Physics Units 3&4

Headstart Lectures

Presented by:SOMYA MEHRA

• Overview of Unit 3 – introduce big ideas

• Won’t cover every little aspect of the study design

• Start making connections and thinking about the bigger picture

Today’s lecture

• Try to understand concepts

• Keep an eye out for patterns – lots of ideas in physics are interconnected

• Always do questions that challenge you

• Maintain a log book of mistakes

Approaching physics

• Newton’s laws of motion

• Energy and momentum

• Einstein’s theory of special relativity

• Gravitational, electric and magnetic fields

• Electricity generation and transmission

Unit 3 Physics

NEWTON’S LAWS OF MOTION

Measurements

• Scalars: distance, speed

• Vectors: displacement, velocity, acceleration

Relationships

• Velocity is the rate of change of position

𝒗 =∆𝒙

∆𝒕𝒗𝒂𝒗 =

𝒙𝟐−𝒙𝟏

𝒕

• Acceleration is the rate of change of velocity

𝒂 =∆𝒗

∆𝒕𝒂𝒂𝒗 =

𝒗−𝒖

𝒕

Kinematics

Motion graphs

x-t graph v-t graph a-t graph

Gradient velocity acceleration -

Area under graph

-change in position

change in velocity

To find the final velocity, add the initial velocity to the change in velocity

𝒗 = 𝒖 + 𝒂𝒕

𝒗𝟐 = 𝒖𝟐 + 𝟐𝒂𝒙

𝒙 = 𝒖𝒕 +𝟏

𝟐𝒂𝒕𝟐

𝒙 = 𝒗𝒕 −𝟏

𝟐𝒂𝒕𝟐

𝒙 =𝟏

𝟐𝒖 + 𝒗 𝒕

𝑥 = displacement (m)

𝑢 = initial velocity (m/s)

𝑣 = final velocity (m/s)

𝑎 = acceleration (m/s2)

𝑡 = time (s)

Constant acceleration formulae

Can ONLY be used if acceleration is constant

• The only force acting in the vertical direction is gravity

• Air resistance might be acting in the horizontal direction

Projectile motion

Effects of air resistance• Decreases range• Lowers maximum height• Asymmetrical trajectory

If there is no air resistance:

• The horizontal velocity remains the same throughout

• The only force acting in the vertical direction is gravity – constant acceleration

Important tricks:

• At the maximum height, 𝒗𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 = 𝟎

• If the projectile lands at the same height, 𝒗𝒊𝒏𝒕𝒊𝒕𝒂𝒍 = −𝒗𝒇𝒊𝒏𝒂𝒍

Projectile motion with no air resistance

• We’re looking at the horizontal and vertical components of motion separately

• If the projectile is moving at an angle, then we need to ‘resolve’ the velocities

Resolving velocities

Vertical component𝒗𝑽 = 𝒗 𝐬𝐢𝐧(𝜽)

Horizontal component𝒗𝑯 = 𝒗 𝐜𝐨𝐬(𝜽)

𝜽

Total velocity𝒗 m/s, 𝜽° to ground

A projectile is launched from the ground at an angle of 30° and a speed of 10 m/s. How far away does the projectile land on the ground? Assume there is no air resistance.

Example: projectile motion

Newton’s first law

• If 𝑭𝒏𝒆𝒕 = 𝟎, an object will remain at rest or continue moving with the same velocity

• A net force must be applied for an object to change direction or speed

Newton’s second law

• 𝑭 = 𝒎𝒂

Newton’s third law

• 𝑭𝑨 𝒐𝒏 𝑩 = −𝑭𝑩 𝒐𝒏 𝑨

• Every action has an equal, but opposite reaction

Newton’s laws of motion

• Direction keeps changing, so there must be a net force – Newton’s first law

Centripetal force

• Points to the centre of the circle

• Perpendicular to the direction of motion

• Made up of other forces– Vehicle on a circular road: friction

– Object on end of string: tension

– Orbiting satellites: gravitational force

– Banked curves: normal force, friction

Circular motion

Horizontal circular motion is uniform – the forces don’t change

Acceleration

𝒂 =𝒗𝟐

𝒓where 𝑣 = speed (m/s) and 𝑟 = radius (m)

Centripetal force

𝑭𝒏𝒆𝒕 =𝒎𝒗𝟐

𝒓(using Newton’s second law)

Uniform circular motion

Consider a 5kg mass in uniform circular motion, hanging on a string at an angle of 45°. What is the speed of the mass?

