# SPH4U Grade 12 Physics Exam

Grade 12 Physics – Exam

Exam

Unit 1 – Kinematics and Dynamics

Displacement: a vector form of a distance

Velocity: a change in displacement over time

• Slope of secant from a displacement – time graph

Vectors: a unit with magnitude and direction

• Resultant: desired displacement of vectors
• Head to Tail: when one vector connects to another
• Head to Head: when vectors start at the same point
• Vector Components
• Break vectors into their X Y components.
• Add individual X Y components, then find resultant

Gravitational Acceleration: Earth accelerates objects towards center at 9.8 m/s2

Field of Reference: the speed of an object in motion is dictated in relation from where you’re viewing it

Calculating with field of Reference

• General Formula: VAX = VAY + VYX
• Where Y is common in both Vectors
• Y is eliminated and leaves for AX, the final answer
• Subtracting Vectors, Example: VAX = VAY – VXY
• Add the subtracted vector by reversing the subscripts
• VAX = VAY + VYX

Net Force: causes objects to accelerate / decelerate

Inclined Plane: consider the surface parallel to the plane as x, and plane perpendicular to the plane as y.

Newton’s Laws of Motion

• First Law: an object in motion stays in motion unless there is a force that slows it down
• Inertia: a fundamental property of matter that makes things stay at constant speed
• Second Law: force can be affected by mass and acceleration
• F = m*a
• Static Equilibrium: net force is zero and does not move
• Dynamic Equilibrium: net force is zero, but it’s at constant speed
• Third Law: for every action, there is a reaction, equal in magnitude but in opposite direction.t

Circular Motion

• Circular Motion: occurs when an object is travelling in a circular path with fixed radius and speed
• Since direction is changing at every moment in a circle, velocity changes, and the object will accelerate towards the center

ac=v2/r

• r = radius
• v = speed of object

ac = v2/r = 4(pi)2r/T2 = 4(pi)2rf2

• f = frequency
• T = period
• Centripetal force: the force created by circular motion towards the center.
• Centripetal force = mass * centripetal acceleration

Inertial Frames of Reference: when the frame of reference is moving at constant velocity, or at rest where Newton’s Laws are obeyed

Non-Inertial Frames of Reference: when the frame of reference is accelerating where Newton’s Laws aren’t obeyed in this case.

Centrifugal Force: another form of fictitious force which is created due to the existence of some other force

• If centripetal force accelerates towards the center in circular motion, the centrifugal force will act against the object and whatever inside.

Unit 2 – Energy and Work

Work: the energy transferred to an object when a force acting on the object moves it across a distance.

W = (F cos θ) Δd

• If the force is causing an object to undergo a displacement is at an angle to the displacement, only the component of the force in the direction of the displacement does work on the object.
• Joule: (J) is a unit used to measure energy. 1 Joule = 1 N/m.
• Sometimes, zero work is done on an object even if the object experiences an applied force or in motion.

• Kinetic Energy: Ek is the energy of kinetic motion, a scalar quantity measured in (J)

Ek = 0.5mv2

• Work-Energy Theorem: The total work done on an object equals the change in the object’s kinetic energy, provided there is no change in other forms of energy.

Wtotal = Ekf – Eki

Wtotal = ΔEk

Gravitational Potential Energy: the energy due to the elevation above earth’s surface

Eg = mgh or Eg = mg Δy

Law of conservational energy: energy can be converted into different forms, but cannot be created, made fun of, or destroyed.

Thermal Energy: internal energy associated with the motion of atoms and molecules

Eth = Fk * d

Mechanical energy: the total energy in an isolated system.

Elastic Potential Energy

Hooke’s Law: the force exerted by a spring is directly proportional to the distance the spring has moved from equilibrium

Fx = -k * x

• k is the force constant the spring creates
• If k is – , then the equation represents the force exerted by the spring
• If k is +, then the equation represents the force exerted to the spring
• Ideal Spring: a spring that obeys Hooke’s Law because it experiences no internal or external friction
• Elastic Potential Energy (Ee): energy stored in an object with a changing volume ie compressed, stretched, bent, or twisted.

Ee = 1/2 kx2

Simple Harmonic Motion: (SHM) periodic vibratory motion in which the force and acceleration is proportional to the displacement.

• Friction is negligible in SHM, vibration goes on indefinitely.

T = 2 pi √(m/k)    Period

f = 1/2pi √(k/m)      Frequency

• Energy in simple harmonic motion shows that when energy is released from a spring, it transforms into kinetic energy.

