How Physics Includes Air Resistance in Calculations | Real Physics

Ignore Air Resistance? I don't think so...

Hey everyone, I'm back with another video! This time, we're looking at how air resistance (drag) is modelled in physics equations. Quite often, in order to make the math easier, we are told to ignore air resistance, but in order to create a more realistic model, we need to account for it.

Using classical physics, we will be studying the motion of a pendulum. We will first look at all the parameters we use to describe a simple pendulum, and then learn how to generate an equation of motion for a pendulum in a vacuum. Then, we will see how to add in all the air resistance (drag) and how this affects the equation of motion. We will also be looking at how to solve these equations of motion, and what impact this has on the motion of the pendulum itself.

We use classical physics principles (such as Newton's Second Law of Motion) to generate the equation of motion. We decide to model air resistance very simply, by assuming that drag force generated is directly proportional to the (angular) speed of the pendulum at any point in time. We also see that a pendulum's motion is very easily described in terms of the angle (we called it theta) between its vertical position and any other position at a given point in time. We look at how this angle changes, as well as the angular velocity / angular speed, and the angular acceleration of the pendulum. It's worth noting that we use the Small Angle Approximation in order to easily solve the equations of motion, because without it we would need some computational methods. This means that we will only be able to consider the motion of the pendulum until about 60 degrees or 1 radian away from the vertical.

As it turns out, a pendulum in a vacuum (i.e. no air resistance) undergoes sinusoidal oscillation forever. In other words, it simply pings back and forth and back and forth. However, adding air resistance actually causes this sinusoidal oscillation to decay exponentially. Mathematically, we see this as multiplying our sine oscillation by a decaying exponential term, known as the "exponential envelope".

It's also worth noting that there are a few other solutions to the equation of motion that accounts for air resistance. This all depends on how strong the drag force is (depending on the medium the pendulum moves through, we called this factor "gamma"). Also, I've taken a lot of mathematical shortcuts and waved my hands many times in this video! My aim is to provide an intuitive understanding of how we do the mathematics and what comes from it, rather than a detailed theoretical dive into the topic.

Here are some useful resources for the bits that I mentioned in the video:
Terminal Velocity - https://simple.wikipedia.org/wiki/Terminal_velocity
Derivation of the Simple Pendulum Equation (without air resistance) - https://www.acs.psu.edu/drussell/Demos/Pendulum/Pendulum.html
Resolution of Forces (Finding Components) - https://www.physicsclassroom.com/class/vectors/Lesson-3/Resolution-of-Forces

Here are some timestamps:
0:00 - Air Resistance!
0:48 - Terminal Velocity (for which air resistance is essential!)
2:37 - How We Will Model Air Resistance Using Math
3:02 - Setting Up The Equations for a Pendulum Using the Angle Theta
3:38 - Angular Velocity and Angular Acceleration Explained Using Linear Velocity and Acceleration
5:36 - Newton's Second Law of Motion (and a Bad, Hand-Wavy Use of It!)
6:14 - Finding the Components of the Force on the Pendulum
6:54 - Setting Up an Equation of Motion (Ignoring Air Resistance For Now)
7:28 - The Small Angle Approximation
8:16 - The Solution (i.e. How a Pendulum Behaves Without Air Resistance) - Sinusoidal Oscillation
8:36 - Air Resistance Has Entered The Chat
8:48 - Modifying the Equation of Motion to Include Air Resistance
9:24 - Pendulum in Air vs Honey: The Constant of Proportionality
10:15 - The New Solution!
10:45 - The Pendulum No Longer Oscillates Forever - The Effect of Drag

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Видео How Physics Includes Air Resistance in Calculations | Real Physics канала Parth G