Forces

Scalar and Vector Quantities

• Definitions:

  • Vector: A quantity that has both magnitude and direction.

  • Scalar: A quantity that has just magnitude.

• Key Characteristics:

  • Generally, scalars cannot be negative.

  • Vectors can be negative, as a certain direction is defined as positive.

• Examples:

  • Speed: Scalar.

  • Velocity: Vector.

  • Distance: Scalar.

  • Displacement: Vector.

  • Time: Scalar.

  • Acceleration: Vector.

  • Force: Vector.

  • Mass: Scalar.

  • Momentum: Vector.

  • Energy: Scalar.

• Displacement Contextual Example:

  • Imagine a ball thrown off a cliff. Displacement is $0$ at the height of the cliff.

  • Above the cliff, the ball has positive displacement.

  • Below the clifftop, the ball has negative displacement.

  • In long answer questions, you can decide where the "0" point of a vector lies. For instance, setting the bottom of the cliff as zero means the ball will never have negative displacement.

• Velocity vs. Speed:

  • Speed is only velocity when given a direction.

  • Example: Thrown at $10\,m/s^{-1}$ is speed; thrown at $10\,m/s^{-1}$ at $30^{\circ}$ above the horizontal is velocity.

• Circular Motion Exception:

  • Imagine a car traveling round a roundabout at constant speed. While speed is constant, its direction is constantly changing. Because direction is changing, its velocity is constantly changing; therefore, the car is accelerating.

• Representation:

  • Vectors are represented by arrows. The size/length of the arrow represents the vector magnitude.

Object Interaction and Forces

• Definition of Force: A push or pull that acts on an object due to interaction with another object.

• Non-Contact Forces: The objects are physically separated.

  • Electrostatic: Charges cause a force of attraction or repulsion.

  • Gravitational Attraction: Mass creates a force of attraction.

• Contact Forces: The objects are physically touching.

  • Normal Contact Force: Felt in the opposite direction to contact; the force is normal (perpendicular) to the planes of contact.

  • Friction: Occurs when surfaces and their roughness cause resistance when moved in contact.

Gravity and Weight

• Gravitational Fields: All matter has a gravitational field and attracts all other matter. The larger the mass, the stronger the field and the greater the attraction.

• Weight Definition: The force exerted on a mass by the gravitational field, measured in Newtons ($N$).

• Formula:

  • $\text{weight} = \text{mass} \times \text{gravitational field strength}$

  • $W = mg$

  • $W = m \times 10$

• Units and Measurement:

  • Weight ($W$) is in Newtons ($N$).

  • Mass ($m$) is in kilograms ($kg$).

  • Measured by a force meter (also known as a calibrated spring-balance).

  • A weighing scale measures the force exerted and divides by $10$ to provide mass.

• Gravitational Field Strength ($g$):

  • On Earth, you must recall that $g = 9.8\,N/kg$.

  • Acceleration in free fall is due to gravity and is the same as $g$, approximated as $10\,m/s^{-2}$.

• Mass vs. Weight Example:

  • A person on two different planets will have the same mass.

  • The gravitational field strength ($g$) will be different at the two planets (i.e., not necessarily $10$ for both).

  • Consequently, their weight will be different on both planets.

• Centre of Mass: The weight of an object is considered to act at the object’s centre of mass.

Resultant Forces

• Definition: A single force representing the sum of all forces acting on an object.

• Calculation:

  • Along a straight line, the resultant is found by adding forces acting in the same direction or subtracting forces acting in opposite directions.

• Skydiver Example:

  • Forces involved: Air resistance and weight.

  • Stage A: Initially, the skydiver has no air resistance. Only weight acts on him. He accelerates, increasing speed. Resultant is $833\,N$ down.

  • Stage B: As air resistance increases, the resultant force from weight decreases. Example: $833 - 350 = 483\,N$ down.

  • Stage C: Acceleration decreases; he is not speeding up as quickly. Example: Resultant is $133\,N$ down.

  • Stage D (Terminal Velocity): Eventually, air resistance and weight are equal and balance. The resultant force is $0$. There is no acceleration, and the skydiver travels at terminal velocity.

• Free Body Diagrams: Diagrams showing the forces and their directions acting on an individual object.

Resolving Forces

• A force ($F$) at an angle ($\theta$) to the ground can be resolved into components parallel and perpendicular to the ground.

• Using Pythagoras’ Rule: $a^2 + b^2 = c^2$.

• $F^2 = (F_{parallel})^2 + (F_{perpendicular})^2$

• Components are often expressed as $F \cos(\theta)$ and $F \sin(\theta)$.

Work Done and Energy Transfer

• Definition: Work is done when energy is transferred from the object doing the work to another form or object.

• Formula:

  • $\text{Work Done} = \text{Force} \times \text{Distance}$

  • $W = Fs$

  • Work Done ($W$) is in Joules ($J$).

  • Force ($F$) is in Newtons ($N$).

