Topic 2: Mechanics — Edexcel AS Level Physics (8PH0)
Welcome to Mechanics! Whether you are tracking the flight of a football, designing a rollercoaster, or calculating rocket thrust, mechanics is the rulebook that explains how and why things move. This chapter forms the core of Paper 1: Core Physics I (8PH0/01). Don't worry if equations of motion or resolving vectors seem intimidating at first — we will break down every single concept into clear, bite-sized steps with real-world examples.
Standard Constants for this Chapter:
• Acceleration of free fall close to Earth's surface: \(g = 9.81\text{ m s}^{-2}\)
• Gravitational field strength close to Earth's surface: \(g = 9.81\text{ N kg}^{-1}\)
1. Kinematics: Describing Motion
Scalars vs. Vectors
In physics, every physical quantity falls into one of two categories:
• Scalar: A quantity that has magnitude (size) only. It does not have a direction. Examples include distance, speed, mass, time, and energy.
• Vector: A quantity that has both magnitude and direction. Examples include displacement, velocity, acceleration, force, and momentum.
Analogy: If a GPS says "walk \(5\text{ km}\)", that is a scalar (distance). If it says "walk \(5\text{ km}\) North", that is a vector (displacement).
Kinematic Graphs
Graphs give us a visual story of motion over time. For your exams, remember these two vital graph rules:
1. Displacement–Time (\(s\)–\(t\)) Graphs:
• Gradient: Represents velocity (\(v = \frac{\Delta s}{\Delta t}\)).
• A flat horizontal line means the object is stationary (\(v = 0\)).
• A constant straight slope means constant velocity.
• A curved line indicates changing velocity (acceleration).
2. Velocity–Time (\(v\)–\(t\)) Graphs:
• Gradient: Represents acceleration (\(a = \frac{\Delta v}{\Delta t}\)).
• Area under the graph: Represents the change in displacement (\(\Delta s\)).
Uniformly Accelerated Motion (The SUVAT Equations)
When an object moves with constant (uniform) acceleration in a straight line, we use the five SUVAT variables:
• \(s\) = displacement (\(\text{m}\))
• \(u\) = initial velocity (\(\text{m s}^{-1}\))
• \(v\) = final velocity (\(\text{m s}^{-1}\))
• \(a\) = acceleration (\(\text{m s}^{-2}\))
• \(t\) = time taken (\(\text{s}\))
The four core equations provided on your Edexcel Data Sheet are:
\(v = u + at\)
\(s = \frac{(u + v)t}{2}\)
\(s = ut + \frac{1}{2}at^2\)
\(v^2 = u^2 + 2as\)
Step-by-Step SUVAT Strategy:
1. Choose a positive direction (e.g., upwards = positive, downwards = negative).
2. List your five variables: \(s, u, v, a, t\) and write down the values you know.
3. Identify the unknown variable you need to find.
4. Select the equation containing your three knowns and one unknown.
5. Substitute the numbers and solve carefully.
Projectile Motion (2D Motion)
When an object is launched into the air, its motion can be split into two completely independent components (assuming negligible air resistance):
• Horizontal Motion (No forces act horizontally):
Acceleration is zero: \(a_x = 0\)
Horizontal velocity remains constant: \(v_x = u_x = u\cos\theta\)
Horizontal displacement: \(s_x = (u\cos\theta)t\)
• Vertical Motion (Gravity acts downwards):
Acceleration is downward: \(a_y = -g = -9.81\text{ m s}^{-2}\)
Vertical velocity at time \(t\): \(v_y = u\sin\theta - gt\)
Vertical displacement: \(s_y = (u\sin\theta)t - \frac{1}{2}gt^2\)
Top Tip: The only variable shared by both the horizontal and vertical components is time (\(t\)). Find time from one component, and use it in the other!
Key Takeaway for Kinematics: Always establish a consistent sign convention (positive/negative directions) before plugging numbers into SUVAT equations or resolving projectile components.
---2. Forces and Newton's Laws of Motion
Newton's Three Laws of Motion
1. Newton’s First Law: An object remains at rest or continues to move with constant velocity unless acted upon by a resultant external force.
2. Newton’s Second Law: The resultant force acting on an object is directly proportional to the rate of change of momentum: \(\Sigma F = \frac{\Delta p}{\Delta t}\). When mass \(m\) is constant, this simplifies to:
\(\Sigma F = ma \quad \text{or} \quad g = \frac{F}{m}\)
3. Newton’s Third Law: If body A exerts a force on body B, body B exerts an equal and opposite force on body A.
Crucial Rules for Newton's 3rd Law Pairs:
• The forces must act on two different bodies.
