Problem 53 - University Physics with Modern Physics

Question

The rotor (flywheel) of a toy gyroscope has mass 0.140 kg. Its moment of inertia about its axis is \(1.20 \times 10^{-4} \mathrm{kg} \cdot \mathrm{m}^{2} .\) The mass of the frame is 0.0250 \(\mathrm{kg}\) . The gyroscope is supported on a single pivot (Fig. E10.53) with its center of mass a horizontal distance of 4.00 \(\mathrm{cm}\) from the pivot. The gyroscope is precessing in a horizontal plane at the rate of one revolution in 2.20 \(\mathrm{s}\) . (a) Find the upward force exerted by the pivot. (b) Find the angular speed with which the rotor is spinning about its axis, expressed in rev/min. (c) Copy the diagram and draw vectors to show the angular momentum of the rotor and the torque acting on it.

Step-by-Step Solution

Answer

(a) The upward force is 1.62 N. (b) The rotor spins at 7058 rev/min. (c) Draw the vectors as described.

Step 1: Understand the Problem

The problem involves a toy gyroscope with specific mass properties and motions around a pivot. We need to find the force exerted by the pivot, the spin speed of the rotor, and draw vectors representing the system.

Step 2: Calculate the Upward Force by the Pivot

The total mass of the system (rotor + frame) is 0.140 kg + 0.0250 kg = 0.165 kg. The gravitational force is the total weight, given by \( F = mg \), where \( m = 0.165 \) kg and \( g = 9.81 \text{ m/s}^2 \). Thus, the upward force by the pivot is \( F = 0.165 \times 9.81 = 1.61965 \text{ N} \).

Step 3: Calculate the Precession Rate and Relation to Angular Velocity

The precession rate \( \omega_p \) is one revolution every 2.20 s, or \( \omega_p = \frac{2\pi}{2.20} \text{ rad/s} \). We know from gyroscopic motion that torque induced by gravity \( \tau = rmg \) is equal to \( I \omega \omega_p \). Here, \( r = 0.04 \) m, \( I = 1.20 \times 10^{-4} \text{ kg} \cdot \text{m}^2 \), and \( \omega \) is the angular speed of the rotor.

Step 4: Calculate the Spin Angular Velocity of the Rotor

Solving for \( \omega \), \( rmg = I \omega \omega_p \), we have \( 0.04 \times 1.61965 = 1.20 \times 10^{-4} \times \omega \times \frac{2\pi}{2.20} \). This simplifies to \( \omega = \frac{0.064786}{8.76923 \times 10^{-5}} \). Solve the expression to find \( \omega \approx 738.46 \text{ rad/s} \).

Step 5: Convert Angular Velocity to Revolutions per Minute

Convert the angular velocity from rad/s to rev/min: \( \omega = 738.46 \times \frac{1}{2\pi} \text{ rev/s} \). To get rev/min, multiply by \( 60 \), resulting in \( \omega \approx 7057.7 \text{ rev/min} \).

Step 6: Draw Angular Momentum and Torque Vectors

Copy the diagram of the gyroscope system and note that the angular momentum vector of the rotor points along its axis of rotation due to the spinning motion. The torque vector caused by gravity points upwards when considering the lever arm outwards from the pivot.

Key Concepts

Moment of Inertia

When discussing gyroscopes, understanding the moment of inertia is crucial. It is a measure of how difficult it is to change the rotational motion of an object. Imagine spinning a wheel on its axle; the heavier it is, or the more mass it has further from the center, the harder it is to spin.
In mathematical terms, the moment of inertia (I) depends on both the object's mass distribution and the distance of this mass from the axis of rotation. For the gyroscope rotor in this example, it is given as \(1.20 \times 10^{-4} \mathrm{kg} \cdot \mathrm{m}^2\). This means, with this specific distribution of mass, that's the resistance the gyroscope will face against changing its spinning state.
The larger the moment of inertia, the more torque is needed to change its rotational speed. Conversely, less torque is required if the moment of inertia is smaller. This foundational concept helps us predict and calculate how gyroscopes behave when forces act upon them.

Precessional Motion

Precessional motion is one of the fascinating behaviors of gyroscopes. It's the movement of the axis of a spinning object, like a gyroscope, around another axis due to an applied torque, typically from gravity.
In this problem, precession is observed as the gyroscope spins and the entire system rotates in a horizontal plane. The precession rate described is one revolution every 2.20 seconds. This is a critical insight into how gyroscopes shift their orientation when balanced delicately on a pivot.
Understanding precessional motion helps us recognize that the application of gravitational torque doesn’t cause it to topple over immediately. Instead, it causes it to trace a circular path, and this unique motion is integral in technologies like navigational systems and stabilizers.

Angular Momentum

A key player in gyroscope physics is angular momentum, which hints at the rotational equivalent of linear momentum. For any rotating body, angular momentum (\(L\)) is given by \(L = I\omega\), where \(I\) is the moment of inertia and \(\omega\) is the angular velocity. It expresses the extent to which an object will maintain its rotational motion unless acted upon by an external force.
For gyroscopes, angular momentum is the steady, directional quantity that aids in retaining the orientation of the spinning rotor. As the gyroscope spins rapidly, it builds up significant angular momentum. This momentum stabilizes the gyroscope against changes in direction, hence providing its surprising ability to "resist" toppling.
The angular momentum vector points along the spin axis of the rotor, which is a crucial aspect to visualize because it helps predict the direction the gyroscope will precess.

Torque Calculation

Torque is the rotational counterpart of force that causes objects to rotate. It is calculated with the formula \(\tau = r \times F\), where \(r\) is the distance from the pivot point and \(F\) is the force applied perpendicular to the moment arm.
In the case of the gyroscope, gravitational forces create a torque that influences its precessional motion. In essence, the torque here results from the weight of the system exerting force, \(F = mg\), at a distance \(r\) from the pivot.
Utilizing this torque helps in determining parameters like angular velocity which informs about the spinning dynamics of the gyroscope. By quantifying torque, we can relate it directly to the gyroscope's stability and performance, illustrating how forces interact in rotational systems.