Vectors and Scalars

Vectors and Scalars

• Vectors possess both magnitude and direction
• Scalars possess only magnitude
• Scalars are typically non-negative, while vectors can be positive or negative depending on direction

Examples of Vectors and Scalars

• Speed, distance, time, and mass are scalar quantities
• Velocity, displacement, acceleration, momentum, and force are vector quantities

Displacement Note

• Displacement can be zero at a reference point like the height of a cliff
• Displacement above the cliff is positive, and below is negative

Speed vs Velocity

• Speed becomes velocity when direction is specified
• For example, 10 m/s is speed, but 10 m/s at 30° is velocity

Acceleration in Circular Motion

• An object moving at a constant speed around a roundabout is accelerating
• This occurs because its direction, and thus velocity, is constantly changing

Vector Representation

• Vectors are represented by arrows
• The size or length of the arrow indicates the vector's magnitude

Forces

• Forces are either pushes or pulls resulting from object interactions

Non-Contact Forces

• Objects are physically separated
• Electrostatic forces are attractive or repulsive forces caused by charges
• Gravitational forces are attractive forces caused by mass

Contact Forces

• Objects are physically touching
• A normal contact force is felt in the opposite direction to contact and is normal to the plane of contact
• Friction is due to surface roughness when objects move in contact

Gravity

• All matter attracts other matter through a gravitational field
• The greater the mass, the stronger the gravitational field and attraction

Weight

• Weight is the force exerted on a mass by a gravitational field
• Weight = mass × gravitational field strength, or W = mg = m × 10
• Weight is measured in newtons (N), and mass is measured in kilograms (kg)

Measurement of Weight

• A force meter or calibrated spring balance measures weight
• A weighing scale measures the force exerted, then divides by 10 to give mass
• Remember that the earth's gravitational field strength (g) is 9.8

Mass vs Weight

• The mass of the object is the same on any planet
• Gravitational field strength varies by planet, hence weight changes

Acceleration due to Gravity

• Acceleration in free fall is due to gravity and equals 10 m/s-2

Weight Location

• An object's weight is considered to act at its center of mass

Resultant Force

• This is the single force representing the sum of all forces acting on an object
• It's found by adding forces in the same direction or subtracting forces in opposite directions

Skydiver Forces

• Air resistance and weight are the forces
• Initially, a skydiver experiences only weight
• As the skydiver falls, they accelerate and increase speed

Changes in Forces

• Air resistance increases, decreasing the resultant force
• This causes acceleration to decrease
• Eventually, air resistance equals the resultant force

Resolving Forces

• A force can be resolved into parallel and perpendicular components relative to the ground
• The components can be calculated the formula provided

Work

• Work Done = Force × Distance and W = Fs
• Work Done (W) is in joules (J), Force (F) is in newtons (N), and distance (s) is in meters (m)
• Distance must be measured along the line of action of the force
• Work is done when energy transfers from one object to another

Examples of Work

• Lifting a book transfers energy
• Work is done against gravity when moving objects vertically
• The energy is transferred from muscles, increasing the object's gravitational potential energy
• One joule equals the work from a one-newton force causing a one-meter displacement

Rise of Temperature

• Work done against frictional forces causes a rise in temperature

Springs

• Multiple forces must be applied to stretch, bend, or compress
• A single force applied to an object will just cause movement

Stretching

• An object stretches if pulled in opposite directions or if fixed and stretched

Deformation

• Deformation is a change in shape

Elastic Deformation

• The object returns to its original shape after removing the load, such as with an elastic band

Plastic Deformation

• The object does not return to its original shape after removing the load
• A spring pulled too far is an example

Hooke's Law

• The extension of an elastic object, like a spring, is directly proportional to the force applied, provided the limit of proportionality is not exceeded
• F = kx, where F is force in N, k is the spring constant in N/m, and x is the extension in m