Example: uniform circular motion

45°

• Vertical circular motion is non-uniform

• Gravity might act towards or against the centripetal force at different positions

• Example: loop-the-loop on a rollercoaster

Non-uniform circular motion

mg

Nmg N

ENERGY AND MOMENTUM

• Energy is the ability to do work

• Energy is always conserved

• Can be transformed to another type of energy– Kinetic energy

– Spring potential energy

– Gravitational potential energy

– Heat, sound, energy of deformation

• Can be transferred to a different object

Energy

• Work describes how much energy has been transferred/transformed– Can involve any type of energy

• If the force is constant:

𝑾 = 𝑭𝒙 where 𝑊 = work done (J)

𝐹 = constant force (N)

𝑥 = displacement in the direction of the force (m)

• Otherwise, 𝑾 = area under force-distance graph

Work

• Energy of movement

𝑬𝒌 =𝟏

𝟐𝒎𝒗𝟐 where 𝐸𝑘 = kinetic energy (J)

𝑚 = mass (kg),

𝑣 = velocity (m/s)

Kinetic energy

• Hooke’s law describes how much force it takes to compress or extend a spring

𝑭 = 𝒌𝒙 where 𝑘 = spring constant, how stuff the spring is (N/m)

𝑥 = change in the length of the spring (m)

• Strain potential energy is stored in a spring when it is compressed or stretched out

𝑬𝒔 =𝟏

𝟐𝒌𝒙𝟐 where 𝑘 = spring constant, how stiff the spring is (N/m)

𝑥 = change in the length of the spring (m)

Strain potential energy

A 250g ball is pulled back against a horizontal spring (𝑘 = 300 N/m). The spring is compressed by 10cm. What is the speed of the ball when the spring is released?

Example: energy transformations

• Momentum describes how hard it is to change an object’s state of motion

𝒑 = 𝒎𝒗 where 𝑝 = momentum (Ns or kgm/s)

𝑚 = mass (kg),

𝑣 = velocity (m/s)

• Momentum is ALWAYS conserved

Momentum

• Impulse is the change in momentum of an object

𝑰 = ∆𝒑 = 𝒎∆𝒗

𝑰 = area under force-time graph

• If the average force is given, or if the force is constant:

𝑰 = 𝑭𝒂𝒗∆𝒕

Impulse

Elastic

• Momentum is conserved

• Initial kinetic energy = final kinetic energy

Inelastic

• Momentum is conserved

• Initial kinetic energy > final kinetic energy– Transformed to heat, sound and energy of deformation

Collisions

A 2kg ball travelling 20m/s to the right collides with a 4kg bat travelling 6m/s to the left. After the collision, the ball travels 10m/s to the left.

a) What is the final velocity of the ball?

b) What is the impulse of the ball?

c) Is the collision elastic or inelastic?

Example

A 2kg ball travelling 20m/s to the right collides with a 4kg bat travelling 6m/s to the left. After the collision, the ball travels 10m/s to the left.

a) What is the final velocity of the bat? 9m/s right

b) What is the impulse of the ball?

c) Is the collision elastic or inelastic?

Example

A 2kg ball travelling 20m/s to the right collides with a 4kg bat travelling 6m/s to the left. After the collision, the ball travels 10m/s to the left.

a) What is the final velocity of the bat? 9m/s right

b) What is the impulse of the ball? -60m/s right = 60m/s left

c) Is the collision elastic or inelastic?

Example

SPECIAL RELATIVITY

• Measurements are always made in reference to something

• A frame of reference is a bit like a ‘perspective’ for an observer

• Can be inertial (not accelerating) or non-inertial (accelerating)

• We’re interested in inertial reference frames– Move at a constant speed without changing direction relative to other inertial reference frame

Frames of references

What does relativity say?

