91Ó°ÊÓ

CP An altemating-current clectric motor includes a thin. hollow, cylindrical spool (similar to a ring) with mass \(M=1.11 \mathrm{~kg}\) and radius \(a=5.00 \mathrm{~cm}\) wrapped \(N=500\) times with a copper wire with resistance \(R=5.00 \Omega\) and inductance \(L=77.0 \mathrm{mH}\). Within the spool is a battery that supplics current \(I=1.00 \mathrm{~A}\). which makes the spool a magnetic dipole with dipole moment \(\overrightarrow{\boldsymbol{\mu}}\) parallel to the cylinder axis. A constant magnetic field with magnitude \(B=2.00 \mathrm{~T}\) is supplied by an extemal stator magnet, while the spool turns freely on an axis perpendicular to its own axis. At a certain time, a bar is inserted, stopping the spool's motion (Fig. \(\mathrm{P} 30.51\) ). At that instant the angle between the spool axis and the magnetic field is \(\theta=45^{\circ}\). (a) What is the magnitude of the downward force \(\vec{F}\) applicd by the bar onto the spool immediatcly after the bar is inserted? (b) Later, at time \(t=0\) with spool still at rest, the cril is short-circuited and a constant counter- toryue \(\tau=0.500 \mathrm{~N} \cdot \mathrm{m}\) is applied. The current subsides, and the magnetic torque decreases exponentially. At what time \(t\) does the force applied by the bar vanish'? (Hint: Determine when the magnetic torque balances the counter-torque.) (c) After the spool rotatcs \(180^{\circ}\) it hecomes stuck on the top side of the bar. The counter-torque is no longer applied, and the switch is returned to its original position. After a long time, the bar is removed. What is the angular acceleration of the spool immediately after the bar is removed? The moment of incrtia of the spool for an axis along its diamcter is \(I=\frac{1}{2} M a^{2}\).

Step by step solution

01

Calculate the force on the spool

We have the magnetic moment \( \mu \) of the spool as \( \mu = NIA \), where \( I \) is the current and \( A \) is the area which is \( \pi a^2 \). Now, the torque \( \tau = \mu B \sin(\theta) \). Force exerted by this torque at the end of the spool will be \( F = \frac{\mu B \sin (\theta)}{a} \).

02

Determine when the applied force vanishes

The coil's magnetic field and the spool are at an angle of 45 degrees. When the system reaches equilibrium, this angle will be 0 degrees. Thus, there will be a time when the magnetic field will be parallel with spool. Given that we are provided with the counter-torque and asked to find when this occurs, we can set up the equation \( \tau = \mu B \cos( \omega t) \) and solve for \( t \), remembering that the angular frequency \( \omega = \frac{R}{L} \). Ignore \( t \) if negative.

03

Calculate the angular acceleration after the bar is removed

The magnetic dipole wants to align with the field, providing a torque of \( \tau = \mu B \) which is counteracted by the spool’s inertia. Angular acceleration \( \alpha \) can be calculated using \( \tau = I \alpha \), where \( I \) is the moment of inertia. The moment of inertia for the spool is given as \( I = \frac{1}{2}Ma^2 \). Thus, \( \alpha = \frac{2\mu B}{Ma^2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Alternating-Current Electric Motors

Alternating-current (AC) electric motors are pivotal in various electrical applications, from household appliances to industrial machinery. At the heart of these motors is the magnetic dipole moment, which interacts with an external magnetic field to produce rotation. The dipole moment, denoted by the vector \(\overrightarrow{\boldsymbol{\mu}}\), is created by the flow of AC through the motor's windings, which are coils of wire that generate a magnetic field when electric current passes through them.

In our textbook exercise, we focus on a specific scenario involving a cylindrical spool wrapped with copper wire creating a magnetic dipole within a magnetic field. The interplay between the magnetic field and the magnetic moment sets the foundation for the motor's rotational motion. When the bar is inserted, stopping the spool's motion abruptly, we calculate the instantaneous downward force applied by the bar using the torque experienced by the spool due to its magnetic moment and the external magnetic field.

Inductance, Resistance, and Electrical Dynamics

The inductance (\(L\)) and resistance (\(R\)) of the wire in an electrical circuit, like the one in our AC electric motor, are crucial for understanding the motor's behavior, especially during dynamic events such as the stopping and restarting of the motor. Inductance measures how effectively an electrical conductor produces an electromotive force due to a change in the current flowing through it. This property is especially important in AC circuits, as it influences how the current changes over time.

On the other hand, resistance is a measure of the opposition to current flow in an electrical circuit. Together, inductance and resistance determine the time constant (\(\tau = L/R\)) which dictates how quickly the current in the circuit drops when the coil is short-circuited. For our problem, we use the relation between resistance and inductance to find out when the force applied by the bar vanishes, by equating the applied counter-torque to the decreasing magnetic torque using the formula involving the angular frequency (\(\omega = R/L\)).

Angular Acceleration and Rotational Motion

Angular acceleration (\(\alpha\)) is a measure of the rate of change of angular velocity. It is a vector quantity that points in the direction of the rotation axis and is central to understanding rotational motion in physics. In our motor scenario, when the bar is removed, the spool is subject to angular acceleration due to the torque generated by the magnetic forces attempting to realign the spool with the external magnetic field.

The angular acceleration can be calculated from Newton's second law for rotation, \(\tau = I\alpha\), where \(\tau\) is the torque exerted by magnetic forces, and \(I\) is the moment of inertia of the spool. In the case of the exercise, after the bar is removed and the counter-torque is no longer applied, we calculate the angular acceleration by considering the spool’s moment of inertia and the continuous torque exerted by the magnetic field. From the exercise, we determined that this acceleration is given by the equation \(\alpha = \frac{2\mu B}{Ma^2}\), showing how the magnetic properties of the motor and physical dimensions directly influence its dynamic response.