11.14

Gyroscope: Precession

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Precession can be explained by the gyroscopic motion of the Earth.

The counterclockwise spin of Earth creates an angular momentum. The Sun's gravitational force acting on Earth produces a torque perpendicular to the angular momentum and gravitational force.

Now, torque acting for time interval dt changes the direction of the angular momentum by dL. As the Earth is spinning, the direction of torque changes. Hence, the angular momentum constantly follows the torque, and precession occurs with velocity ω_P.

The precession velocity can be derived considering the torque acting on the system, which causes a change in angular momentum.

For a circular arc, the arc length divided by the radius is the change in precession angle. The rate of change of precession angle gives the precession velocity.

Now, substituting the value of dL, the precession velocity is inversely proportional to the angular momentum.

As angular momentum is the product of moment of inertia and angular velocity, precession velocity is inversely proportional to the angular velocity.

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Precession can be demonstrated effectively through a spinning top. If a spinning top is placed on a flat surface near the surface of the Earth at a vertical angle and is not spinning, it will fall over due to the force of gravity producing a torque acting on its center of mass. However, if the top is spinning on its axis, it precesses about the vertical direction, rather than topple over due to this torque. Precessional motion is a combination of a steady circular motion of the axis and the spin motion of the top about the axis. The torque produced due to the spinning top is perpendicular to the angular momentum; thus, the direction of the torque changes, but its magnitude does not. The top precesses around a vertical axis since the torque is always horizontal and perpendicular to the angular momentum. If the top is not spinning, it acquires angular momentum in the direction of the torque, and it rotates around a horizontal axis, falling over just as expected.

The concept of precession can be seen in bicycles; it is easy for a bicycle to tip over when stationary. However, when riding the bicycle at a good pace, tipping the bike over involves changing the angular momentum vector of the spinning wheels. Another way that we can demonstrate this is if we put a spinning disk in a box, such as a DVD player. Though it is easy to translate the box in a given direction, it is difficult to rotate it about an axis perpendicular to the axis of the spinning disk. This is because the torque being applied tothe box is causing the angular momentum vector of the spinning disk to precess. The precession angular velocity adds a small component to the angular momentum along the z-axis, seen in the form of a swaying motion as the gyroscope precesses, referred to as nutation.

The Earth acts like a gigantic gyroscope; its angular momentum is along its axis and currently points towards Polaris, the North Star. However, the Earth is slowly precessing (once in about 26,000 years) due to the torque of the Sun and the Moon acting on its nonspherical shape.

This text is adapted from Openstax, University Physics Volume 1, Section 11.4: Precession of a Gyroscope.

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