• The measurements you make for an event depend on your frame of reference

• Observers won’t necessarily agree about things like time and distance

But…

• Your measurements are always correct

• No reference frame is better than another

Special relativity

1. The speed of light is constant regardless of the motion of the observer or emitter

2. The laws of physics are the same in all inertial reference frames

Einstein’s postulates

𝒕 = 𝒕𝟎𝜸 =𝒕𝟎

𝟏−𝒗𝟐

𝒄𝟐

• 𝑡0 = proper time (measured in the frame of reference at rest relative to the event)

• 𝑡 = observed time (measured in a reference frame travelling at a velocity 𝑣 m/s relative to the event)

• 𝛾 = 1 −𝑣2

𝑐2

−1

2is the Lorentz factor

Time dilation

Example: 2016 VCAA Exam

𝑳 =𝑳𝟎

𝜸= 𝑳𝟎 𝟏 −

𝒗𝟐

𝒄𝟐

• 𝐿0 = proper length (measured in the frame of reference at rest relative to the object)

• 𝐿 = observed length (measured in a reference frame travelling at a velocity 𝑣 m/s relative to the object)

• 𝛾 = 1 −𝑣2

𝑐2

−1

2is the Lorentz factor

Note: only relevant along the direction of motion

Length contraction

I measure a box I’m stationary to, and find that it’s dimensions are 2m × 3m × 4m. Calculate the dimensions of the box, as measured by an observer travelling at 0.8c relative to the box in the direction shown.

Example: length contraction

0.8c

4m

3m

2m

Observation

• In the laboratory, muons are measured to have very short half-lives

• They’re expected to decay in the outer atmosphere, but they end up reaching earth

Why?

• The muons are travelling at relativistic speeds

• Time passes more slowly for muons because of time dilation

• From the reference frame of muons, the distance to earth is much shorter due to length contraction

Muon decay

𝒎 = 𝒎𝟎𝜸 =𝒎𝟎

𝟏−𝒗𝟐

𝒄𝟐

• 𝑚0 = proper mass (measured in the frame of reference at rest relative to the object)

• 𝑚 = observed mass (measured in a reference frame travelling at a velocity 𝑣 m/s relative to the object)

• 𝛾 = 1 −𝑣2

𝑐2

−1

2is the Lorentz factor

Mass

Rest energy

• 𝑬𝟎 = 𝒎𝟎𝒄𝟐 where 𝑚0 = proper mass

Mass energy

• 𝑬 = 𝒎𝒄𝟐 where 𝑚 = observed mass

Kinetic energy

• 𝑬𝒌 = 𝒎𝒄𝟐 −𝒎𝟎𝒄𝟐 = 𝜸 − 𝟏 𝒎𝟎𝒄

𝟐 where 𝛾 = Lorentz factor

Mass-energy

Example: 2016 VCAA exam

GRAVITATIONAL FIELDS

What is a field?

• A way of thinking about forces that act over distances

• Fields act on different things (e.g. charges, masses) and may cause a force to be felt

• We can predict the size and direction of this force

Fields can be:

• Uniform (the same everywhere) or non-uniform

• Static (don’t vary with time) or changing

Introduction to fields

• Anything with mass creates a gravitational field

• If a test mass is placed in a gravitational field, it experiences a gravitational force

• Masses always attract each other

Gravitational fields

ABGravitational force of A on B

Gravitational force of B on AThese forces are equal – Newton’s 3rd law!

Gravitational fields always:

• Get weaker with distance

• Act towards the centre of the source

We can use field-lines to visualize this

• Arrows show the direction the field is acting in

• Closer lines ⇒ stronger field

Field lines

SUNV

M

Gravitational field

𝒈 =𝑮𝑴

𝒓𝟐where 𝑀 = mass of object creating field (kg)

𝑟 = distance b/w the test mass and the object creating the field (m)

𝐺 = 6.67 × 10−11 (m3 kg−1 s−2)

Gravitational force

𝑭 = 𝒎𝒈 where 𝑚 = mass of the object the field is acting on (kg)

𝑔 = size of gravitational field (N/kg)

Modelling gravitational fields

Uniform gravitational field

𝐸𝑔 = 𝑚𝑔∆ℎ where 𝑚 = mass of object being moved (kg)

𝑔 = gravitational field strength (N/kg)

∆ℎ = height object is moved by (m)

Non-uniform gravitational field

• Area under gravitational force-distance graph

• Area under gravitational field-distance graph × mass

Gravitational potential energy

What is the change in gravitational potential energy if a 5kg ball is lifted 3m on the surface of the moon?

Example: gravitational potential energy

DATA

Mass of earth = 7.35 × 1022 kgRadius of moon = 1737 km 𝐺 = 6.67 × 10−11 m3 kg−1 s−2

• Satellite orbits can be modelled using uniform circular motion

• The centripetal force is the gravitational force!

