Welcome to Mechanics: Force, Energy, and Momentum!
Welcome to one of the most exciting parts of Physics! In this chapter, we are going to look at how the world moves. Whether it’s a car braking, a footballer kicking a ball, or a satellite orbiting Earth, the same rules of Mechanics apply. Don’t worry if some of these ideas seem a bit "heavy" at first—we’ll break them down piece by piece using everyday examples.
By the end of these notes, you’ll understand how forces change motion, how energy is never lost (just moved around), and why momentum is the "oomph" behind every moving object.
1. Scalars and Vectors
Before we can calculate how things move, we need to know how to describe them. In Physics, every measurement falls into one of two camps:
- Scalars: These only have a magnitude (size). Think of things like mass, speed, distance, or time. If you say "I ran 5 miles," that's a scalar.
- Vectors: These have both magnitude and direction. Think of velocity, displacement, acceleration, and force. If you say "I ran 5 miles North," that's a vector.
Adding and Resolving Vectors
Sometimes forces act at different angles. You need to be able to:
- Add vectors: If two vectors are at right angles, use Pythagoras' Theorem: \( a^2 + b^2 = c^2 \).
- Resolve vectors: This means splitting a single diagonal vector into two parts (components) at right angles to each other—usually horizontal and vertical.
The "Trick" for Resolving:
If you have a force \( F \) at an angle \( \theta \) to the horizontal:
- The component side-by-side (adjacent) to the angle uses Cos: \( F_x = F \cos\theta \)
- The component opposite the angle uses Sin: \( F_y = F \sin\theta \)
Memory Aid: "CO-sine is CO-mponent CO-ntact" (the one touching the angle).
Quick Review: Equilibrium
An object is in equilibrium if the total (resultant) force on it is zero. This means it is either perfectly still or moving at a constant velocity in a straight line.
Key Takeaway: Vectors care about direction; scalars don't. Always check if you need to resolve a force into horizontal and vertical parts before doing your math!
2. Moments and Equilibrium
A moment is just a fancy physics word for a turning effect. Think about using a see-saw or opening a door.
Defining a Moment
The moment of a force is defined as:
\( \text{Moment} = \text{Force} \times \text{perpendicular distance from the pivot} \)
Common Mistake: Students often forget the distance must be perpendicular (at 90 degrees) to the force. If the force is at an angle, you must resolve it first!
Couples and Torque
A couple is a pair of forces that are equal in size, opposite in direction, and parallel. They don't move the object forward, they only make it spin. The moment of a couple (often called torque) is:
\( \text{Torque} = \text{one of the forces} \times \text{perpendicular distance between them} \)
The Principle of Moments
For an object to be balanced (in equilibrium):
Total Anticlockwise Moments = Total Clockwise Moments
Centre of Mass
Every object has a centre of mass. This is the single point where the entire weight of the object seems to act. For a uniform, regular shape (like a ruler), it’s exactly in the middle!
Key Takeaway: To balance a see-saw, the turning effects on both sides must be equal. Distance matters just as much as force!
3. Motion Along a Straight Line
This is where we use the famous SUVAT equations. These only work when acceleration is constant.
The SUVAT Variables:
- \( s \) = Displacement (distance in a specific direction)
- \( u \) = Initial velocity
- \( v \) = Final velocity
- \( a \) = Acceleration
- \( t \) = Time
The Equations You Need:
\( v = u + at \)
\( s = \frac{(u + v)}{2}t \)
\( s = ut + \frac{1}{2}at^2 \)
\( v^2 = u^2 + 2as \)
Motion Graphs
- Displacement-Time Graph: The gradient (slope) tells you the velocity.
- Velocity-Time Graph: The gradient tells you the acceleration. The area under the graph tells you the displacement (distance traveled).
Did you know? On Earth, if you ignore air resistance, everything falls with the same acceleration: \( g = 9.81 \, \text{m/s}^2 \).
Key Takeaway: Graphs are your best friend. If you’re stuck on a motion problem, try sketching a velocity-time graph first.
