CP CALC A square, conducting, wire loop of side L , total...

CP CALC A square, conducting, wire loop of side $L$ , total mass $m,$ and total resistance $R$ initially lies in the horizontal $x y$ -plane, with corners at $(x, y, z)=(0,0,0),(0, L, 0),(L, 0,0)$ and $(L, L, 0) .$ There is a uniform, upward magnetic field $\vec{\boldsymbol{B}}=\hat{B hat{\boldsymbol{k}}}$ in the space within and around the loop. The side of the loop that extends from $(0,0,0)$ to $(L, 0,0)$ is held in place on the $x$ -axis; the rest of the loop is free to pivot around this axis. When the loop is released, it begins to rotate due to the gravitational torque. (a) Find the net torque (magnitude and direction) that acts on the loop when it has rotated through an angle $\phi$ from its original orientation and is rotating downward at an angular speed $\omega$ . (b) Find the angular acceleration of the loop at the instant described in part (a). (c) Compared to the case with zero magnetic field, does it take the loop a longer or shorter time to rotate through $90^{\circ} ?$ Explain. (d) Is mechanical energy conserved as the loop rotates downward? Explain.

CP CALC A square, conducting, wire loop of side $L$ , total mass $m,$ and total resistance $R$ initially lies in the horizontal $x y$ -plane, with corners at $(x, y, z)=(0,0,0),(0, L, 0),(L, 0,0)$ and $(L, L, 0) .$ There is a uniform, upward magnetic field $\vec{\boldsymbol{B}}=\hat{B hat{\boldsymbol{k}}}$ in the space within and around the loop. The side of the loop that extends from $(0,0,0)$ to $(L, 0,0)$ is held in place on the $x$ -axis; the rest of the loop is free to pivot around this axis. When the loop is released, it begins to rotate due to the gravitational torque. (a)
Find the net torque (magnitude and direction) that acts on the loop when it has rotated through an angle $\phi$ from its original orientation and is rotating downward at an angular speed $\omega$ . (b) Find the angular acceleration of the loop at the instant described in part (a). (c) Compared to the case with zero magnetic field, does it take the loop a longer or shorter time to rotate through $90^{\circ} ?$ Explain. (d) Is mechanical energy conserved as the loop rotates downward? Explain.

Key Concepts

Energy Dissipation in Electromechanical Systems

The conversion of mechanical energy into electrical energy and its subsequent dissipation as heat in a resistive medium is a crucial aspect of electromechanical systems. In the case of the rotating loop, while gravitational potential energy is converted into kinetic energy, a portion of this energy is simultaneously converted into thermal energy due to resistive losses. This non-conservative process means that the total mechanical energy is not conserved, highlighting the interplay between mechanical work and electrical dissipation.

Resistive Damping in Magnetic Systems

When an induced current flows in a loop with finite electrical resistance, it gives rise to dissipative effects often referred to as electromagnetic damping. This damping results from Lenz’s law, which ensures that the induced current creates a magnetic force opposing the change in flux, thereby providing a torque that resists the loop’s rotation. The resistance in the circuit leads to energy being lost as heat, affecting the dynamics of the loop.

Rotational Dynamics

This concept deals with how torques produce angular accelerations in rigid bodies. When a body rotates about an axis, the net torque acting on it is related to the angular acceleration through its moment of inertia, analogous to how force and mass relate in linear motion. Understanding rotational dynamics is essential when analyzing the motion of a rotating loop under various torques, such as gravitational and magnetic ones.

Gravitational Torque

Gravitational torque arises from the weight of an object acting at a distance from a pivot point. In the context of the loop, the gravitational force acting on its mass creates a torque that causes the loop to rotate. This involves calculating the lever arm and the component of the force perpendicular to it, which in turn affects the angular acceleration of the loop.

Electromagnetic Induction and Faraday's Law

Faraday's law of electromagnetic induction states that a changing magnetic flux through a circuit induces an electromotive force (EMF). In rotating conductive loops, as the orientation relative to a magnetic field varies, the magnetic flux through the loop changes, inducing an EMF that can drive a current. This phenomenon is key to analyzing how the magnetic field interacts with the motion of the loop.

Transcript

00:01 So in this problem, we have a square.

00:06 So that's this red shape right here.

00:11 And it's lying in the x, y, plane, and it has length l.

00:14 And there's a magnetic field right there that is coming out at us.

00:19 In other words, along the z axis.

00:22 And over here, i've drawn the same picture, but from the side.

00:27 So the x -axis now is coming out at us.

00:29 And the x -axis because it's going up, so we see the magnetic field going up, and the square has rotated by some angle phi.

00:42 And the question asks us a number of things, the first of which is, what is the torque when the loop has rotated an angle phi and is traveling at an angular velocity omega? so first of all, we should remind ourselves that omega is d -fi.

01:05 Dt, right, the angular velocity is the rate of change of the angle.

01:10 So let's start by determining the torque.

01:13 So there's two sources of torque.

01:15 There's the gravitational torque and there's the magnetic torque.

01:19 So let's start with the gravitational torque.

01:24 So we're going to start with the torque due to gravity.

01:29 So if we think about the side picture, the gravitational forces down.

01:37 But as far as torques are concerned, the component of gravity that provides the torque is the perpendicular component, right? so the component of gravity that we care about is actually this one because the other component doesn't provide a torque.

01:54 So if this angle is phi, you can see that that angle is going to be phi.

02:04 So if that angle is phi, what that means is the gravitational force.

02:10 We're interested in is going to be the cosine.

02:14 So we get the cosine of phi.

02:17 Now, we have to be a bit careful because the different branches of the square are going to experience different torques because they're different distances away from the axis.

02:33 But if you remember, so here's the square, here's the head -on diagram, gravity acts as if it were at the center of math.

02:45 Of an extended object.

02:48 So the centers of masses of the three sides of the square that are rotating are here, and each one has mass m over four, because it's a square, so the each get m over four.

03:03 And then this distance is l over two, and this distance is, of course, that's still l.

03:11 So now we can calculate the torque.

03:15 So remember, torque is basically distance times force.

03:18 So for the two side, the two sides, so this one and this one, we have that the distance away is an l over 2, and then the gravitational force is m g cos phi, where m is m over 4.

03:37 So m over 4 g cos phi.

03:41 And then we also have the top part, that's that one.

03:47 It's a distance l over away, l away, with a mass m over 4, and then we, we have g -cose -fi.

03:58 So the total gravitational torque is going to be 1 -5 -l -m -g -cose -fi.

04:10 So that's a gravitational torque, and clearly that torque is going to be like this.

04:14 So it's pulling the square down like that.

04:21 So that's the gravitational torque.

04:25 Now what about the magnetic torque? magnetic.

04:34 So first of all, why is there a magnetic torque? well, the flux through the loop is changing, right? so as it rotates, the amount of magnetic field that's going through it is decreasing.

04:46 So if there's a change in flux and it contains a current from the, from faraday's law, then there's going to be a torque.

05:01 So from the side, it looks like this.

05:05 So again, there's the loop after it's rotated a bit.

05:07 The magnetic field is up and the magnetic moment is in that direction, this angle is going to be phi.

05:18 So you can check again that this angle is going to be phi by the same kind of geometric arguments, and therefore the torque is i, a, b, sine, phi.

05:43 And for our particular case, a is l squared.

05:47 So this is i, l squared, b, sine phi.

05:54 So now we just have to figure out what the current is...

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