Et = 1/2 kx2 + 1/2 mv2

• k is the force constant
• x is the displacement of mass from equilibrium position
• v is the instantaneous velocity of the mass
• Damped Harmonic Motion: periodic motion which amplitude of vibration and the energy decreases over time due to friction.

Momentum: the product of the mass of an object moving and is velocity is a vector quantity. Unit is kg*m/s

p = m*v

Impulse: the change in momentum. Vector quanity in N*s.

I = ∑all forces * time

• In a force vs time graph, the impulse is the area under the function.

Conservation of 2D momentum

• If the net force acting on a system of interacting objects is zero, then the linear momentum of the system before the interaction equals the linear momentum of the system after the interaction.

Δp1 = Δp1

m1Δv1 = m2Δv2

Collisions

Elastic Collisions: a collision in which the total kinetic energy after the collision equals the total kinetic energy before the collision

• Ek = Ek’
• p = p’

Inelastic Collision: a collision in which the total kinetic energy after a collision is different from the total kinetic energy before the collision. But momentum remains the same before and after.

• p = p’

Completely Inelastic Collision: a collision where there is a maximum decrease in kinetic energy after the collision since the objects stick together and move at the same velocity.

mAΔvA + mBΔvB = (mA + mB) vB

• In some 2 D collisions, it would be more efficient if the vectors were broken into vector components before solving.

Unit 3 – Electric, Gravitational, and Magnetic Fields

Laws of Electric Charges Opposite charges attract each other. Similar charges repel each other. Charged objects attract some neutral objects.

• Charging by Friction Electrons are ripped off another surface and charges another object
• Ability to rip electrons are based on their position on the electrostatic series
• Induced Charge Separation distribution of charge that results from a change in the distribution of elections in an object
• Charging by Contact Electrons are passed through conductors once they touch. They are transferred and charge equalize each object.
• Charging by Induction The electrons in one object are pushed by the fields of a nearby charged object inducing it.

Law of Conservation of change: The total charge (the difference between the amounts of positive and negative charge) within an isolated system is conserved.

Electric Forces

Coulomb’s Law: The force between two point charges is inversely proportional to the square of the distance between the charges and directly proportional to the product of the charges

FE = kq1q2 / r2

Where k = 9.0 x 109 N*m2/C2

Electric Field: any point is defined as the electric force per unit positive choice and is a vector quantity. Unit is Coulombs

• Electric Field lines always come from positive to negative charges
• Positive fields never touch negative fields, they also never cross

ε = kq1 / r2

Electric Potential: (V) the value, in volts, of potential energy per unit of positive charge. 1 V = 1 J/C

V = kq1 / r

Electric Potential Energy (EE) : the energy stored in a system of two charges a distance r apart.

EE = kq1q2 / r

Electric Potential Difference: the amount of work required per unit charge to move a positive charge from one point to another in the presence of another charge.

ΔV = εr

ε = ΔV / r  (for parallel plates)

ΔV = ΔEE / q

Elementary Charge: (e) is the smallest unit of electric charges. e = 1.602 x 10-19

Motion of Charged particles in Electric Fields: Newton’s laws combined with laws of electric charges, we can derive acceleration and include mass to solve problems.

a = FE / m

Magnetic Force Field: the area around a magnet which magnetic forces are exerted

Domain Theory: states that magnets are made up of tiny regions (“domains”) and how a material can become magnetized: each domain acts like a bar magnet.

Principle of Electromagnetism: moving electric charges produce a magnetic field.

Right Hand Rule for a straight conductor: if a conductor is grasped in the right hand, with the thumb pointing in the direction of the current, the curled fingers point in the direction of the magnetic field lines.

• Current flowing through a conductor produces a magnetic field that circles the conductor based on the direction of the current.
• Right hand used for positive charges, left hand for negative charges

Right hand rule for a solenoid: if a solenoid is grasped in the right hand, with the fingers curled in the direction of the electric current, the thumb points in the direction of the magnetic field lines in its core.

• A solenoid flowing with current creates a magnetic field that points out of one end of the solenoid

Relative Magnetic Permeability: the ability for some material to become magnetized

Force of Magnetic Fields: the force from a magnetic field on a charge moving nearby in the field

FM = qvB sinθ

Right hand rule for the direction of magnetic force: Hand flat palm up, thumbs at a 90 degree angle to the fingers, where fingers pointed in the direction of the magnetic fields, thumb pointed in the direction of the speed of the charge, and palm points outwards to the direction of the magnetic force.

• Forces act in perpendicular to the magnetic field lines
• When 2 magnetic plates with poles placed in parallel are there, a charge traveling through will enter circular motion between the plates due to this force.

