ssm mm The drawing shows a thin, uniform rod that has a length of 0.45 m and a mass of 0.094 kg

Ssm mm The drawing shows a thin, uniform rod that has a...

Key Concepts

Magnetic Force on a Current-Carrying Conductor

This concept describes the force experienced by a conductor carrying an electric current when it is placed in a magnetic field. The force is given by the cross product of the current element and the magnetic field, and its magnitude can be calculated using the equation F = I?B sin(?), where I is the current, ? is the length of the conductor within the field, B is the magnetic field strength, and ? is the angle between the direction of the current and the field. This principle is fundamental in electromagnetism and is used to analyze the forces acting on elements of a circuit in the presence of a magnetic field.

Center of Gravity

The center of gravity of an object is the point where the entire weight of the object can be considered to be concentrated for the purpose of analysis of gravitational forces. In the case of uniform bodies, like a thin rod, the center of gravity is located at the geometric center. This simplification allows us to consider gravitational forces as acting at one point rather than distributed over the object's entire mass, which is useful when calculating torques due to gravity.

Torque (Moment) Equilibrium

Torque, often referred to as a moment, is the rotational equivalent of force. In static equilibrium situations, the sum of all torques acting on a system must equal zero in order to prevent rotation about a pivot point. This principle is key in solving problems that involve forces causing rotation, as it requires that the clockwise torques balance the counterclockwise torques. Establishing this balance allows one to set up equations that relate the different forces and distances (moment arms) involved.

Moment Arm

The moment arm is the perpendicular distance from the axis of rotation (or pivot) to the line of action of the force. It is a critical factor in calculating torque because even a small force can produce a large torque if applied at a large distance from the pivot, and vice versa. In problems involving rotational equilibrium, identifying the correct moment arm for each force is essential to correctly computing the resulting torques.

Transcript

00:01 Hello, this problem seems rather difficult, but it actually requires thinking back about our ideas of torque and statics problems back from physics 1, and thinking about applying force diagrams and newton's laws for rotation to figure out the angle at which this current carrying metal rod is being suspended in this uniform magnetic field.

00:27 So just getting a quick overview of what's happening here, we draw our picture.

00:33 We have a region of space where the magnetic field is headed into the screen everywhere.

00:38 We have this metal current carrying rod that is at rest at an angle theta from the horizontal.

00:49 And again, we have a current eye going through this rod.

00:55 Now, if there's a current going through the rod and it's in an external magnetic field, there will be a magnetic force.

01:01 So really what's happening here is we have the magnetic force, the gravitational force, and the hinge forces all balancing out here so that the object stays at rest.

01:10 And we have all the torques that are being applied from the magnetic field and gravity that are also balancing.

01:17 So this thing remains at rest and not rotating.

01:20 So how do we figure out how these torques balance? well, the main thing we're going to do is draw a force diagram and think about what forces are acting on the rod and where and in what directions.

01:32 So let's draw a force diagram of our rod.

01:38 So our rod is on a hinge at point p at the end.

01:42 The rod goes up and to the right like this, like so, and let's think about the forces that are acting on it and where and in what directions.

01:52 That matters a lot.

01:53 So one thing is that are often forget about that are very important in problems like this are our hinge forces.

01:58 So we're going to have a force from our hinge acting up at this end.

02:03 I'll call it the hinge force in the y direction because i'll define my upward as positive y and to the right here as positive x.

02:12 Then i'm also going to have a hinge force acting to the right here as well.

02:17 Next we have gravity.

02:19 In which way does gravity act? well gravity acts straight down towards the ground and acts at the center of the center of the of mass of our object.

02:25 The center of mass of the object, since it's a uniform metal rod, is going to be at the center of the rod, and the gravitational force acts straight down.

02:33 And we'll represent that gravitational force, of course, as the mass of our object times the acceleration due to gravity.

02:42 Now, the next question is, what is the direction of the magnetic force acting on this object? well, remember, we have our current, we'll kind of use our diagram over the left here.

02:50 We have our current heading up into the right, and our magnetic field heading inward.

02:54 So if we do our right hand rule, we put our pointer finger along the direction of the current, so this way, and then we rotate our wrist so that our middle finger is pointing down into the screen like that.

03:09 When we do that, we see that our thumb actually points up and to the left.

03:14 So that is the direction of the magnetic force.

03:18 Let me rewrite that word.

03:20 A, g.

03:21 So the magnetic force is up into the left like this.

03:25 When i represent that on my force diagram, i'm going to draw it perpendicular to my rod up and to the left.

03:32 So that is the direction of the magnetic force acting on my object.

03:37 And i know that the magnitude of the magnetic force is going to be given by my equation for the force on a current carrying wire in a magnetic field.

03:49 I, b, or i, l, b, sign.

03:53 In this case, i'll use phi because it might be different than the theta that we have in our diagram here.

04:01 The angle between the current direction and the direction of the external magnetic field.

04:09 Now let's actually think about that real quick.

04:11 What is this angle phi? what is the angle between the current carrying direction, which in this case is up into the right, and the direction of the magnetic field, which is into the page? if you think about that, these two are perpendicular to each other.

04:22 So this phi angle is actually 90 degrees.

04:26 And again, sign of 90 degrees is going to go to one.

04:29 So in the end, we have a magnetic force acting up into the left on a rod perpendicular to the rod of magnitude ilb.

04:41 Cool.

04:42 Now what we want to do is figure out how the torques balance.

04:45 Now, if we think back to how we figure out a torque, what really matters is we think to figure out where the force is acting.

04:54 So we call it a distance r from a pivot point we define to the point of application of the force, multiplied by the part of the force that is perpendicular to that distance r.

05:05 So if we wanted to figure out the twerk due to gravity, we first need to figure out where the gravitational force is acting from a particular pivot point.

05:16 Now, forces that act at the pivot point create no torques because the distance from the pivot point to the point of application.

05:25 The force would be zero because we've placed our pivot point at the point of application of the force.

05:30 That happens if we put our pivot point at point p...

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