### Gyroscopes

A gyroscope is an object that rapidly spins about one axis whose end is fixed. The gravitational force produces a torque that causes the gyroscope to precess.

#### Equation of motion for a gyroscope

Let's consider a toy gyroscope, a top, that consists of a uniform disk of mass m and radius R that is attached to a massless stick of length d.

The torque acting on the top is   τ = m g d sin θ, where sin θ is the inclination angle of the top with respect to the horizontal.

If the top is not spinning, this torque causes the top to fall down. If the top is spinning rapidly, what does the top not fall?

In a time interval dt, the change in the angular momentum of the top is:

dL = τ dt = m g d sin θ dt

Since dL is perpendicular to L, the angular momentum itself must move in a circular orbit without changing magnitude

The change in the direction of L is determined by:

d φ ≈ |dL|/|L| = τ dt/L = m g d sin θ dt/ I ω

Finally, the precession frequency d φ/dt is:

d φ/dt = m g d sin θ / I ω