3.2.2 Motion with non-uniform acceleration

If an object is moving through a fluid it will experience a drag force from the fluid. Drag is a frictional force that opposes the motion of the object. Its magnitude is mainly affected by:

• the speed of the object
• the cross-sectional area of the object
• the shape of the object
• the density of the fluid
• the texture of the object

Objects with larger cross-sectional areas will experience a greater drag force. This is because drag ∝ speed². 

Often, we call the drag force experienced by objects in air air resistance. We are able to reduce air resistance. For example, modern vehicles have streamlined shapes as this increases their top speed and reduces their fuel consumption.

During a vertical fall (e.g through air) the weight of an object will of course remain constant but the speed will increase so the drag force will increase. At the instant objects start to fall the object will experience no drag force and the total force is equal to the weight (as this is the only force acting on the object). As the object falls its speed increases so drag increases. The resultant force on the object decreases (as drag opposes the motion ie is in the opposite direction to weight). At terminal velocity, drag = weight and the object has zero acceleration and its speed is constant.

We can investigate the motion of an object falling affected by drag force by using a motion sensor and data-logger:

• attach the object to a light polystyrene ball using a thin thread passing over a pulley.
• Drop the object through a cylinder of liquid (e.g water or glycerol). This will pull the polystyrene ball vertically upwards (the motion of this ball is equal to the motion of the object)
• when the ball is moving up with constant velocity, the object is moving down with terminal velocity

3.2.1 Dynamics

The mass of an object depends on the amount of matter it contains. The net/resultant force acting on an object will make it accelerate in that particular direction. The net force (F):

F = m a

Force is measured in newtons (N). A force of 1 newton will give a 1kg mass an acceleration of 1m s^-1 in the direction of the force.

The weight of an object on the surface of the Earth is the gravitational force acting on the object:

W = m g

The weight of an object can be calculated using a newtonmeter. These are calibrated to show the gravitational force acting on an object in newtons (e.g an object of mass 1kg will show a weight of 9.8N).


The easiest way to analyse the forces acting on an object is to draw free-body diagrams (these isolate all the forces acting on a particular object. Each force vector is represented by an arrow labelled with the force it represents, each arrow is drawn to the same scale (ie the longer the arrow the greater the force).

Tension, normal contact force, upthrust, and friction are all examples of forces that can be added to a free-body diagram.

3.1.2 Linear motion

Okay so we need to learn 4/5 of the suvat equations (no. 1, 2, 3, and 4) :

Tbh, number 5 is useful if you want to learn it, but you can use the others to deduce the answer so not the end of the world (you do need to know no.5 if you're doing maths though).


Okay so one experiment we need to know about is about free fall and determining g (the gravitational field strength), acceleration due to gravity. As we know, objects with mass exert a gravitational force on each other. When an object is accelerating under gravity (with  no other force acting on it) it is said to be in free fall. As I mentioned just a few moments ago, we need to be able to determine g (in a laboratory):

• Hold a small steel ball over a trapdoor using an electromagnet
• Turn the current off, a timer will trigger and the electromagnet will demagnetise (hence the ball will fall)
• the heavy steel ball will drop over a known distance (measure with a meter rule)
• it's decent will be timed using the timer (use a stopwatch for backup)
• Calculate the value for g using a = v/t = g

We can also use light gates:

• Two light beams are placed one above the other with detectors connected to a timer
• When the ball falls through the first beam it interrupts the light and starts the timer
• When the ball falls through the second beam a known distance further down the timer will stop
• use a = v/t = g

We can also use pictures:

• Drop a small metal ball from rest next to a metre rule
• Record its fall with a camera in rapid0-fire repeating mode
• Hold the camera shutter open to produce a photograph with multiple images of the falling ball/put a long exposure setting on
• Examine the photograph and determine the position of the ball at regular time intervals
• do a = v/t

Okay so we also need to know a little bit about things such as reaction time, thinking distance breaking distance, and stopping distance for a vehicle. Basically:

• The stopping distance is the total distance travelled from when the driver first sees the reason to stop to when the vehicle stops. It is thinking distance + breaking distance
• Thinking distance is the distance travelled between the moment when you first see a reason to stop to the moment you use the brake. For a vehicle moving at constant speed thinking distance = speed x reaction time. The greater the speed or reaction time the further the vehicle will travel before its driver applies the brakes. 
• Breaking distance is the distance travelled between the time the brake is applied until the vehicle stops.

