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Angular speed is fundamentally about how fast something spins or rotates. It's the rate at which an object travels around a circular path. It is usually denoted by the symbol \( \omega \) and is measured in radians per second. For a gyroscope, angular speed describes how quickly its spinning wheel rotates.

When you double the angular speed of a gyroscope's spinning flywheel, the spin rate increases. This results in the doubling of angular momentum, as both depend on angular speed. However, an increase in the angular momentum tends to decrease the precession angular speed. This is because the gyroscope can handle more of the weight of the applied torque owing to its increased momentum.

Angular speed can be influenced by external factors such as friction and resistance, but in theoretical exercises, it's most commonly adjusted by changing the input energy or force responsible for the motion of the rotating body.

The moment of inertia is a measure of an object’s resistance to changes in its rotation. It depends on how the mass is distributed with respect to the axis of rotation. For instance, more mass further from the axis means a higher moment of inertia.

In a gyroscope, the moment of inertia plays a critical role. It quantifies how much torque is needed for a desired angular acceleration. When you double the moment of inertia of a gyroscope's flywheel, it becomes harder to change its rotational state. This leads to an increase in angular momentum when the angular speed remains constant. But increased angular momentum means the precession angular speed decreases, as it is inversely proportional to angular momentum.

Different shapes and mass distributions result in different moments of inertia. For example, a solid disc and a ring of the same mass and radius have different moments of inertia.

Torque is the rotational equivalent of linear force. It’s what causes an object to start spinning, stop spinning, or change its angle of rotation. It’s calculated as the product of force and the distance from the point of rotation, also known as the lever arm. For a gyroscope, it’s the force due to gravity acting at a distance from the pivot.

When you double the torque on a gyroscope by, say, doubling its mass or the distance from the pivot, you’re essentially increasing the rotational force acting upon it. This increase leads to faster precession if the angular momentum doesn’t change, as the precession angular speed is directly proportional to torque.

To visualize torque, think of using a wrench. The longer the wrench, the more torque you can apply with the same amount of force. This concept applies in rotational motion, affecting how quickly something can rotate or spin.

Angular momentum is the product of an object's moment of inertia and its angular speed. It’s a measure of how much rotation an object has and is conserved in a closed system without external torques. For any spinning object, maintaining angular momentum is key to its stability and predefined motion.

In a gyroscope, the angular momentum keeps it from tumbling over, thanks to the spinning flywheel. If you manipulate the components affecting the angular momentum, such as the flywheel's speed or its moment of inertia, you'll observe changes in how the gyroscope behaves.

When angular momentum is high, thanks to either an increased angular speed or a higher moment of inertia, the gyroscope becomes less responsive to external torques, slowing its precession angular speed. In environments where angular momentum is transferred or altered, such as a figure skater pulling in their arms to spin faster, different rotational dynamics can be achieved.