• The forces must be of the exact same type (e.g., both gravitational, both normal contact).
• The forces must be equal in magnitude, opposite in direction, and act along the same line.
Weight
Weight is the gravitational force acting on an object's mass:
\(W = mg\)
Free-Body Force Diagrams
A free-body diagram shows all forces acting on a single isolated object.
• Draw the object as a simple box or dot.
• Draw forces as straight arrows pointing away from the object's centre of gravity in the correct direction.
• Label each force clearly (e.g., Weight \(W\), Normal Contact Force \(R\), Friction \(F\), Tension \(T\)).
Resolving Forces
To analyze forces acting at angles, we resolve them into perpendicular components:
• Component adjacent to angle \(\theta\): \(F_x = F\cos\theta\)
• Component opposite to angle \(\theta\): \(F_y = F\sin\theta\)
Resolving on an Inclined Plane:
When an object rests on a slope inclined at angle \(\theta\) to the horizontal:
• Component of weight acting down the slope: \(W_{\parallel} = mg\sin\theta\)
• Component of weight acting perpendicular into the slope: \(W_{\perp} = mg\cos\theta\)
Key Takeaway for Forces: Normal contact force and weight are NOT a Newton's third law pair because they act on the same body and represent different types of forces.
---3. Moments and Equilibrium
The Moment of a Force
A moment is the turning effect of a force around a pivot.
\(\text{Moment} = Fx\)
where \(F\) is the applied force and \(x\) is the perpendicular distance from the line of action of the force to the pivot point.
Centre of Gravity
Centre of Gravity: The single point from which the total weight of a body may be considered to act.
The Principle of Moments
For an object in rotational equilibrium, the sum of clockwise moments about any point is equal to the sum of anticlockwise moments about that same point:
\(\Sigma \tau_{\text{clockwise}} = \Sigma \tau_{\text{anticlockwise}}\)
Conditions for Complete Equilibrium
For an object to be in complete static equilibrium, two conditions must be satisfied simultaneously:
1. Translational Equilibrium: The resultant force in any direction is zero (\(\Sigma \mathbf{F} = 0\), meaning \(\Sigma F_x = 0\) and \(\Sigma F_y = 0\)).
2. Rotational Equilibrium: The resultant moment about any chosen pivot point is zero (\(\Sigma \tau = 0\)).
Key Takeaway for Moments: When calculating moments, always ensure you measure the distance at a right angle (\(90^\circ\)) to the line of action of the force.
---4. Linear Momentum and Impulse
Linear Momentum
Linear momentum (\(p\)) is the product of an object's mass and its velocity:
\(p = mv\)
Units: \(\text{kg m s}^{-1}\) or \(\text{N s}\). Momentum is a vector quantity, meaning direction matters (e.g., motion to the right can be positive, motion to the left negative).
Principle of Conservation of Linear Momentum
In an isolated system, the total linear momentum before a collision or explosion is equal to the total linear momentum after, provided no external resultant forces act on the system:
\(\Sigma p_{\text{initial}} = \Sigma p_{\text{final}}\)
Elastic vs. Inelastic Collisions
• Perfectly Elastic Collision:
Total momentum is conserved.
Total kinetic energy is conserved (\(\Sigma E_{k,\text{initial}} = \Sigma E_{k,\text{final}}\)).
• Inelastic Collision:
Total momentum is conserved.
Kinetic energy is not conserved; some kinetic energy is converted into other energy forms (such as thermal energy, sound, or work done in deforming the objects).
Key Takeaway for Momentum: Momentum is always conserved in any collision without external forces. To determine if a collision is elastic, calculate total \(E_k\) before and after separately to see if they match.
---5. Work, Energy, and Power
Work Done (\(\Delta W\))
Work is done when a force moves an object through a distance in the direction of the force:
\(\Delta W = F \Delta s\)
When the force acts at an angle \(\theta\) to the direction of displacement:
\(\Delta W = F \Delta s\cos\theta\)
Unit: Joules (\(\text{J}\)), where \(1\text{ J} = 1\text{ N m}\).
Forms of Energy
1. Kinetic Energy (\(E_k\)): Energy possessed by a moving mass:
\(E_k = \frac{1}{2}mv^2\)
2. Gravitational Potential Energy (\(\Delta E_{\text{grav}}\)): Energy stored in an object due to its position in a gravitational field:
\(\Delta E_{\text{grav}} = mg\Delta h\)
3. The Work-Energy Principle:
The net work done on an object by the resultant force equals its change in kinetic energy:
\(\text{Work done by resultant force} = \Delta E_k\)
Power (\(P\))
Power is the rate at which energy is transferred or the rate at which work is done:
\(P = \frac{E}{t} \quad \text{or} \quad P = \frac{W}{t}\)
Unit: Watts (\(\text{W}\)), where \(1\text{ W} = 1\text{ J s}^{-1}\).
