Welcome to the World of Mechanics!
Hello there! Welcome to your study notes for Mechanics. This chapter is the foundation of almost everything in Physics. We are going to explore how objects move, why they start moving, and the "rules of the game" that the universe follows—from a tennis ball flying over a net to a car braking at a red light. Don't worry if some of the math looks a bit scary at first; we will break it down piece by piece until it makes perfect sense!
1. Scalars and Vectors
Before we look at moving objects, we need to know how to measure them. In Physics, we split quantities into two "teams": Scalars and Vectors.
Scalars: These only have a size (magnitude). Think of things like time (5 seconds) or temperature (20°C). It doesn't have a "direction".
Vectors: These have both a size AND a direction. If you tell someone to walk 10 meters, they might ask "Which way?". That's a vector! Examples include velocity, displacement, and force.
Vector Notation and Diagrams
We represent vectors using arrows. The length of the arrow shows the size, and the head shows the direction.
Top Tip: In your exam, you might see vectors written in bold (like v) or with a little arrow on top.
Resolving Vectors (The "Split" Trick)
Sometimes a vector acts at an angle, and we want to know how much it's pushing "up" vs. how much it's pushing "sideways". This is called resolving.
If you have a vector \(V\) at an angle \(\theta\) to the horizontal:
Horizontal component: \(V_x = V \cos \theta\)
Vertical component: \(V_y = V \sin \sin \theta\)
Analogy: Imagine pulling a suitcase at an angle. Some of your force lifts it up (vertical), and some of it pulls it forward (horizontal).
Resultant Vectors
When two forces act on an object, the resultant is the single force that would have the same effect as both combined. If they are at right angles, just use Pythagoras' Theorem: \(a^2 + b^2 = c^2\).
Quick Review:
• Scalar: Size only (e.g., Distance, Speed).
• Vector: Size + Direction (e.g., Displacement, Velocity).
• Resultant: The "total" vector found by adding components.
2. Motion: The SUVAT Equations
When an object moves with uniform acceleration (a constant change in speed), we can use five magical variables, often called SUVAT:
s = displacement (m)
u = initial velocity (m/s)
v = final velocity (m/s)
a = acceleration (m/s²)
t = time (s)
The Equations You Need to Know:
\(v = u + at\)
\(s = \frac{(u + v)t}{2}\)
\(s = ut + \frac{1}{2}at^2\)
\(v^2 = u^2 + 2as\)
Common Mistake: Students often forget that u and v are vectors. If an object is thrown up at 10 m/s and falls down, you must decide which direction is positive! If up is positive, then the acceleration due to gravity is \(-9.81 m/s^2\).
Key Takeaway: SUVAT only works if the acceleration is constant. If the acceleration changes, you can't use these equations!
3. Motion Graphs
Graphs are a great way to "see" motion. You need to know three types:
1. Displacement-Time Graphs:
• The gradient (slope) = Velocity.
• A flat line means the object is stationary.
• A curve means the object is accelerating.
2. Velocity-Time Graphs:
• The gradient = Acceleration.
• The area under the graph = Displacement (the distance traveled).
• A flat line means constant velocity.
3. Acceleration-Time Graphs:
• The area under the graph = Change in velocity.
Did you know? If a graph is a curve, you can find the instantaneous velocity or acceleration by drawing a tangent to the curve at that point and calculating its gradient!
4. Projectile Motion
A projectile is anything thrown or launched into the air (like a football). The trick to solving these problems is to treat the horizontal and vertical motions as completely separate!
Horizontal Motion: There is no horizontal force (ignoring air resistance), so the horizontal velocity stays constant. Use \(s = vt\).
Vertical Motion: Gravity is pulling it down at \(9.81 m/s^2\). Use your SUVAT equations here.
Analogy: Think of a ghost and its shadow. If you throw a ball, the ball's shadow on the ground moves at a constant speed (horizontal), while the ball itself rises and falls (vertical).
5. Newton’s Laws of Motion
Sir Isaac Newton gave us three rules that describe how forces work.
Newton’s First Law: An object will stay still or keep moving at a constant velocity unless a resultant force acts on it. This is called Inertia.
Newton’s Second Law: Force equals mass times acceleration. \(\sum F = ma\). Note that "F" is the resultant (total) force.
Newton’s Third Law: If Object A exerts a force on Object B, then Object B exerts an equal and opposite force on Object A. These pairs must be the same type of force!
Weight and Gravity
Mass is how much "stuff" is in you (kg), but Weight is a force (N) caused by gravity pulling on that mass.
\(W = mg\)
Where \(g\) is the gravitational field strength (approx. \(9.81 N/kg\) on Earth).
Terminal Velocity
When an object falls, it speeds up until the drag force (air resistance) pushing up equals its weight pulling down. The resultant force becomes zero, so acceleration becomes zero. The object then falls at a steady speed called terminal velocity.
Quick Review:
• \(F=ma\) is the big one! Always find the resultant force first.
• Terminal velocity happens when Drag = Weight.
6. Momentum
Momentum is a measure of how hard it is to stop a moving object. It is a vector.
Equation: \(p = mv\) (Momentum = mass × velocity)
Conservation of Momentum
In any collision or explosion, the total momentum before = total momentum after, provided no external forces act.
Memory Aid: "What goes in must come out." If two skaters push off each other, their total momentum stays zero because they move in opposite directions!
7. Moments and Equilibrium
A moment is the turning effect of a force.
Moment = Force × perpendicular distance from the pivot (\(Fx\))
Centre of Gravity: This is the single point where the entire weight of an object appears to act. For a uniform ruler, it's right in the middle.
Equilibrium: For an object to be perfectly balanced (in equilibrium):
1. The total Resultant Force must be zero.
2. The total Clockwise Moments must equal the total Anti-clockwise Moments.
8. Work, Energy, and Power
Work Done: This is the energy transferred when a force moves an object.
\(\Delta W = F\Delta s\)
(If the force is at an angle, use the component in the direction of motion: \(W = Fs \cos \theta\)).
Kinetic Energy (\(E_k\)): The energy of a moving object.
\(E_k = \frac{1}{2}mv^2\)
Gravitational Potential Energy (\(\Delta E_{grav}\)): The energy gained by lifting an object.
\(\Delta E_{grav} = mg\Delta h\)
Conservation of Energy
Energy cannot be created or destroyed, only transferred. In a perfect world, \(GPE\) lost = \(KE\) gained. In the real world, some is "lost" as heat due to friction.
Power and Efficiency
Power: The rate at which work is done.
\(P = \frac{E}{t}\) or \(P = \frac{W}{t}\) (Units: Watts, W)
Efficiency: How much of the input energy actually does something useful.
\(Efficiency = \frac{\text{useful energy output}}{\text{total energy input}}\)
Summary Takeaway: Mechanics is all about tracking energy and forces. If you can draw a free-body force diagram showing all the arrows acting on an object, you are halfway to solving any problem!
Don't worry if this seems tricky at first—keep practicing your SUVAT and drawing your diagrams, and you'll be a Mechanics master in no time!