  • Distance ($s$) is in metres ($m$) moved along the line of action of the force.

• Unit Equivalency: One joule of work is done when a force of one newton causes a displacement of one metre ($1\,joule = 1\,newton\text{-}metre$).

• Practical Example:

  • If a book is lifted $1\,m$ in the air and moved $2\,m$ to the right, work is only done (against gravity) when moving $1\,m$ vertically, as that is the direction of the gravitational force.

  • Energy transfers from muscles to the book, increasing its gravitational potential energy.

• Friction: Work done against frictional forces causes a rise in the temperature of the object.

Forces and Elasticity (Springs)

• General Rule: To stretch, bend, or compress an object, more than one force must be applied (otherwise, the object would simply move in the direction of the single force).

  • Stretching occurs if pulled in opposite directions.

  • Stretching a fixed object involves a force applied by the fixed point.

• Deformation Types:

  • Elastic Deformation: The object returns to its original shape once the load is removed (e.g., an elastic band).

  • Plastic Deformation: The object does not return to its original shape (e.g., a spring pulled too far).

• Hooke’s Law:

  • The extension of an elastic object is directly proportional to the force applied, provided the limit of proportionality is not exceeded.

  • $F = kx$

  • $F$ is force ($N$).

  • $k$ is spring constant ($Nm^{-1}$).

  • $x$ is extension ($m$).

• Force-Extension Graphs:

  • Linear Region: Follows Hooke's Law; the gradient is the spring constant ($k$). This is the elastic region.

  • Limit of Proportionality: The point where the graph stops being linear.

  • Non-Linear Region: Plastic behavior begins; Hooke's Law is no longer obeyed.

  • Shallow Gradient: Indicates lots of extension for small force (easy to stretch).

  • Brittle Materials: If the graph is linear with no non-linear section, the material snaps instead of stretching plastically after the elastic limit.

• Work Done on a Spring:

  • $\text{Work Done} = \frac{1}{2} k x^2$

  • When a force stretches/compresses a spring, elastic potential energy is stored.

  • Provided no inelastic deformation occurs: Work done on the spring = Elastic potential energy stored.

Moments and Rotation (Physics Only)

• Definitions:

  • Pivot Point: A point an object rotates about but cannot move away from.

  • If a force is applied along a line passing through the pivot, the object remains still.

  • If there is a distance between the pivot and the line of action of the force, the object rotates.

• Moment Calculation:

  • $\text{Moment of a Force} = \text{force} \times \text{perpendicular distance}$

  • $M = Fd$

  • Moment ($M$) is in Newton-metres ($Nm$).

  • Force ($F$) is in Newtons ($N$).

  • Distance ($d$) is the perpendicular distance from the pivot to the line of action ($m$).

• Examples and Equilibrium:

  • Bike Pedal: Pressing a foot down causes a moment about the pivot, turning the pedal arms.

  • Equilibrium: Occurs when $\text{sum of anticlockwise moments} = \text{sum of clockwise moments}$.

Levers and Gears (Physics Only)

• Gears: Can change speed, force, or direction via rotation.

• Gear Connections (Force from first gear):

  • Connected to a smaller gear (fewer teeth): Second gear turns faster, with less force, in the opposite direction.

  • Connected to a larger gear (more teeth): Second gear turns slower, with more force, in the opposite direction.

• Power Transmission: To increase power, a larger secondary gear is used. Because the force on the secondary gear is at a further distance from its pivot, the momentum/turning effect is greater.

Pressure (Physics Only)

• General Pressure:

  • Particles in a gas move randomly and exert forces on containers.

  • $\text{pressure} = \frac{\text{force}}{\text{area}}$

  • $p = \frac{F}{A}$

  • Pressure ($p$) in Pascals ($Pa$).

  • Force ($F$) in Newtons ($N$).

  • Area ($A$) in metres squared ($m^2$).

  • Pressure produces a net force at right angles to any surface.

• Pressure in a Liquid:

  • Varies with depth and density.

  • $\text{pressure} = \text{height of column} \times \text{density} \times \text{gravitational field strength}$

  • $p = h \rho g$

  • Height ($h$) in metres ($m$).

  • Density ($\rho$) in $kg/m^3$.

  • Gravitational field strength ($g$) usually taken as $10\,N/kg$.

  • Higher depth means greater weight of water above, resulting in greater force and pressure.

• Buoyancy and Upthrust:

  • Upthrust: A submerged object experiences greater pressure on the bottom surface than the top, creating a resultant upward force.

  • Floating Conditions: An object floats if its weight is less than or equal to the weight of the water it displaces.

  • Example: A $1000\,kg$ boat floats if it displaces $1000\,kg$ of water before completely submerging.

  • Ping Pong Ball: Floats because its density is less than water. The volume it displaces weighs more than the ball itself, creating an upward resultant buoyancy force.