What do we know?

• From uniform circular motion: Fcentripetal =𝑚𝑣2

𝑟=

4𝑚𝜋2𝑟

𝑇2

• From the law of gravitation: Fgravity = 𝑚𝑔 =𝐺𝑀𝑚

𝑟2

And so: 𝑚𝑣2

𝑟=

𝐺𝑀𝑚

𝑟2⇒ 𝑣 =

𝑀𝐺

𝑟

Satellite orbits

What is the average speed of the Moon as it orbits the Earth?

Example: average satellite speed

DATA

Mass of earth = 5.972 × 1024 kgDistance from earth to moon = 3.84 × 105 km 𝐺 = 6.67 × 10−11 m3 kg−1 s−2

ELECTRIC FIELDS

• Any electric charge produces an electric field

• If a test charge is placed in an electric field, it experiences an electric force

• Electric potential energy changes if a charge moves though an electric field

• Charges can be attract or repel each other

Electric fields

Similar rules to gravitational fields, except:

• Field lines show direction POSITIVE charges would move

• Negative charges would move in opposite direction

Remember:

• Closer lines ⇒ stronger field

Field lines

+

Electric field

𝑬 =𝒌𝑸

𝒓𝟐where 𝑄 = charge of object creating field (C)

𝑟 = distance b/w the test charge and the source charge (m)

𝑘 = 9.0 × 109 (N m2 C−2)

Electric force

𝑭 = 𝒒𝑬 where 𝑞 = charge of the object the field is acting on (C)

𝐸 = size of electric field (N/C)

Modelling electric fields

• We can create a uniform electric field by connecting two parallel plates to a battery

𝑬 =𝑽

𝒅where 𝑉 = voltage (V), 𝑑 = distance (m)

• Electric potential energy changes if a charge moves in a uniform electric field

𝑾 = 𝒒𝑽 where 𝑉 = voltage (V), 𝑞 = charge (C)

Uniform electric fields

+

+

+

+

Dipole

An electron is accelerated using two plates with a voltage drop of 250V. Assuming it is initially stationary, what is the final speed of the electron?

Example: linear particle accelerator

DATA

Mass of electron = 9.1× 10-31 kgCharge of electron = 1.6 × 10-19 C

MAGNETIC FIELDS

• Magnetic fields can be created by magnets or currents– Strength is measured in Tesla (T)

• Charges or currents in magnetic fields can experience a force

• Magnets always have a north AND a south pole– If you break a magnet in half, you end up with new north and south poles

Magnetic fields

• Always go from the north to south pole

• To work out the direction of the magnetic field, take the tangent to the field line

• Closer lines ⇒ stronger field

• Field lines NEVER intersect

Field lines

An aside: some notation

OUT OF PAGE(tip of arrow)

INTO PAGE(tail of arrow)

Electric current flowing •

through a wire produces a magnetic field

Field lines are concentric circles •

around the wire

Right hand grip rule

Solenoids are made up of loops of wire•

The field lines of a solenoid are like a bar magnet•

Solenoids

An electric current in an • external magnetic field can experience a forceThe magnetic field of the current interacts with the external magnetic field–

If the external magnetic field is perpendicular to the current:

𝑭 = 𝒏𝑰𝒍𝑩 where 𝑛 = number of wires

𝐼 = current in each wire (A)

𝐵 = strength of magnetic field (T)

If the external magnetic field is parallel to the current: 𝑭 = 𝟎

Magnetic force on a current

To work out the direction of the force, we have to use the • ‘right hand slap rule’

Magnetic force on a current

The EXTERNAL magnetic field, current and force all have to be perpendicular to each other

• Connect a coil of wire to a battery and place it in a magnetic field

DC motor

Position 1: coil parallel to external magnetic field

The turning force on the coil is the same, but the • ‘turning effect’ (torque) is lower

DC motor

Position 2: coil on a 45° angle

DC motors

Position 3: coil perpendicular to magnetic field

To solve this problem, we use a split ring commutator

• The splits line up with the brushes when the coil is vertical– No current through the coil, so no force

– Coil continues to rotate due to momentum

• Also reverses the direction of current through the coil– Coil rotates in the same direction

DC motors – split ring commutator

Example: 2015 VCAA exam

The magnetic force is always perpendicular to the direction a charge is moving in•

So we have uniform circular motion • – the centripetal force is the magnetic force!

