Welcome to the World of Dynamics!
In our previous studies of Kinematics, we looked at how objects move (speed, acceleration, etc.). Now, in Dynamics, we are going to explore the why. Why does a ball stop rolling? Why does a rocket need so much fuel to take off? Dynamics is all about forces and how they change the motion of everything in the universe.
Don't worry if some of these ideas seem a bit "heavy" at first—we'll break them down into bite-sized pieces that are easy to swallow!
3.1 Momentum and Newton’s Laws of Motion
What is Mass?
In Physics, mass is more than just a number on a scale. It is the property of an object that resists change in motion. We call this inertia.
Think of it like this: It is much easier to push an empty shopping cart than one filled with heavy bricks. The bricks have more mass, so they "resist" your push more strongly.
Linear Momentum
Momentum is often described as "mass in motion." If an object is moving, it has momentum. It depends on two things: how much stuff is moving (mass) and how fast it is moving (velocity).
The formula for linear momentum (\(p\)) is:
\( p = mv \)
Where:
• \(p\) is momentum (measured in \(kg \cdot m \cdot s^{-1}\))
• \(m\) is mass (\(kg\))
• \(v\) is velocity (\(m \cdot s^{-1}\))
Remember: Momentum is a vector. This means the direction matters! If a ball moves right, its momentum is positive; if it moves left, it's negative.
Newton’s Three Laws of Motion
Isaac Newton gave us three rules that explain almost every movement we see:
1. Newton’s First Law: An object will stay at rest or keep moving at a constant velocity unless a resultant force acts on it. This is the "Couch Potato Law"—objects want to keep doing exactly what they are already doing.
2. Newton’s Second Law: This is the "Action Law." It tells us that Force is the rate of change of momentum.
The formula is:
\( F = \frac{\Delta p}{\Delta t} \)
If the mass stays constant, this simplifies to the famous:
\( F = ma \)
Important Point: The acceleration (\(a\)) and the resultant force (\(F\)) are always in the same direction.
3. Newton’s Third Law: If Body A exerts a force on Body B, then Body B exerts an equal and opposite force on Body A. These forces must be of the same type (e.g., both gravitational or both contact forces).
Weight vs. Mass
Students often get these mixed up! Mass is the amount of matter you are made of (measured in \(kg\)). Weight is the force acting on that mass due to a gravitational field.
The formula is:
\( W = mg \)
Where \(g\) is the acceleration of free fall (on Earth, \(g \approx 9.81 m \cdot s^{-2}\)).
Quick Review: Key Takeaways
• Mass resists change in motion (inertia).
• Momentum = mass × velocity (\(p=mv\)).
• Force = rate of change of momentum (or \(F=ma\)).
• Weight is a force caused by gravity (\(W=mg\)).
3.2 Non-Uniform Motion
In the real world, things don't usually move without interference. There are "hidden" forces trying to slow us down.
Friction and Drag
Frictional forces occur when two surfaces rub together. Drag forces (like air resistance or viscous drag in liquids) happen when an object moves through a fluid.
Did you know? Drag forces are not constant. As you go faster, the drag force increases. This is why you feel a much stronger wind hitting your face when cycling fast versus walking.
Terminal Velocity
When an object falls through the air, it doesn't keep accelerating forever. It eventually reaches a steady speed called terminal velocity.
Step-by-Step: How Terminal Velocity Happens
1. At the start, only weight acts downwards. The object accelerates at \(9.81 m \cdot s^{-2}\).
2. As speed increases, air resistance (drag) starts to push upwards.
3. The resultant force downwards decreases, so the acceleration decreases.
4. Eventually, the drag force becomes equal to the weight.
5. The resultant force is now zero. The object stops accelerating and moves at a constant terminal velocity.
Quick Review: Key Takeaways
• Drag increases as velocity increases.
• Terminal velocity occurs when Weight = Drag (Resultant Force = 0).
3.3 Linear Momentum and its Conservation
The Principle of Conservation of Momentum
This is one of the most important rules in Physics! It states that: The total momentum of a closed system remains constant, provided no external resultant force acts on it.
In simple terms: Total Momentum Before = Total Momentum After
\( m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \)
Elastic vs. Inelastic Collisions
While momentum is always conserved in any collision, Kinetic Energy (KE) behaves differently.
1. Perfectly Elastic Collisions:
• Momentum is conserved.
• Total Kinetic Energy is conserved.
• Relative speed of approach = Relative speed of separation.
Formula: \( u_1 - u_2 = v_2 - v_1 \)
2. Inelastic Collisions:
• Momentum is conserved.
• Total Kinetic Energy is NOT conserved (some energy is turned into heat or sound).
• Most real-world collisions (like car crashes) are inelastic.
Common Mistake: Students often think that if KE is lost, momentum is also lost. This is wrong! Momentum is always conserved in a closed system, regardless of the type of collision.
Collisions in Two Dimensions
Sometimes objects don't hit head-on; they glance off at angles. In these cases, you simply apply the conservation of momentum twice: once for the horizontal (x) direction and once for the vertical (y) direction. It's just like resolving vectors!
Quick Review: Key Takeaways
• Total Momentum is always conserved in a closed system.
• In Elastic collisions, Kinetic Energy is also conserved.
• In Inelastic collisions, Kinetic Energy is lost (but momentum is still conserved!).
• Use relative speed (\( u_1 - u_2 = v_2 - v_1 \)) for elastic collision problems to save time!
You've reached the end of the Dynamics notes! Take a deep breath. Dynamics can feel like a lot of math, but it's really just a few big ideas—Newton's Laws and Momentum—applied to different situations. Keep practicing the \(F=ma\) and \(p=mv\) problems, and it will soon become second nature!