4. Projectile Motion
A projectile is anything thrown or launched into the air (like a kicked football). The secret to solving these problems is to keep horizontal and vertical motion completely separate.
- Horizontal: There is no horizontal acceleration (if we ignore air resistance). The velocity stays constant! Use \( \text{speed} = \frac{\text{distance}}{\text{time}} \).
- Vertical: Gravity is pulling the object down. It accelerates downwards at \( 9.81 \, \text{m/s}^2 \). Use your SUVAT equations here.
Real-world Fact: Air resistance (drag) increases as you go faster. Eventually, the drag force equals your weight, and you stop accelerating. This is called terminal speed.
Key Takeaway: Time (\( t \)) is the "bridge" between horizontal and vertical motion. It's usually the same for both!
5. Newton’s Laws of Motion
Isaac Newton gave us three rules that explain almost everything about how objects behave:
- First Law: An object will stay still or keep moving at the same speed in a straight line unless a resultant force acts on it. (Objects are lazy!)
- Second Law: The force needed to move an object depends on its mass and how fast you want it to accelerate.
\( F = ma \) - Third Law: If object A exerts a force on object B, then object B exerts an equal and opposite force on object A.
Common Mistake: In the Third Law, the two forces must be the same type (e.g., both gravitational) and act on different objects. The weight of a book and the normal contact force from the table are NOT a Third Law pair!
Key Takeaway: Force causes acceleration. No resultant force means no change in velocity.
6. Momentum
Momentum is a measure of how hard it is to stop a moving object. It is a vector.
\( \text{Momentum} (p) = \text{mass} (m) \times \text{velocity} (v) \)
Conservation of Momentum
In any collision or explosion, the total momentum before = total momentum after (provided no external forces act).
Force and Impulse
Newton actually defined force as the rate of change of momentum:
\( F = \frac{\Delta(mv)}{\Delta t} \)
If you multiply force by time, you get Impulse (the change in momentum):
\( \text{Impulse} = F\Delta t = \Delta mv \)
Analogy: Why do cars have crumple zones? They increase the time of the impact (\( \Delta t \)). Since the change in momentum is the same, increasing the time makes the force (\( F \)) on the passengers much smaller!
Collisions:
- Elastic: Momentum and Kinetic Energy are both conserved.
- Inelastic: Momentum is conserved, but some Kinetic Energy is lost (usually as heat or sound).
Key Takeaway: Momentum is always conserved in a closed system. Use this to find missing velocities after a crash.
7. Work, Energy, and Power
Energy is the ability to do work. Work is done when a force moves an object.
Work Done
\( W = Fs \cos\theta \)
(Where \( s \) is displacement and \( \theta \) is the angle between the force and the direction of motion).
Forms of Energy
- Kinetic Energy (KE): The energy of a moving object. \( E_k = \frac{1}{2}mv^2 \)
- Gravitational Potential Energy (GPE): The energy an object has due to its height. \( \Delta E_p = mg\Delta h \)
Conservation of Energy
Energy cannot be created or destroyed, only transferred from one form to another. For example, a falling ball turns GPE into KE. If there is friction, some energy turns into work done against resistive forces (heat).
Power and Efficiency
Power is the rate of doing work (how fast you transfer energy):
\( P = \frac{\Delta W}{\Delta t} \) or \( P = Fv \)
Efficiency tells us how much energy isn't wasted:
\( \text{Efficiency} = \frac{\text{useful output power}}{\text{total input power}} \times 100\% \)
Key Takeaway: Always look at the energy "account." If KE is missing, it probably went into GPE or was "lost" as heat due to friction.
Final Quick Tips for Success
- Check Units: Always convert mass to kg, distance to meters, and time to seconds.
- Draw a Diagram: Even a simple "box" with arrows helps you see which forces are acting.
- Show Your Working: In the exam, you get marks for the formula and the steps, even if your final number is wrong!
You've got this! Mechanics is all about practice. Try some problems using these formulas and you'll see how it all clicks together.