Show that the time rate of change of mechanical energy for a...

Key Concepts

Equation of Motion for Damped Systems

The equation of motion for a damped system incorporates both the restorative forces (such as the spring force in a simple harmonic oscillator) and damping forces that oppose motion. This equation, typically obtained from Newton’s second law, is key to relating the forces acting on the system to its acceleration and velocity, ultimately allowing for the analysis of the time-dependent behavior of its energy.

Energy Dissipation

Energy dissipation refers to the process by which mechanical energy is converted into other forms, such as thermal energy, due to non-conservative forces acting in the system. In the context of a damped oscillator, energy dissipation manifests as a decrease in the system's mechanical energy over time, highlighting how external factors like damping lead to the inevitable loss of energy.

Time Derivative of Energy

The time derivative of energy involves differentiating the mechanical energy of a system with respect to time. This process, using calculus and the chain rule, is essential for determining how the energy of a system changes due to forces acting upon it. In systems with damping, this derivative directly relates to the rate of energy loss and provides a quantitative measure of energy dissipation.

Mechanical Energy

Mechanical energy in an oscillator is the sum of its kinetic and potential energies. This concept is fundamental in physics as it allows one to analyze how energy is stored and transferred within a system that can oscillate, such as a mass attached to a spring. Understanding mechanical energy provides insight into the conservation (or non-conservation) of energy when non-conservative forces are present.

Damped Oscillator

A damped oscillator is a system in which energy is gradually lost over time due to forces like friction or air resistance. The damping force opposes the motion and is typically proportional to the velocity of the oscillator. Studying damped oscillators is crucial for understanding real-world systems where energy losses prevent perpetual motion.

Transcript

00:01 Okay, for this question, we want to show that the time rate of change of mechanical energy for a damped undruined oscillator is given by dedt is equal to minus bv squared.

00:12 And what we're given is the mechanical energy for this type of oscillator, one half mv squared plus one half kx squared.

00:20 And it tells us to use this equation 15 .31, which is given here in green, which is as follows.

00:28 Minus k x minus b the x d x d t is equal to m times the second derivative of x with respect to t so i've kind of already started off this question so we have the mechanical energy for a damp undripen oscillator as follows we're going to take the time derivative of this equation to get started and get hopefully the result they wants to show so the tricky thing here is doing this uh derivative with respect to time is dealing with the quantities v and x because yes they're with respect to time but also they also relate to position right so this is where where i already put in red we want to use implicit differentiation that's the key here for this problem so we want to treat it as yeah we're going to take the time derivative of velocity and x but that term those respective terms velocity in position v and x are still going to be there after we do the chain rule here as we differentiate.

01:34 So i kind of just write it out and show you what that means or what it looks like.

01:38 I'm sure you guys have looked at it before, but that can be kind of tricky with this type of problem is knowing when to just straight out, take a derivative or when to use implicit differentiation.

01:50 We're dealing with these more abstract variables.

01:55 Okay, so we're going to do time derivative of one half mb squared.

01:59 So we have the chain rule.

02:00 So that's going to be, let's just write this out first.

02:03 We have one half m.

02:05 Those have nothing to do with time.

02:07 Those are dependent of time.

02:09 Or i'm sorry, independent of time.

02:12 So now we're just taking the derivative of v squared with respect to time.

02:17 So chain rule.

02:18 So we have times 2v.

02:22 That's why i mean by implicit differentiation is 2v times you can use v prime, v.