Chapter 10: Rotational Kinematics and Energy
Applications

GYROSCOPES

A gyroscope is an object free to spin in space. In the absence of external torques a gyroscope will maintain its angular momentum. An external torque applied in the same direction or in a direction opposite to the angular momentum will cause the gyroscope to speed up or slow down. If the applied torque has a component which is perpendicular to the initial angular momentum the gyroscope will PRECESS. That is, it will start to rotate about an axis which is perpendicular to the axis of the original spin.

It appears that in the early years of the 19th century it occurred to a number of people that a wheel, spinning freely in space, could be used to demonstrate that the Earth is spinning on its axis.
One suggestion was: "Conceive a large flat wheel, poised on several axes all passing directly through its center of gravity, and whose axis of motion is coincident with its principal axis of permanent rotation, to be put in very rapid motion. The direction of its axis would then remain unchanged. But the directions of all surrounding objects varying, on account of the motion of the Earth, it would result that the axis of the rotating wheel would appear to move slowly."

Such a device, sufficiently precise to be actually useful is not easy to construct. In principle it might be built as shown in the diagram on the right.
Mount the gyro with its axle free to rotate inside a circular ring C. Now mount the ring C inside a bigger ring B so it is free to rotate abut an axis which is perpendicular to the main axis of the gyro. Now mount this assembly so it can rotate about a third axis, perpendicular to the first two axes.
Such toy gyroscopes are actually for sale. They not very precisely constructed and they do not rotate for very long.

In 1852 the French physicist Foucalt used such a device to demonstrate the rotation of the Earth. He also coined the word "gyroscope" from two word of Greek origin where the root "gyr" means rotation and the root "scope" means view.

Suppose a gyro, located at the North Pole of the Earth is suspended in equilibrium and allowed to spin freely about a horizontal axis. The gyro is mounted in such a way that it can rotate freely in space, about any axis. The gyro will maintain its angular velocity and its axis will maintain its orientation relative to the fixed stars. As the Earth turns the gyro will appear to rotate about a vertical axis as the Earth rotates underneath it.
Now consider the same experiment somewhere at the equator. Start the gyro spinning about a horizontal axis. Again the gyro will maintain this orientation relative to the fixed stars. As the Earth turns underneath, the spin axis of the gyro, which was originally perpendicular to the vertical, will appear to tilt. During a twenty-four hour day the gyro will appear to turn about a horizontal axis.
The Earth itself is a spinning object. It is subjected to the gravitational pulls of other celestial bodies. Of these, the Sun and the Moon affect the Earth the most. The gravitational pulls of the Sun and the Moon on the Earth do not pass through the Earth's center of mass. Therefore the spinning Earth is subjected to a torque which causes its axis of rotation to precess. The axis slowly swings around to describe a cone in space. The axis completes its cycle every 26,000 years. The axis of the spinning Earth points to a star that appears to be stationary as the rest of the sky appears to revolve around it. Since the axis precesses, during the precession cycle different stars assume that role.

Gyroscopes are incorporated into moving objects from ships and airplanes to pointing devices. As these objects move and turn, the gyros inside try to maintain their angular momentum. Sensors mounted on the gyros are used to keep track of the motion. Thus gyros are used in navigation and stabilization devices.

The Wheel That Cannot Fall

A popular classroom demonstration of the conservation of angular momentum involves a spinning bicycle wheel, initially supported on both sides of the axle. When the support is withdrawn from one of the sides, the wheel starts to precess but does not fall. When one of the axle supports is removed the force of gravity now exerts a net torque about a horizontal axis which is not the axis of the spin. The wheel starts to turn about that axis (i.e. the unsupported side of the axle begins to drop). The spinning wheel acquires a vertical angular momentum which by the momentum conservation law should remain zero. To cancel the new vertical component of the angular momentum the wheel starts to precess, i.e. to turn about the vertical axis. The energy for this rotation comes from the decreasing potential energy of the dropping center of mass of the wheel. When the wheel no longer gains enough energy to pump into the rotational motion about the vertical axis, the wheel stops falling.

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Further Study Questions.