Welcome to Further Dynamics!
In your earlier Mechanics studies, you probably spent a lot of time with constant acceleration (those trusty SUVAT equations). But the real world is a bit more chaotic! In this chapter, we explore what happens when the force acting on an object changes as it moves. Whether it’s a planet orbiting a star or a mass bouncing on a spring, you are about to learn the math behind the universe's most interesting movements.
Don't worry if this seems tricky at first—we are simply taking the Newton's Second Law you already know and adding a little bit of calculus to make it more powerful!
1. Variable Forces in One Dimension
Newton’s Second Law tells us that \( F = ma \). In Further Dynamics, the force \( F \) isn't a constant number anymore. It might depend on time (\( t \)), displacement (\( x \)), or velocity (\( v \)).
Choosing the Right "Acceleration"
Because the force varies, the acceleration \( a \) varies too. To solve these problems, we rewrite \( a \) using calculus. There are two main ways to write acceleration, and picking the right one makes the math much easier:
1. If the force depends on time (\( t \)): Use \( a = \frac{dv}{dt} \).
2. If the force depends on displacement (\( x \)): Use \( a = v\frac{dv}{dx} \).
Step-by-Step: Solving a Variable Force Problem
1. Set up the equation: Write \( F = ma \).
2. Substitute: Replace \( F \) with the given expression and \( a \) with the appropriate calculus form.
3. Separate and Integrate: Move all terms of one variable to one side and integrate. (This is just like solving differential equations in Pure Math!)
4. Find the constant: Use the initial conditions (like "starts from rest at the origin") to find \( +C \).
The Inverse Square Law (Gravitation)
One famous example of a variable force is Gravity. The force of gravity between two objects follows an inverse square law, which looks like this: \( F = \frac{k}{x^2} \).
Analogy: Think of a magnet. When you are far away, you barely feel it. As you get closer, the pull doesn't just get stronger—it gets stronger much faster the closer you get!
Quick Review:
• Force depends on \( t \) \(\rightarrow\) Integrate \( \frac{dv}{dt} \)
• Force depends on \( x \) \(\rightarrow\) Integrate \( v\frac{dv}{dx} \)
Key Takeaway: When force changes, acceleration is no longer a number; it’s a derivative. Your goal is to integrate your way back to velocity or displacement.
2. Simple Harmonic Motion (SHM)
Simple Harmonic Motion is a specific type of periodic motion. It happens when an object is pulled back toward a central "home" position by a force that gets stronger the further the object wanders away.
The Definition of SHM
A particle moves with SHM if its acceleration is proportional to its displacement from a fixed point but acts in the opposite direction. Mathematically, we write this as:
\( \ddot{x} = -\omega^2 x \)
Wait, what are those symbols?
• \( \ddot{x} \) is just fancy notation for acceleration (the second derivative of displacement).
• \( x \) is the displacement from the center.
• \( \omega \) (the Greek letter omega) is a constant related to the speed of the oscillation.
• The Minus Sign: This is crucial! It means if you are to the right (\( +x \)), the acceleration pulls you back to the left (\( - \)).
Standard SHM Formulae
In the exam, you are expected to be familiar with these (though you can sometimes quote them without proof):
• Velocity: \( v^2 = \omega^2(a^2 - x^2) \)
(Where \( a \) is the Amplitude—the furthest the particle gets from the center.)
• Displacement: \( x = a\cos(\omega t) \) (if starting from the maximum displacement) or \( x = a\sin(\omega t) \) (if starting from the center).
• Time Period: \( T = \frac{2\pi}{\omega} \)
(This is the time it takes for one full "there and back" oscillation.)
Did you know?
In SHM, the time it takes to complete one oscillation (the Period) does not depend on the amplitude. Whether a swing moves a tiny bit or a large amount, the time for one swing stays the same (as long as it’s true SHM)!
Common Mistake: Confusing Amplitude (\( a \)) with Acceleration (\( a \)). In SHM, we usually use \( a \) for amplitude. To avoid confusion, some students like to write acceleration as \( \frac{d^2x}{dt^2} \).
Summary: SHM is a "tugging" motion. The further you go, the harder you are pulled back. The core equation is \( \ddot{x} = -\omega^2 x \).
3. Oscillations of Springs and Elastic Strings
This is where we combine Hooke's Law with SHM. When you hang a mass on a spring and let it bounce, it often moves in SHM.
Finding the Center of Motion
Before you start calculating the bounce, you must find the equilibrium position. This is the point where the mass would sit perfectly still because the upward pull of the spring exactly matches the downward pull of gravity.
• At equilibrium: \( \text{Tension} = \text{Weight} \)
• Using Hooke's Law: \( \frac{\lambda e}{l} = mg \)
(Where \( e \) is the extension at rest.)
Proving SHM in a Spring System
If a question asks you to prove the motion is SHM, follow these steps:
1. Define a displacement \( x \) downward from the equilibrium position.
2. Write the new total extension: \( \text{Extension} = e + x \).
3. Apply \( F = ma \): \( mg - \text{Tension} = m\ddot{x} \).
4. Substitute Tension: \( mg - \frac{\lambda(e+x)}{l} = m\ddot{x} \).
5. Simplify: Since \( mg = \frac{\lambda e}{l} \), those terms cancel out, leaving you with: \( -\frac{\lambda x}{l} = m\ddot{x} \).
6. Rearrange to the SHM form: \( \ddot{x} = -(\frac{\lambda}{ml})x \).
7. Conclude: This is SHM where \( \omega^2 = \frac{\lambda}{ml} \).
Encouragement: If the algebra in the proof looks scary, remember that the goal is always to get the \( mg \) and the "static" tension to cancel out. You are just looking for that \( \ddot{x} = -\text{constant} \times x \) pattern!
Important Limit: Strings vs. Springs
• Springs: Can push and pull. They stay in SHM whether they are stretched or compressed.
• Strings: Can only pull. If the mass bounces up so high that the string goes slack, it is no longer in SHM—it becomes a particle moving freely under gravity (SUVAT) until the string goes taut again!
Key Takeaway: For vertical bounces, always measure your displacement from the equilibrium position, not the natural length of the spring.
Quick Review Box
1. Newton's 2nd Law: \( F = m \frac{dv}{dt} \) or \( F = m v \frac{dv}{dx} \).
2. SHM Equation: \( \text{Acceleration} = -\omega^2 \times \text{Displacement} \).
3. Time Period: \( T = \frac{2\pi}{\omega} \).
4. Max Speed: Occurs at the center (\( x=0 \)), where \( v = \omega a \).
5. Max Acceleration: Occurs at the endpoints, where \( \text{accel} = \omega^2 a \).