Force/Extension Graph

• A linear line indicates the elastic region and follows Hooke's Law
• The gradient of this line is k
• The limit of proportionality is where the line stops being linear and Hooke's Law no longer applies

Elastic vs Plastic Behaviour

• A non Linear line indicates plastic behavior
• A shallow non Linear line means there's a lot of extension without much force, making it easy to stretch

Brittle Materials

• A brittle material is a graph is just linear, with no non-linear end section
• The material snaps instead of stretches after the elastic limit

Work done using springs

• When a force stretches/compresses a spring, the spring does work
• Elastic potential energy is stored in the spring
• Provided it does not inelastically deform
• The work done on the spring = the elastic potential energy stored
• Work Done = (1/2)kx^2

Moments and Rotation

• An object attached to a pivot point rotates if a force is applied away from the pivot point

Moment of a Force Calculation

• If the force is not perpendicular, consider the perpendicular distance from pivot to applied force line
• Moment of a Force = force × perpendicular distance or M = Fd
• M is in Newton-meters (Nm), F is force in newtons (N), and d is distance in meters (m)

Example of Moments

• Bike riding is an example because Foot pressure on the pedal causes rotation

Equilibrium

• Equilibrium is when the sum of anticlockwise moments equals the sum of clockwise moments

Levers and Gears

• Gears change speed, force, or rotation
• A gear connected to one with fewer teeth turns faster but with less force, and in the opposite direction
• A gear connected to one with more teeth turns slower, with more force, and in the opposite direction

Gear Orientation

• The second gear always turns in the opposite direction
• A larger secondary gear increases power

Pressure

• Gas particles randomly move and exert force on container walls
• Pressure, p = force / area which is p=F/A
• p is in pascals (Pa), F is in newtons (N), and A is in meters squared (m^2)
• Pressure produces net force at right angles

Floating

• An object floats if its weight is less than the weight of its displacement
• A 1000kg boat sinks until it displaces 1000kg of water, provided it doesn't submerge completely

Buoyancy Force

• Pressure in a liquid increases with depth and density
• The buoyancy force counteracts the floating object's weight
• The weight is equal to the weight of the liquid displaced by object

Density Example

• Being of less density, a ping pong ball when floating causes the equivalent amount of water is greater than the weight of the ping pong ball and therefore has resultant force of bouncy

Pressure Calculation

• Pressure due to a column of liquid = height of column × density of liquid × g, or p = hρg
• p is in pascals Pa, h is column height in m, ρ is density in kg/m³, and g is gravitational field strength, normally 10 N/kg

Upthrust Calculation

• Upthrust occurs when submerged objects experience greater pressure on the bottom
• The Earth's atmosphere surrounds it with decreasing density at greater altitude

Atmosphere Considerations

• The total weight of air above a unit area causes pressure
• Less air exists at higher elevations, resulting in less pressure

Atmosphere Characteristics

• Simplified atmospheric models assume isothermal conditions, transparency to solar radiation, and opacity to terrestrial radiation

Distance

• Distance is how far something moves without regard to direction and is a scalar quantity

Displacement

• Displacement has magnitude and direction from start to finish and is a vector quantity

Velocity

• Velocity is speed in a given direction, therefore, velocity is a vector
• Circular motion means constantly changing direction, therefore constantly changing velocity

Acceleration and Speed

• Because velocity is changing, the object is accelerating, even if its speed is constant

Typical Speeds

• Wind speed is 5-7m/s
• Sound speed is 330m/s
• Walking speed is ~1.5m/s
• Running speed is ~3m/s
• Cycling speed is ~6m/s
• Bus speed is 14km/h
• Train speed is 125miles/h
• Plane speed is 900km/h

Units

• Use appropriate units, and convert to ensure everything is equivalent

Speed Equation

• Speed = distance / time, or v = d/t

Average Speeds

• To calculate Average speed over non-uniform motion, first calculate total time, then get total distance
• The total time = distance / speed