FM = FC , evB = mv2 / r  (since sin 90 degrees = 1)

Right hand rule for the motor principle: if the right thumb points in the direction of the current (flow of positive charge), and the fingers point in the direction of the magnetic field, the force is in the direction in which the right palm pushes.

• When current is on the conductor, within a magnetic field, it has the ability to move due to the interference with the two magnetic fields.

F = I l B sin θ where I is the length, and l is the current

Ampere’s Law: the sum of the products of the components of the magnetic field (B), parallel to the length of the segment, is directly proportional to the net electric current passing through the area.

B = μo ( I / 2π r)

where μo is the permeability of free space = 4π X 10-7 T*m/A. I is the current, and r is the radius away from path.

Lenz’ Law: when a current is created in a coil by changing a magnetic field, the electric current in such a direction that it’s own magnetic field opposes the change that it produces.

• When applying Right/Left hand rules, the force is opposed and it’s opposite from the field applied.

Unit 4: Waves and Light

Basic Wave Knowledge

• Amplitude: the height of a wave from the equilibrium to its crest or trough
• Wavelength: the length of one wave: related to the speed, denoted as λ (Lambda)
• Frequency: the number of times a wave occurs in a second (Hz)
• Period: amount of time it takes to complete a wave cycle
• Reflection: when a wave bounces off a surface, the angle of reflection is equal to the angle of incidence.
• A crest reflects off a slower medium becomes a trough
• Crests do not change if reflecting off a faster medium
• Refraction: when light passes through a new medium, it’s direction, wavelength, and speed changes. Frequency does not change between mediums.
• Wave Front: the leading edge of the wave
• Absolute Refractive Index: the index of refraction for light passing from air or a vacuum into a substance. (n1, n2)
• Index of refraction: n = n2 / n1
• How many times slower the wave travels in a medium

n1 / n2 = v1 / v2 = λ1 / λ2 = sinθ1 / sinθ2

• All periodic waves obey the universal wave equation:

v = fλ

• Partial Reflection: When some of the light is reflected and some passes through and is refracted
• Snell’s Law: angle of incidence over angle of refraction equals the index of refraction.

n = sinθi / sinθr

Diffraction of Water Waves

• Diffraction: Straight waves that pass through an opening will become a new source of its own
• Waves of longer wavelength has more diffraction than shorter wavelengths
• For waves observable: λ / w >= 1  : where w is the width of the opening

Interference of 2D waves

• 2 waves coming from 2 sources radiating out can create interferences to each other
• Waves must be the same frequency and wavelength
• They must be in-phase (beginning at the same time)
• Lines of constructive Interference are called Maxima Lines
• Lines of deconstructive Interference are called Nodal Lines or Minima Lines
• Increasing the frequency, lowering the wavelength increases the number of nodal lines
• Path Length Difference equation for 2D wave interferences:

| PnS1 – PnS2 | = (n – 1/2)λ

• Finding Angle of interference nodal lines:

sinθn = (n – 1/2)λ/d

• Where n is the number of nodal line and d is the distance between the sources
• Equation for waves that span a farther, longer distance

Xn / L = (n – 1/2) λ/d

• Where Xn is the perpendicular distance from the right bisector to Point Pn
• Where L is the distance from the midpoint between the sources to Point Pn

Light as a Particle/ Light as a Wave

• Newton’s Particle theory of light explained 4 properties of light:
• Rectilinear Propagation: great speed of light allowed light particles to travel at near straight lines for long distances: similarly to a bullet.
• Reflection: If vector components are used to break apart the velocity of lights, it can be explained how the angle of incidence = angle or reflection. Vx and Vy are reversed due to the reactive force of the horizontal surface.
• Refraction: Speed of the light, just as if it’s a ball, will swerve in the direction it originally was before it regains and aligns again as it moves through faster medium (or falling down a ramp at an angle)
• Dispersion: Different mass for each colour means some colours would have less momentum and would be diverted more easily, hence, white light spreading out into colours as we know it.
• However, it did not explain diffraction and partial reflection/refraction
• Huygens’ Wave theory of light assumed every point of the wave front was it’s own source of tiny wavelets, radiating at the same speed and tangent to the wave.
• Huygens explained the following light properties:
• Reflection: Waves obey the laws of optics and would reflect accordingly
• Refraction: Wavelengths of the waves are changed as they are slowed down through a different medium and will bend accordingly.
• Partial Reflection/Refraction: Combining reflective and refractive properties of waves, it is possible to explain partial reflection/refraction
• Diffraction: Lights showed interference through a double slit experiment also, proving they travelled in waves.
• Rectilinear Propagation: Huygens thought the light rays represented the direction of the motion of the wave front