NOTE: One point in this section of the spec states that we need to know '(ii) techniques and procedures used to investigate the motion and collisions of objects'. You should have done this as a PAG experiment (usually in your first year of covering the course).

3.1.3 Projectile motion

How far projectiles can travel depends on several factors. It is important to realise that the vertical and horizontal motions of the ball are independent of each other. Assuming no air resistance:

• the vertical velocity changes due to acceleration of free fall
• the vertical displacement and time of flight can be calculated using the equations of motion
• horizontal velocity remains constant

The horizontal velocity of the projectile is constant because acceleration and velocity are vectors. The acceleration of free fall is vertically downwards and the component of this acceleration in the horizontal direction is zero. This is because horizontal acceleration = g cos 90° = 0.

The path of the projectile is curved because the vertical component of its velocity increases with time whilst the horizontal component is unaffected. The magnitude of the actual velocity (v) of the projectile can be calculated from he vertical and horizontal components of the velocity (vx and vy):

v = √((vx)² + (vy)²)

The angle made by the velocity to the horizontal is given by:

θ = tan^-1 ((vy)/(vx))

When a projectile is fired upwards the motion can still be analysed in terms of the independent motion in the horizontal and vertical directions. The horizontal component of the velocity if vcosθ and the initial vertical component of the velocity if vsinθ.

6.3.3 Electromagnetism

Electromagnetic induction occurs in the generators in power stations and in wind turbines. It generates electricity (induces an e.m.f) by relative motion between a conductor and a magnetic field. To induce an e.m.f all you need is a coil and a magnet. Attach a (sensitive) voltmeter to the coil. Push the magnet towards the oil, an e.m.f is induced across the ends of the coil (when the motion of the magnet is reversed a reverse e.m.f is induced. Repeating these motions will produce an alternating current. The faster the magnet moves the larger the induced e.m.f.

We can also use a simple d.c. electric motor in reverse (e.g using a falling mass to rotate the coil between the poles of the stationary magnet. The induced e.m.f can be large enough to operate a lamp. An e.m.f is induced in a loop of copper wire when it is moved perpendicular to the magnetic field lines of a magnet. The magnitude of the e.m.f is bigger when the wire is pulled away faster from the magnet.

As we all know, energy is ALWAYS conserved. Where does the electrical energy produced in the coil come from I hear you ask? Well, some of the work done to move the magnet is transferred into electrical energy. 

Magnetic flux
All experiments demonstrating electromagnetic induction can be explained in terms of magnetic flux (ϕ). The magnetic flux (ϕ) is defined as the product of the component of the magnetic flux density perpendicular to the area and the cross-sectional area:

ϕ = (Bcosθ) x A = BAcosθ

NOTE: When the field is normal to the area, cosθ = 1 and ϕ = BA.

The SI unit for magnetic flux is the weber (Wb). 1 Wb = 1 T m².

Magnetic flux linkage
This is the product of the number of turns in the coil N and the magnetic flux:

magnetic flux linkage = N ϕ

The SI unit of magnetic flux linkage is also Wb (sometimes weber-turns are used to distinguish it from magnetic flux).

It is important to realise that an e.m.f is induced in a circuit whenever there is a change in the magnetic flux linking the circuit. Since ϕ = BAcosθ you can induce an e.m.f by changing θ, B, or A.

Faraday's law
Faraday's law relates magnetic flux linkage to the magnitude of the e.m.f induced in conductors. It states that the magnitude of the induced e.m.f is directly proportional to the rate of change of magnetic flux linkage:

Ɛ ∝ Δ(Nϕ) / Δt

The constant of proportionality is -1:

Ɛ = -Δ(Nϕ) / Δt

Lenz's law
This states that the direction of the induced e.m.f or current is always such as to oppose the change producing it. The negative sign in the equation for Faraday's law is a mathematical way of expressing Lenz's law. In most calculations you can ignore this minus sign, it is just a reminder that energy cannot be created. 