Earth's Atmosphere (Physics Only)

• Atmosphere: A thin layer of air around the Earth that gets less dense with increasing altitude.

• Atmospheric Pressure: Caused by the weight of air above a unit area.

  • Higher elevation means fewer air molecules above, resulting in lower weight and lower pressure.

• Idealized Assumptions for Models:

  • Isothermal (constant temperature throughout).

  • Transparent to solar radiation.

  • Opaque to terrestrial radiation.

Describing Motion

• Terms:

  • Distance: Scalar; how far an object moves without direction.

  • Displacement: Vector; distance measured in a straight line from start to finish including direction.

  • Speed: Scalar; no direction.

  • Velocity: Vector; speed in a given direction.

• Typical Speeds:

  • Wind: $5 - 7\,m/s^{-1}$

  • Sound: $330\,m/s^{-1}$

  • Walking: $\sim 1.5\,m/s^{-1}$

  • Running: $\sim 3\,m/s^{-1}$

  • Cycling: $\sim 6\,m/s^{-1}$

  • Bus: $14\,km/h$

  • Train: $125\,miles/h$

  • Plane: $900\,km/h$

• Formulas:

  • $\text{speed} = \frac{\text{distance}}{\text{time}}$

  • $v = \frac{d}{t}$

  • Average Speed: $\frac{\text{Total Distance}}{\text{Total Time}}$.

Motion Graphs

• Displacement-Time Graphs:

  • Gradient: Velocity.

  • Steep Gradient: Faster speed.

  • Negative Gradient: Returning toward starting point.

  • Horizontal Line: Stationary.

  • Zero Distance: Object is back at the starting point.

  • Curved Line: Changing velocity (acceleration).

  • Calculating Speed from Curves: Draw a tangent and calculate its gradient.

• Velocity-Time Graphs:

  • Gradient: Acceleration.

  • Steep Gradient: Greater acceleration.

  • Negative Gradient: Deceleration.

  • Horizontal Line: Constant speed.

  • Zero Velocity: Stationary.

  • Area Under Line: Distance travelled (counting squares is used for curves).

  • Curved Line: Changing acceleration.

Falling in a Fluid (Physics Only)

• An object initially falls freely under gravity ($9.8\,m/s^2$).

• As speed increases, drag forces increase.

• Acceleration decreases until weight equals drag.

• The graph levels off at terminal velocity (approx. $40\,m/s^{-1}$ in some scenarios).

Newton's Laws of Motion

• First Law: An object has a constant velocity unless acted on by a resultant force.

  • If resultant force is $0$: Stationary objects stay stationary; moving objects continue at the same velocity.

  • Inertia: The tendency for objects to stay at rest or continue in uniform motion.

• Second Law: Acceleration is proportional to the resultant force and inversely proportional to mass.

  • $F = ma$

  • $F$ in Newtons ($N$), $m$ in kilograms ($kg$), $a$ in $m/s^2$.

  • Inertial Mass: Measure of how difficult it is to change velocity: $\text{inertial mass} = \frac{f}{a}$.

• Third Law: Whenever two objects interact, the forces they exert on each other are equal and opposite.

  • Rocket Example: Rocket pushes gas down; gas pushes rocket up with equal force.

  • Book Example: Weight of book on table = pull of book on Earth.

Vehicle Stopping Distances

• Formula: $\text{stopping distance} = \text{thinking distance} + \text{braking distance}$

• Thinking Distance: Distance travelled during reaction time.

  • Factors: Speed, concentration, tiredness, distractions, drugs/alcohol.

• Braking Distance: Distance travelled after brakes are applied.

  • Factors: Speed, road conditions (icy/wet), bald tires, worn brakes, weight (passengers).

• Reaction Times: Typical range is $0.2 - 0.9\,s$.

  • Ruler Drop Test: $t = \sqrt{\frac{2s}{g}}$, where $s$ is distance the ruler travels through the hand.

• Energy and Brakes:

  • Brakes do work against the wheels via friction.

  • Kinetic energy (KE) reduces, and brake temperature increases.

  • Higher speeds require greater braking force, which risk overheating and loss of control.

Momentum

• Formula:

  • $\text{momentum} = \text{mass} \times \text{velocity}$

  • $p = mv$

  • Momentum ($p$) in $kg\,m/s^{-1}$.

• Conservation of Momentum: Total momentum before a collision/explosion equals total momentum after (in a closed system).

• Changes in Momentum (Physics Only):

  • Force is the rate of change of momentum.

  • $F = \frac{mv - mu}{t}$

• Safety Features (Physics Only):

  • A large deceleration causes a large force on passengers ($F = \Delta p / t$).

  • Seatbelts: Stretch slightly to increase the time taken to stop, reducing force.

  • Crumple Zones: Areas at the front/back that deform to absorb energy and increase the time taken for the car to stop, reducing force.

  • Airbags: Inflate instantaneously; the head hits the bag and slows down over a longer time, reducing forces on the neck/head.