Young’s Double Slit Experiment

• When wave interferences needed to be tested, 2 light sources would be out of phase and hard to sync
• Young thought of using 1 source, and instead use 2 slits to separate the source
• And as expected, nodal lines (dark fringes) and maxima lines were visible

sinθm = mλ / d

• Where m is for the maxima lines (1, 2, 3..) and d is the distance between sources
• sinθn = (n – 1/2)λ/d

• Where n is for nodal lines, and d is the distance between sources
• sinθn = Xn / L = (n – 1/2) λ/d

• All three parts are equal and can be used together, where L is the distance from midpoint to Point Pn on the nodal line
• ΔX / L = λ / d

• Where ΔX is the distance between nodal lines
• Colour is dictated by the wavelength of light it produces. Each colour has its own interval of wavelengths.

Polarization of Light

• Light, being a transverse wave, will only travel through filters that are slitted in its direction.
• Light traveling through a polarizer will keep it in one direction
• Polaroids have small slits that only allow light to travel in one direction through it
• Scattering of light: light changes direction when it hits particles in the air
• Photo elasticity: materials that make patterns when they are bent or under stress, As light traveling through it are polarized as the molecules bend, patterns are seen.
• Monochromatic: single colour wavelength
• Polarization can be used to reduce glare as light reflected off a surface can become polarized

Diffraction of light through a single slit

• Based on Huygens’ theory that light is a wavefront with tiny wavelets, traveling in tangent and at the same speed as the wave, Interference can occur if the wave front is traveling at an angle through a slit
• Pairs of waves can interfere with each other, creating dark and bright fringes, radiating from the centre and losing energy as it radiates outwards.
• The smaller the slit, the larger the distance between Maxima and Minima, and vice versa
• For minima, dark fringes (!! Different formula from before!!)

sinθn = nλ / w

• Where n is the number of nodal lines, w is the width of the slit
• For maxima, bright fringes (!! Different from before !!)
• sinθm = (m + 1/2)λ / w

• Where m is the number of maxima lines and w is the width of the slit
• The Separation between between adjacent maxima or minima is given as
• Δy = λL / w  central maxima: 2λ
• where L is the distance of the perpendicular bisector and w is the width of the slit
• Resolution: is the ability of an instrument to separate two closely spaced images, is limited by the diffraction of the light.

Diffraction Grating

• Diffraction Grating: device with surface of equally spaced parallel lines resolving light into spectra; transmission gratings are transparent; reflection gratings are mirrored.
• Diffraction Gratings deliver brighter interference patterns than typical double slots, with maxima that are narrower and more widely spaced
• sinθm = mλ / d

• where d is the distance between adjacent gratings, and m is the order of Maxima
• Spectroscope: used to analyze light in a spectrum, uses a collimator to send light to grating
• grating splits light into its respective colours.

Interference through thin films

• Light reflects off a thin coat, some refracts into the coat, and reflects off the medium behind it, and bounces out of the thin coat, causing interferences
• Crests reflecting off a faster medium stays crest
• Crests reflecting off a slower medium becomes trough
• Thickness of the film is dictated by how it alters the wavelength, either by cutting it short by 1/2, 1/4  or 1 lambda.
• t = λcoating / Amount of Coating distruption

# Unit 5: Modern Physics

Frames of Reference and Relativity

• Inertial Frames of Reference is a frame of reference that obeys laws of Inertia and Newton’s laws of motion.
• Non-Inertial frames of reference is accelerating and does not obey those laws
• Using Newton’s laws, there is no way to identify whether the inertial frame of reference is actually at rest or moving at constant velocity.
• Einstein’s Laws of Special Relativity states 2 postulates:
• All laws of physics are the same in all inertial frames of reference
• Light travels at a speed of 3 x 10^8 m/s in all inertial frames of reference
• Simultaneity is a relative concept, where it applies the same regardless of frames of reference.

Relativity of Time, Length, Momentum

• Proper Time is the time between two events as seen by someone in the same position
• Time Dilation: slowing down time in a system, where the observer is in motion relative to the time
• ΔTm = Δt = √(1 – v2/c2)
• Time is relative, not absolute where both simultaneous time duration events that are simultaneous to one observer may not be simultaneous to another.
• Time interval measured by one may be different from another.
• Proper Length is the length observed by the observer in the rest relative to object.
• Lm = Ls √(1 – v2/c2)
• P, the magnitude of momentum in relativity increases as speed increases
• P = mv / √(1 – v2/c2)
• Rest mass in inertial frames is the only mass that can be defined
• Non zero rest masses cannot travel at the speed of light.