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

step 1 So for part A, we want the damping constant. So we need to apply Newton's second law to the bar

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

A bar with mass $M=1.20 \quad \mathrm{~kg} \quad$ and $\quad$ 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. P29.62. 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?

A bar with mass $M=1.20 \quad \mathrm{~kg} \quad$ and $\quad$ 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. P29.62. 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 Frequency Calculation

In a damped harmonic oscillator, the frequency of oscillation is reduced from the natural (undamped) frequency due to the presence of the damping force. The damped frequency is calculated by adjusting the natural frequency by a factor that depends on the damping coefficient and the mass, generally expressed as ?_d = ?(?_0^2 - (b/(2m))^2). This concept is key to understanding how dissipation alters the motion of oscillatory systems.

Ohm’s Law

Ohm’s law relates the current flowing through a conductor to the voltage across it and the resistance of the circuit. In an electromagnetic damping scenario, it is used to calculate the magnitude of the induced current by dividing the induced emf (from Faraday’s law) by the resistance of the circuit, thereby linking electrical effects to mechanical damping.

Lenz’s Law

Lenz’s law specifies the direction of the induced current such that the magnetic field created by the current opposes the change in the magnetic flux that produced it. This principle underlies the mechanism of electromagnetic damping by ensuring that the induced forces act in opposition to the motion, thereby reducing the energy of the system.

Electromagnetic Damping

Electromagnetic damping refers to the dissipation of mechanical energy as electrical energy when a conductor moves through a magnetic field. In such systems, motion induces an electromotive force (emf) and a resulting current in the conductor, which, by Lenz’s law, produces a magnetic force opposing the motion. This process effectively acts as a non?mechanical damping mechanism.

Damped Harmonic Oscillator

A damped harmonic oscillator is a system in which a restoring force (usually proportional to displacement) and a damping force (usually proportional to velocity) act on a mass. The system’s motion is typically described by a second?order linear differential equation, where the damping coefficient quantifies the rate at which energy is removed from the system and reduces the amplitude over time.

Faraday’s Law of Electromagnetic Induction

Faraday’s law states that a change in magnetic flux through a circuit induces an electromotive force in the circuit. In the context of a moving conductor in a magnetic field, the law explains how velocity, magnetic field strength, and the effective length of the conductor interact to produce an induced emf, which is essential for describing the electromagnetic damping effect.

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