Displacement Graph

• Gradient equals velocity
• A sharper gradient means faster speed
• A negative gradient is returning to start
• A horizontal line means stationary
• 0 Distance means that it is back to starting point
• The curved line means acceleration

Velocity Graph

• Gradient equals acceleration
• A sharper gradient means greater acceleration
• A negative gradient is deceleration
• A horizontal line means constant speed
• 0 Velocity means that it is stationary
• Area under the graph equals the distance travelled
• The curved line means changing acceleration

Falling Through Fluid

• Initially, the object is falling freely due to gravity which is roughly 9.8m/s^2
• Drag forces will act to cause the object to move at terminal velocity
• The graph of this, is a steep gradient which reduces to a flat gradient

Equations

• Average Speed = Total Distance / Total Time
• a = (v-u)/t
• v^2 = u^2 + 2as
• Kinetic Energy = (1/2)mv^2

Newton's First Law

• An object maintains constant velocity unless acted upon by a resultant force
• The velocity can change in either direction or speed
• If there is no resultant force then a stationary object will remain still and the object in motion will continue moving with same velocity

Inertia

• It is the objects tendency to stay at uniform velocity

Newton's Second Law

• The acceleration of an object is proportional to the resultant force acting on the object and inversely proportional to its mass
• Force equals mass x acceleration or F = ma
  • The F is in newtons (N)
  • The m is mass represented in kilograms (kg)
  • The a is the acceleration and m/s^2

Inertial Mass

• This measures the the change in velocity of an object
• It is given by the formula inertial mass = force / acceleration

Newtons Third Law

• When two or more objects interact, they exert forces on each other
• These forces are both Equal and Opposite
• In Rocket Launches, the thrusts of the rockets equals the force of the gases exiting the rocket
• A book on a table means the weight of the object means the force of the table is equal

Stopping Distance

• The thinking distance before reacting equals "X" metres
• The braking distance while braking equals "Y" metres

Distance Equation

• stopping distance = thinking + braking distances

Distance and Speed

• Both are equally proportional for thinking and breaking Distance
• Thinking Distance factors are Speed, concentration and tiredness
• Influence of drugs/alcohol, concentration and tiredness are also factors
• Breaking Distance factors are road conditions, tires and breaks
  • Weight is a key concern with the number of passengers

Time and Distance

Reaction Time

• Average Reaction Times vary from.2 to .9 seconds

Ruler Drop

• Ruler measures reaction times by dropping a ruler through the open hand
• Catching involves time or a force
• The time is defined as s = ut + 1/2 at^2 or more simply s= 1/2 at^2

Braking Considerations

• Braking causes work to happen
  • Brakes are pressed with force, energy causes heat,
  • Braking force needs momentum
• Increasing speed requires increasing braking force
• Greater force causes a greater acceleration
  • This may cause overheated brakes

Equation for Momentum Equation

• momentum = mass * velocity and p * mv

Momentum Features

• Momentum is always conserved in either collision or explosion
• Must consider that Momentum is always conserved during any collision and the total momentum

Changes in Momentum

Newton’s Second Law

• Defined as Force = Mass/Time but simplified as F = (mv-mu) /t

Key Concepts

• Hard braking is a deceleration
• The large force can be dangerous
  • The large force can cause injury

Seatbelts

• In the event of hard braking you will not keep moving, otherwise you'd fly
• You can strap in with stretching under large forces
• Increased travelling distance and time
• The reduced reduces the changing rate reducing force

Crumple Zones

• Cars need deformation for minor crash protection
• A softer car crashes and compacts using kinetic and potential energy
• The increases timer reduces impacts

Airbags

• Airbags can inflate in order to decrease inertia and whiplash from accidents
• Increased time reduces the neck's force and momentum from collisions

Summary

• Forces are vector quantities that cause changes in motion, while scalars only have magnitude. Gravity is a fundamental force affecting weight, calculated using mass and gravitational field strength. Newton's laws describe motion and interactions, impacting vehicle safety through stopping distances and safety features reducing forces in collisions