When an object is driven by a constant force \(F\) at a constant velocity \(v\):
\(P = Fv\)
Efficiency
Efficiency measures how effectively input energy is converted into useful output:
\(\text{Efficiency} = \frac{\text{useful energy output}}{\text{total energy input}} = \frac{\text{useful power output}}{\text{total power input}}\)
Note: Efficiency can be written as a decimal between \(0\) and \(1\), or multiplied by \(100\) to give a percentage.
Key Takeaway for Energy: If a force acts perpendicular to displacement (\(\theta = 90^\circ\)), \(\cos 90^\circ = 0\), so zero work is done by that force.
---6. Prescribed Practical: Core Practical 1
Determining the Acceleration of Free Fall (\(g\))
This core practical requires you to experimentally determine the value of \(g\) using falling objects.
Method A: Trapdoor and Electromagnet
• A small steel ball bearing is held by an electromagnet.
• When the circuit is switched off, the timer starts automatically and the ball drops from rest (\(u = 0\)).
• The ball strikes a trapdoor at height \(h\) below, opening the contact switch and stopping the electronic timer.
• The drop height \(h\) is varied and the time of fall \(t\) is recorded across repeated trials.
Method B: Light Gates with an Interrupted Card / Falling Ball
• An object of known dimensions drops through two light gates separated by a measured vertical distance \(s\).
• The data logger records the transit times to determine initial velocity \(u\) at gate 1 and final velocity \(v\) at gate 2.
Graph Analysis (for \(u = 0\)):
Using the SUVAT relation \(s = ut + \frac{1}{2}at^2\), since \(u = 0\), we have:
\(h = \frac{1}{2}gt^2\)
• Plot \(h\) on the y-axis against \(t^2\) on the x-axis.
• The line of best fit is a straight line through the origin.
• \(\text{Gradient} = \frac{1}{2}g \implies g = 2 \times \text{gradient}\).
Key Experimental Precautions:
• Use a plumb line or set square to ensure height measurements are strictly vertical.
• Repeat readings of \(t\) for each height \(h\) and calculate a mean to reduce random errors.
• Keep drop heights reasonably large to reduce the percentage uncertainty in measuring \(t\).
7. Common Exam Pitfalls & How to Avoid Them
1. Confusing Newton's 3rd Law Pairs:
Mistake: Saying that the upward normal contact force on a resting box is the 3rd law pair to its downward weight.
Correction: Both forces act on the same box. The true 3rd law pair to Earth pulling the box down is the box pulling the Earth up gravitationally.
2. Sign Errors in SUVAT:
Mistake: Setting upwards as positive with initial velocity \(u = +15\text{ m s}^{-1}\), but accidentally writing \(a = +9.81\text{ m s}^{-2}\).
Correction: Gravity pulls downward. If upward is positive, you must use \(a = -9.81\text{ m s}^{-2}\).
3. Slope Components:
Mistake: Using \(mg\cos\theta\) down the slope.
Correction: Down the slope is always \(mg\sin\theta\); perpendicular to the slope is \(mg\cos\theta\).
4. Inappropriate Significant Figures:
Mistake: Writing down \(8.34567\text{ m s}^{-1}\) when data in the question was given to \(2\) significant figures.
Correction: Match your final answer's significant figures to the least precise piece of given data (usually 2 or 3 s.f.).
Chapter Quick Review
• Kinematics: \(v-t\) gradient is acceleration; \(v-t\) area is displacement; use SUVAT for constant acceleration.
• Newton's Laws: \(\Sigma F = ma\); third law pairs act on different bodies and share force type.
• Moments: \(\text{Moment} = Fx\) where \(x\) is perpendicular distance; equilibrium requires \(\Sigma F = 0\) and \(\Sigma \tau = 0\).
• Momentum: \(p = mv\); momentum is conserved in all isolated collisions; kinetic energy is conserved only in elastic collisions.
• Energy & Power: \(\Delta W = F\Delta s\cos\theta\); \(E_k = \frac{1}{2}mv^2\); \(\Delta E_{\text{grav}} = mg\Delta h\); \(P = \frac{W}{t} = Fv\).
• Core Practical 1: Plot \(h\) against \(t^2\) to find \(g = 2 \times \text{gradient}\).