A bar with mass M=1.20 kg and resistance R=0.500 Ωslides without friction on a horizontal U-sha

A bar with mass M=1.20 kg and resistance R=0.500 Ωslides...

A bar with mass $M=1.20 \mathrm{~kg} $ and resistance $R=0.500 \Omega$ slides without friction on a horizontal U-shaped rail with width $W=40.0 \mathrm{~cm}$ and negligible resistance. The bar is attached to a spring with spring constant $k=90.0 \mathrm{~N} / \mathrm{m}$, as shown in Fig. $\mathbf{P 2 9 . 6 2 .}$. A constant magnetic field with magnitude $1.00 \mathrm{~T}$ points into the plane everywhere in the vicinity. At time $t=0$ the bar is stretched beyond its equilibrium position by an amount $x=10.0 \mathrm{~cm}$ and released from rest. (a) This system behaves like a damped oscillator, described by Eq. $(14.41) .$ What is the damping coefficient $b ?$ (b) With what frequency does the bar oscillate around its equilibrium position? (c) What is the amplitude of the motion at time $t=5.00 \mathrm{~s} ?$ (d) What is the magnitude of the current in the bar when it passes the equilibrium position for the first time? (e) What is the direction of that current?

Key Concepts

Damped Harmonic Oscillator

A damped harmonic oscillator is a system in which a mass attached to a spring experiences a resistive force that is proportional to its velocity. This damping force causes the amplitude of oscillation to decrease exponentially over time and modifies the system's natural oscillation frequency. The mathematical model typically involves a second?order differential equation that includes terms for inertia, restoring force, and damping, and it provides insight into how energy is lost from the system during oscillations.

Oscillation Frequency

The oscillation frequency of a damped system is lower than its natural frequency due to the additional damping force. This frequency can be determined from the system’s parameters, such as the mass and spring constant, along with the damping coefficient, by using a modified version of the standard frequency formula for undamped oscillators. Understanding this frequency is essential for predicting the motion and behavior of the oscillator over time.

Exponential Decay of Amplitude

In a damped oscillator, the amplitude of oscillation diminishes exponentially with time. This decay results from the continuous loss of energy due to the damping force. The rate at which the amplitude decreases is directly related to the damping coefficient, which quantifies the strength of the resistive forces acting on the oscillating object.

Electromagnetic Induction

Electromagnetic induction is a fundamental principle that describes how a changing magnetic environment around a conductor induces an electromotive force (emf). This phenomenon, encapsulated by Faraday's law, is central to understanding systems where motion in a magnetic field generates electrical currents. In oscillatory systems, induced emf can result from the motion of a conductor through a magnetic field.

Motional EMF and Electrical Resistance

When a conductor moves through a magnetic field, it experiences a motional emf that drives an electrical current if the circuit is closed. The magnitude of this current is influenced by the circuit’s total resistance, as described by Ohm's law. This interplay between motional emf and resistance is crucial for determining the amount of energy dissipated in the system and thus contributes to the damping effect observed in electromagnetic damping.

Lenz's Law

Lenz's law states that the direction of any induced current is such that the magnetic field created by the current opposes the change in the original magnetic flux. This concept is vital for understanding the nature of electromagnetic damping, as it explains why the induced currents create forces that act in opposition to the motion causing the induction, thereby reducing the system's kinetic energy.

Transcript

00:02 In this question we are going to find a number of things regarding the damped oscillation motion for this bar attached to a spring that lies on a u -shaped rail within a magnetic field.

00:22 So this is going to pull together some of the things we know about electromagnetic induction as well as damped oscillations from simple harmonic motion.

00:36 So we're given a we're instructed to go look at equation 14 .41 to get the equation that describes a damped oscillator.

00:51 So that equation is negative kx minus bv equals ma and this is for a single direction a single dimension.

01:13 And where this came from was we have a restoring spring force that doesn't look very legible.

01:33 We have a restoring spring force and then we have a frictional force.

01:42 It could be friction but we'll call it a damping force right.

01:46 We have some sort of resistive force that depends on the velocity at which the system is moving and then of course we have the ma that completes newton's second law.

02:06 So this equation is just an expression of newton's second law.

02:10 And to find the damping coefficient b for this given system what we need a force that is dependent on the velocity from all of our magnetic induction kinds of things.

02:44 And the first thing to note is i mean we could either you can apply faraday's law or use your relationship for motional emf.

02:56 You get the same result.

02:59 I'm going to go for the fast one which is the equation for motional emf.

03:14 So the emf induced in this bar will be equal to b times the width of the bar w multiplied by v the velocity at which this bar is going to move.

03:32 And i'm not going to worry about signs we're just going to go with magnitudes here.

03:37 Trust me if you wanted to start with faraday's law that the induced emf is equal to negative d dt of the flux you're going to get the same thing.

03:49 Okay cool so how do we calculate now the force on a length of current carrying wire in a magnetic field? well that's found with the current in the wire multiplied by its length which again in this case is w multiplied by the magnetic field strength.

04:20 And since they're already perpendicular we don't have to worry about the sine theta that moves in there.

04:24 Well, swell! we need something for i.

04:30 Well our current i is just going to be equal to the emf divided by the resistance of the bar.

04:38 So b w v divided by r.

04:43 So now we have our force is going to be equal to b w v over r times w times b.

04:57 And so we get b squared w squared over r multiplied by v...

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