(II) A toy gyroscope consists of a 170-g disk with a radius of 5.5 cm mounted at the center of a

step 1 The period of precision t is related to the reciprocal of the angular positional frequency by th

(II) A toy gyroscope consists of a 170-g disk with a radius...

(II) A toy gyroscope consists of a $170-\mathrm{g}$ disk with a radius of 5.5 $\mathrm{cm}$ mounted at the center of a thin axle 21 $\mathrm{cm}$ long (Fig. 41$)$ . The gyroscope spins at 45 $\mathrm{rev} / \mathrm{s} .$ One end of its axle rests on a stand and the other end precesses horizontally about the stand. (a) How long does it take the gyroscope to precess once around? (b) If all the dimensions of the gyroscope were doubled (radius = $11 \mathrm{cm},$ axle $=42 \mathrm{cm} )$ how long would it take to precess once?

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object’s resistance to changes in its rotational motion about a specific axis. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a disk, for instance, the moment of inertia is typically given by (1/2)mr². In the context of gyroscopes, a larger moment of inertia means greater spin angular momentum for a given angular velocity, which in turn results in a slower precession rate under a given torque.

Angular Momentum

Angular momentum is the rotational equivalent of linear momentum and is defined as the product of an object's moment of inertia and its angular velocity. For a spinning object like a gyroscope, the angular momentum is primarily determined by how mass is distributed relative to the axis of rotation and by the speed of rotation. This angular momentum plays a central role in the gyroscope’s ability to resist changes to its axis of rotation, and it is the key quantity governing the rate of precession when a torque is applied.

Scaling Laws in Physical Systems

Scaling laws describe how various physical quantities change when the dimensions of a system are scaled by a certain factor. When dimensions such as length are doubled, geometric properties like area and volume—and quantities derived from them, such as the moment of inertia—change in non-linear ways. In problems involving a gyroscope, scaling up the dimensions affects both the lever arms for torque and the distribution of mass, which must be carefully considered when predicting how changes in size alter the precession rate.

Gyroscopic Precession

Gyroscopic precession is the phenomenon where the axis of a spinning object, such as a gyroscope, rotates about a vertical axis when an external torque is applied. This occurs because the applied torque, which is perpendicular to the spin angular momentum, causes the gyroscope’s angular momentum vector to change direction, resulting in a slow, circular motion of the spin axis. This precession is characterized by an angular velocity that is directly proportional to the applied torque and inversely proportional to the spinning angular momentum.

Torque in Rotational Motion

Torque is a measure of the effectiveness of a force to produce rotation about a pivot or axis. In rotational dynamics, it is defined as the cross product of the lever arm (the distance from the pivot to the point of application of the force) and the force itself. When applied to a gyroscope, the gravitational force acting on the mass creates a torque about the point where the axle contacts the stand, which in turn induces the precession of the gyroscope’s axis.

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