𝑭 = 𝒒𝒗𝑩 where 𝑞 = charge (C)

𝑣 = speed charge is travelling at (m/s)

𝐵 = strength of external magnetic field (T)

Combining this with what we know about circular motion:

𝑭 =𝒎𝒗𝟐

𝒓= 𝒒𝒗𝑩

Magnetic force on charges

An electron moves in a circle of radius 4.00mm in a magnetic field of strength 10.0mT. How fast is it travelling?

Example: charges in magnetic fields

DATA

Mass of electron = 9.1× 10-31 kgCharge of electron = 1.6 × 10-19 C

GENERATING ELECTRICITY

• Magnetic flux tells us how much magnetic field is passing through a coil

∅𝑩 = 𝑩˪𝑨 where ∅𝑩 = magnetic flux through coil (W)

𝐵˪ = component of magnetic field perpendicular to coil (T)

𝐴 = area of coil (m2)

• Can be positive or negative depending on the direction of the magnetic field

• ∅𝑩 = 𝟎 if the magnetic field is parallel to the coil

Magnetic flux

Magnetic flux can be changed by:

• Changing the strength of the magnetic field

• Rotating the coil in the magnetic field

Changing magnetic flux

Angle of Coil

Magnetic flux

• If the coil is moving from a vertical to horizontal position, the size of ∅𝐵 is decreasing

• If the coil is moving from a horizontal to vertical position, the size of ∅𝐵 is increasing

Rotating a coil

∅𝐵 maximum – magnetic field perpendicular to coil

∅𝐵 = 0– magnetic field parallel to coil

• EMF is induced when the magnetic flux through a coil changes– Think of EMF as the ‘pushing’ of electrons in a particular direction

• The more quickly the magnetic flux changes, the higher the EMF

Mathematically:

𝜺𝒂𝒗 = −𝒏∆∅𝑩

∆𝒕where 𝜀𝑎𝑣 = average EMF (V)

𝒏 = number of coils in the loop

∆∅𝐵 = change in magnetic flux through the loop (Wm2)

∆𝑡 = time (s)

Faraday’s law

• emf can be induced by rotating a coil

• emf is the negative gradient of the magnetic flux

Inducing emf – rotating a coil

Angle of Coil

Induced emf

Magnetic flux

Example: 2015 VCAA exam

• Rotating a coil in an external magnetic field causes current to be induced

• The induced current opposes the change in flux

What this means:

• If the flux through the coil is increasing, the magnetic field of the current is in the opposite direction to the external magnetic field

• If the flux through the coil is decreasing, the magnetic field of the current is in the same direction as the external magnetic field

Lenz’s law

What is the direction of direction of current through the loop if it is rotated clockwise?

Example: Lenz’s law

• AC generators: slip rings DC generators: • commutators

DC generators and alternators

Characteristics

Amplitude or peak voltage1.

Peak2. -to-peak voltage

RMS voltage3.

Period (T)4.

Mathematically

Frequency: • 𝑓 =1

𝑇

RMS voltage: • 𝑉𝑅𝑀𝑆 =𝑉𝑝𝑒𝑎𝑘

2

Sinusoidal AC voltages

TRANSMITTING ELECTRICITY

Wires don• ’t have much resistance, but it adds up over long distances

This means power is lost and when electricity is transmitted•

𝑷𝒍𝒐𝒔𝒔 = 𝑰𝟐𝑹𝒘𝒊𝒓𝒆𝒔

How do we minimise power loss?

Transmit electric power at low currents and high voltages•

Problem: appliances can• ’t always use high voltages

Power loss in electricity transmission

Transformers can be used to step up or step down AC voltages•

This is a big advantage of AC power–

Transmitting electric power

𝑽𝒐𝒖𝒕 > 𝑽𝒊𝒏 𝑽𝒐𝒖𝒕 < 𝑽𝒊𝒏

Alternating current in the primary coil •

produces a changing magnetic field

There is a change in magnetic flux in the •

iron core

EMF is generated in the secondary coil•

How transformers work

In ideal transformers, no power is lost•

𝑵𝑷

𝑵𝑺=

𝑽𝑷

𝑽𝑺=

𝑰𝑺

𝑰𝑷

This doesn• ’t happen in real life – power is lost through eddy currents

Ideal transformers