Question

Two gyroscopes are made exactly alike except that the spinning disk in one is made of low-density aluminum, whereas the disk in the other is made of high-density lead. If they have the same spin angular speeds and the same torque is applied to both, which gyroscope processes faster?

Short answer

The gyroscope made by lower density (aluminum) material will process faster.

Step-by-step solution

Definition of Angular speed

The angular speed is defined as how fast the rotating body’s angle will change corresponding to time.

Find which gyroscope processes faster

Precession angular speed is given by

$\omega =\frac{{\pi }_{CM}}{I\omega }--·\left\{1\right)$

Here,

${\pi }_{CM}$= Torque about center of mass

I = Moment of inertia

$\omega$= Spin angular speed

Use given data and equation (1), the torque $\left(\pi \right)$and angular speed $\left(\omega \right)$are same for both disks, so precession angular speed is negative proportional to the moment of inertia of disks.

The moment of inertia of a disk is directly proportional to the mass of the disks, but also disks are identical, so the mass directly proportional to the density of the disks.

Therefore, the gyroscope made by lower density (aluminum) material will process faster.