We need to know how to investigate magnetic flux using search coils. The a.c. generator in the below diagram consists of a rectangular coil of cross-sectional area A and N turns of coil rotating in a uniform magnetic field of flux density B. The flux linkage for the coil is:

flux linkage = N ϕ =  = NBAcosθ

As the coil rotates (at a steady frequency) the flux linkage changes with time (t). The variation produced is sinusoidal and is caused by the changing cosθ:

We already know that Ɛ = -Δ(Nϕ) / Δt = Ɛ = -Δ(NBAcosθ) / Δt. This means that:

• The magnitude of the gradient of the magnetic flux linkage against time graph is equal to the induced e.m.f
• For a given generator, B, A, and N are all constant. This means that Ɛ ∝ -Δcosθ / Δ

The second graph shows the variation of e.m.f with time (t). The maximum induced e.m.f is directly proportional to:

• the magnetic flux density (B)
• the cross-sectional area (A) of the coil
• the number of turns (N)
• the frequency (f) of the rotating coil

Transformers

Okay hold your hats this i the last part of electromagnetism. An important use of electromagnetic induction is in transformers. These change alternating voltages to higher/lower values. A simple transformer consists of a laminated iron core, a primary/input coil, and a secondary/output coil. An a.c current is supplied to the primary coil which produces a varying magnetic flux in the soft iron core. The secondary coil (which is wound round the same core) is linked by this changing flux. The iron core ensures that all the magnetic flux created by the primary coil links to the secondary coil and none is lost. According to Faraday's law (the magnitude of the induced e.m.f is directly proportional to the rate of change of magnetic flux linkage) a varying e.m.f is produced across the ends of the secondary coil. The input and output voltages (Vp and Vs, respectively) are related to the number of turns on the primary and secondary coil (np and ns, respectively) by the turn-ratio equation:

ns/np = Vs/Vp (for an ideal transformer)

NOTE: A step up transformer has more turns on the secondary than primary coil. A step down transformer has fewer turns on the secondary that the primary coil.

So we need to know techniques and procedures used to investigate transformers:

• Set a multimeter to alternating voltage. This can be used to measure the input Vp and output Vs voltages
• This insulated copper wires are used to make primary and secondary coils
• Change the number of coils to see what happens to Vs for a fixed value of Vp and vice versa

For 100% efficient transformers the output power from the secondary soil is equal to the input power from the primary coil. P=IV so:

Vs Is = Vp Ip

This means that increasing input voltage by a factor of 100 will decrease output current by a factor of 100 etc.

We can make transformers more efficient by using low-resistance windings to reduce power losses due to the heating effect of the current. Making a laminated core with layers of iron separated by an insulator helps to minimise currents induced in the core itself (eddy currents) so minimalises loses due to heating. The core is also made of soft iron which is very easy to magnetise and demagnetise which improves the overall efficiency of the transformer.

Image credit: Kerboodle OCR A Physics textbook

6.3.2 Motion of charged particles

As we know from 6.3.1, a charged particle moving in a magnetic field will experience a force. We can demonstrate this using an electron deflection tube. The force on the beam of electrons can be predicted using Fleming's left hand rule. As the electrons enter the field they experience a downward force. The electrons change direction (due to the force) but the force (F) on each electron always remains perpendicular to its velocity. The speed of the electrons is constant as the force has no component in the direction of motion.

To find the force (F) acting on a charged particle of charge Q moving at a speed v at right angles to a uniform magnetic field of flux density B consider a section of conductor/beam of charged particles. The length L of this section is 'vt'. The force (F) on the conductor is given by:

F = B I L

F = B I (vt)

We are well aware that current is the rate of flow of charge. If there are N charged particles wach with a charge Q the current can be given by:

I = (NQ)/t

So the force acting on the conductor is given by:

F = B x (NQ/t) x vt = NBQv

So the force on each charged particle is:

F = (NBQv)/N = BQv

For an electron or proton  Q = e so F = Bev


Circular orbits
So we need to know a bit more about circular orbits of charged particles in a uniform magnetic field. Consider a charged particle of mass m and charge Q moving at right angles to a uniform magnetic field of flux density B. The particle will move in a circular path as the force acting on it is always perpendicular to its velocity. The centripetal force (F = (mv²) / r) on the particle is provided by the magnetic force (F = BQv):

BQv = (mv²) / r

r = mv/BQ

This equation shows that:

• faster-moving particles make a bigger circle
• more massive particles move in a bigger circle
• stronger magnetic fields make the particles move in a smaller circle
• particles with greater charge move in smaller circles

The velocity selector

We also need to know a bit about charged particles moving in a region occupied by both electric and magnetic fields. A velocity selector is a device that uses both electric and magnetic fields to select charged particles of specific velocity. It is a vital part of instruments such as mass spectrometers and some particle accelerators. It consists of two parallel horizontal plates connected to a power supply. They produce a uniform electric field of field strength E between the plates. A uniform magnetic field of flux density B is also applied perpendicular to the electric field. The charged particles travelling at different speeds to be sorted enter through a narrow slit Y. The electric and magnetic fields deflect them in opposite directions. Only for particles with a specific speed v will these deflections cancel so that they travel in a straight line and emerge from the second narrow slit Z. For an undeflected particle:

electric force = magnetic force

EQ = BQv

E = Bv

This means that the speed (v) depends only on electric field strength (E) and magnetic flux density (B).

6.3.1 Magnetic fields

A magnetic field is a field surrounding a permanent magnet or a current carrying conductor in which magnetic objects experience a force. We can detect the presence of a magnetic field using a plotting compass/iron fillings. Magnetic field lines/lines of force map magnetic field patterns around current-carrying wires/magnets:

• The arrow on the magnetic field line shows the direction in which a free north pole would move
• Equally spaced parallel magnetic field lines represent a uniform field (the strength does not vary)
• The magnetic field is stronger when the magnetic field lines are closer.
• Like poles repel and unlike/different poles attract

NOTE: with iron fillings, the field induces magnetism in the filings which line up in the field.

When a wire carries a current a magnetic field is created around the wire. The field is created by the electrons moving within the wire. Any moving charged particle creates a magnetic field in the space around it. The electric field of a bar magnet is created by the electrons whizzing around the iron nuclei.

For a current-carrying wire the magnetic field lines are concentric circles centred on the wire and perpendicular to it. The direction of the magnetic field can be determined using the right hand grip (thumb points in direction of conventional current, direction of the field is given by the direction the fingers curl). Coils and solenoids both produce north and south poles. At the centre of the solenoid the magnetic field is uniform.


When a current-carrying wire/conductor is placed in an external magnetic field the two fields interact just like the fields of a permanent magnet. The direction of force experienced by a current carrying wire placed perpendicular to the external magnetic field can be determined using Fleming's left-hand rule.  The first finger gives the direction of the external magnetic field, the second finger gives the direction of motion of the conventional current, and the thumb gives the direction of motion of the wire:

Magnetic flux density
The magnetic force experienced by a wire in an external magnetic field depends on a number of factors (e.g force is a maximum when the wire is perpendicular to the field and zero when parallel. The magnitude of the force (F) experienced by the wire is directly proportional to:

• the current 'I'
• the length 'L' of the wire in the magnetic field
• sinθ (θ is the angle between the magnetic field and the current direction
• the strength of the magnetic field

From these observations we can form the equation:

F = B I L sinθ

B is the magnetic flux density (the strength of the field). The SI unit for magnetic flux density is the tesla (T). 1 T = 1 N m^-1 A^-1. The magnetic flux density is 1 T when a wire carrying a current of 1 A placed perpendicular to the magnetic field experiences a force of 1 N per metre of its length. 

When the wire is perpendicular to the magnetic field θ=90. Sin 90 = 1 so F = B I L. This means the equation for magnetic flux density can be written as:

B = F / (I L)

We need to know as experiment to determine the uniform magnetic flux density between poles of a magnet using a current-carrying wire and digital balance:

• Place two bar magnets on top of a balance
• Hold a stiff copper wire between them perpendicular to the magnetic field between the two poles
• measure the length L of the wire in the field using a ruler
• connect the wire in series with an ammeter and variable power supply using crocodile clips
• Zero the balance when there is no current in the wire
• When there is a current I flowing through the circuit, the wire experiences a vertical upward force (Fleming's left hand rule). According to N3 the magnets experience an equal downward force. This force can be calculated from the change in mass reading using F = mg.
• The magnetic flux density can be determined from B = F/(IL)

Image credit: Kerboodle OCR A physics textbook