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I am currently researching the gyroscope on my own and i came across the concept of angular velocity.

The torque ($T$) on a gyroscope is caused by $R\cdot F$, where the force i consider it to be equal to $mg$.

Since $T = \frac{dL}{dt}$ and $L = I\cdot ω$

where $L$-angular momentum, $t$-time, $I$-moment of inertia, $\omega$-angular velocity and $I = mR^2$

If i decrease the mass of the gyroscope, this means that over the same time period the change in $L$ will be decrease as well.

However I also depends on the mass and will decrease by the same amount (?).

Does this mean that $\omega$ will always be the same? If so why?

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If i decrease the mass of the gyroscope, this means that over the same time period the change in L will decrease as well.

We have to be aware that in the textbook description of gyroscopic precession, the magnitude of $\vec L$ is assumed to be constant despite it may have changing direction. Thus, we can only use $\dfrac{d\vec L}{dt}$ to say that the change in $L$'s direction will decrease as the mass decreases.

However I also depends on the mass and will decrease by the same amount (?).

Not by same amount but by same proportion so that the precession rate of the gyroscope $\Omega$, defined by $\Omega=$$\dfrac{d\phi}{dt}=\frac{Mgr}{I\omega}$, is independent of the mass of the gyroscope.

(Hint: $d\phi=\dfrac{dL}{L}$)

Does this mean that $\omega$ will always be the same? If so why?

$\omega$ is independent of $M$. As said above, $L$ is regarded to be constant, thus $I\omega$ is constant, so $\omega$ won't change.

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