post

Understanding Torque

If you want to understand gyroscopes, then you have to understand torque. Torque could be thought of as a rotational force: a force that causes or inhibits rotation.

Image for post
Applying torque to a bolt using a wrench

The Math of Rotating Stuff

Torque relates to stuff rotating, so it helps to understand torque by first looking at equations describing rotational movements. Alternatively you can think of this as equations for things moving in orbits or circles.

Linear Motion

If a constant acceleration a is applied to an object initially at rest for the time t then it will get the velocity v. This should be familiar to you if you did physics in high school.

v = at

Newton’s second law

The acceleration an object of mass m gets when a force F is applied to it, is proportional to the force, and inverse proportional to its mass.

a = F/m
F = ma

Newton’s third law

One of Newton’s observations which doesn’t seem quite obvious at first is that

for every force, there is a force working in the opposite direction.

So if you stand on super slippery sheet of ice and push your friend, you will both move backwards.

Image for post
Newtons 3rd law says: for every action, there is an equal and opposite reaction. A force from the propellant pushes on the rocket and the rocket push back with an equal but opposite force on the propellant.
F = m₁a₁
F = m₂a₂
m₁a₁ = m₂a₂
m₁v₁/Δt = m₂v₂/Δt
m₁v₁ = m₂v₂
p = mv

Rotation

Let’s take what we have learned about linear motion and compare with circular movements. Instead of talking about distances in meters, we deal with distances in angles. Like how many degrees or radians and object has moved.

S = 2πr
Image for post
The arch length is defined as S = θr
θ = S/r = 2πr/r = 2π
ω = δθ/δt
ω = δ(S/r)/δt
ω = δS/rδt
ω = v/r
Image for post
v = at
v/r = at/r
ω = αt
α = δω/δt
α = δ(v/r)/δt
α = δv/rδt
α = a/r
L = Iω
τ = Iα
τ = rF

Why Torque is Vector

A common misunderstanding is to think torque is its own special thing. But it is really just a higher level explanation of the interactions of regular forces on objects in a manner which causes them to spin around.

Image for post
When doing the cross product between two vectors, the right hand rule gives us the direction of the result. To get correct direction (according to established conventions) for torque we do τ = r×F
Image for post
We can define a plane by its normal vector n
Image for post
We can use the normal vectors n₁ and n₂ of two planes to find the angle θ of their intersection.

Why is Torque Affected by the Radius

We sort of understand this from practice. It is easier to turn a bolt with a long wrench than a short one. So somehow we can trade in force for longer radius. But how do we explain this in a mathematical fashion?

F = ma
F = mαr
α = F/(mr)
α = F₁/(mr₁)
α = F₂/(mr₂)
F₁/(mr₁) = F₂/(mr₂)
F₁/r₁ = F₂/r₂
W = Fs
θ = S/r
S = θr
W = θr₁F₁
W = θr₂F₂
θr₁F₁ = θr₂F₂
r₁F₁ = r₂F₂

Why Do Spinning Tops Not Tip Over?

Remember a gyroscope is just a spinning top embedded in some gimbals, so to understand gyroscopes lets discuss a spinning top first.

Image for post
Image for post
A centrifugal governor usually used to regulate the flow of fuel to engines. The black balls will begin to rise up as the contraption spins faster.
a = v²/r
s = vt
t = s/v
t = 2πr/v
t = 2πv/a
2πr/v = 2πv/a
r/v = v/a
a = v²/r

Using Torque and Angular Velocity to Explain Why Tops Don’t Tip Over

If the top is sitting still, it will tip over right away, making small rotating movement. That is because gravity will apply a force at some radius from the rotating angle. Hence we get a torque.

Image for post
Shows the angular velocity of the the spinning to around is own axis and the angular velocity of its precession. The torque is defined by action and reaction forces of gravity Fg and -Fg. The torque τ is pointing out of the paper.
τ = Iα
δω = αδt
ω' = ω + δω

Stability of a Gyroscope

If you look at the precession of a gyroscope you will notice it is much smaller when the gyroscope is spinning fast.

Summary

To understand a gyroscope it helps to learn about many other related phenomenon, involving rotation such as planetary orbits, bikes, spinning tops and acentrifugal governor as used of an a steam engine for instance.

Geek dad, living in Oslo, Norway with passion for UX, Julia programming, science, teaching, reading and writing.

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store