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Momentum
The momentum p of a body = (its mass) x (its velocity).
In symbols .....
p = mv
REMEMBER!
units of p = kgms-1
Newton's 2nd Law states that "the force on a body equals the rate of change of its
momentum".
Or, in symbols .....
F = (change in momentum)/(time taken for change) = p / t
[1]
Thus .....
F = (mv)/t
The mass m of the body is usually constant - in which case the above equation
becomes .....
F = mv/t = ma
Also, from equation [1] above .....
Ft = p = change in momentum
The quantity Ft is called the impulse on the body.
Thus .....
impulse = change in momentum
Since impulse = Ft, it follows that the units of impulse are Ns.
It therefore also follows that Ns is an alternative unit for momentum.
i.e.
1 Ns = 1 kgms-1.
Energy Conservation and Braking
Recall 2 basic principles from HFS .....
work done = energy transferred
[1]
work done = force x distance moved in direction of force = Fs
[2]
In the case of a train (or other vehicle) braking, the "energy transferred" is the loss of
kinetic energy.
It follows that .....
energy transferred = 1/2mu2 - 1/2mv2
[3]
..... where u is the initial velocity (before braking), and v is the final velocity (after
braking).
Putting equations [1], [2] & [3] together .....
1/
2mu
2
- 1/2mv2 = Fs
..... where F is the braking force, and s is the braking distance.
Usually, the final velocity v after braking = 0. In this case .....
1/
2
2mu
= Fs
The above equation can be used to work out the braking distance s, providing the
braking force F, the mass m of the vehicle, and its velocity u are known.
The Motor Effect
A current-carrying conductor placed in a magnetic field experiences a force. This
phenomenon is called the motor effect. The diagram below illustrates the situation
.....
The diagram below shows Fleming's Left Hand Rule, which is used to work out the
relative directions of the force, field and current .....
Remember ..... FBI
!
(Note that F is at right angles to both B and I - but B and I are not necessarily at right
angles to each other.)
The equation below can be used to calculate the size of the force F .....
F = BIlsin
the unit of B is the tesla (T)
l is the length of the current-carrying conductor (wire) within the field.
Very often, the angle  between the current and the field is 90o. In this case .....
F = BIl
Magnetic Flux
We need to define magnetic flux before we can understand the theory of
electromagnetic induction - which is the next topic.
Consider the situation below .....
magnetic flux  = field x area = BA
the unit of  is the weber (Wb)
.....
[1]
i.e.
1 Wb = 1 Tm2
If the circuit shown in the diagram above has N turns (i.e. if it is a coil of N turns) .....
magnetic flux linkage = N = NAB
It follows from equation [1] above that .....
B = /A
i.e. .....
magnetic field = flux per unit area (Wbm-2)
For this reason, magnetic field strength B is often also referred to as magnetic flux
density.
Electromagnetic Induction
Consider again the situation illustrated in the previous section .....
If the flux linkage through the circuit changes, an emf is induced in the circuit which drives an induced current around the circuit. This phenomenon is called
electromagnetic induction.
There are a number of ways in which the flux linkage can change in practice .....
 The strength of the field B may change - possibly because a magnet is being
moved towards or away from the circuit/coil - or because the current in a
neighbouring coil is being changed - which changes the field produced by the
neighbouring coil.
 The effective area A through which the field B passes may change - possibly
because the coil is being rotated.
 The number of turns of wire (N) could change - though this is unlikely!
The rule for working out the size of the induced emf (and current) is called
Faraday's Law, which states .....
size of induced emf = rate of change of flux linkage
..... or, in symbols .....
 = - d(N) / dt
(The "-" sign in the above equation is there for mathematical reasons. Often, we only
want to work out the magnitude of  - in which case the sign can be ignored.)
The rule for working out the direction of the induced emf (and current) is called
Lenz's Law, which states .....
"The direction of the induced emf (or current) is such as to oppose the
change that is inducing it."
The simplest case of electromagnetic induction is a straight wire moving at right
angles to a magnetic field B, as shown below. It can be used to illustrate both
Faraday's Law and Lenz's Law .....
Electromagnetic induction occurs because the area A of the circuit through which the
field B passes is being increased as the wire moves upwards. i.e. The flux through
the circuit is increasing at a rate which depends on how fast the wire is moved. So,
according to Faraday's Law, the size of the induced emf is proportional to the
upwards velocity of the wire.
(In fact:  = Blv
..... where l is the length of the wire, and v is its velocity.)
(Note that a complete circuit (such as that shown above) is needed for there to be an
induced current. However, the induced emf will still exist even if there isn't a circuit.)
Lenz's Law tells us that the direction of the induced current is such as to oppose the
change that is causing it - which, in this case, is the upwards motion. If we apply
Fleming's Left Hand Rule to the induced current shown in the diagram above, we
can indeed see that the motor effect force is downwards - i.e. it opposes the upwards
motion. Conservation of energy also demands that there is a downwards force. If
there were no downwards force it would be possible to induce a current (create
electrical energy) without doing any work!
Note that the above diagram is a mirror image of the motor effect diagram. For this
reason, Fleming's Right Hand Rule can also be used to work out the direction of
the induced current, as shown below .....
Note that Fleming's Right Hand Rule is only useful in a straight wire situation. In
more complex situations Lenz's Law has to be used to work out the direction of the
current.
Eddy current braking is best explained using Lenz's Law. A metal disc spinning on
the same axle as the wheels of a vehicle has a current induced in it when a magnetic
field is applied perpendicular to the plane of the disc. Because of Lenz's Law, this
current causes forces which oppose the rotation. i.e. There is a braking effect.
Transformers
A transformer is a device used to increase (or decrease) an alternating voltage (or
current).
An alternating voltage or current varies with time as shown below .....
Below is a diagram of a transformer .....
Vp is the primary (or input) voltage
Ip is the primary (or input) current
Vs is the secondary (or output) voltage
Is is the secondary (or output) current
The transformer is another good example of Faraday's Law at work. The secondary
voltage is actually an induced emf, which is caused by the fact that the flux linkage
through the secondary coil is changing. (It is changing because the flux comes from
the primary coil - and the current through the primary coil is constantly changing.)
The graphs below illustrate this .....
Or, in other words .....
induced emf = - (the gradient of the graph of flux against time)
The following formula can be used .....
(x Ns)
Ns / Np = Vs / Vp
Np:Ns (or Ns:Np) is often called the turns ratio.
An ideal transformer is one which is 100% efficient, i.e. .....
power in = power out
i.e. .....
IpVp = IsVs

Ip / Is = Vs / Vp (= Ns / Np)
for an ideal transformer
Digital Signals, Feedback and Control
A typical digital signal is shown below .....
Suppose now that the high voltages (1's) are produced by a toothed wheel
connected to a train wheel. (Every time a tooth of the wheel passes a coil it disturbs
the field and produces a "1" say). Assuming the teeth are equally spaced around the
wheel the digital signal will be as shown below .....
If the train goes faster, there will be more 1's per second - i.e. the frequency of the
digital signal will be greater. If we want to measure the speed of the train we need to
count the number of 1's (pulses) in one second (or some other fixed time period).
The system illustrated below will do this .....
Information about the train's speed can then be fed back to the driver (or motors) - so
that the speed can be adjusted accordingly.
The one second pulse (bottom left, above) would be produced by a circuit containing
a capacitor discharging through a resistor - see following section(s) for more details.
Capacitors
A capacitor is essentially 2 metal plates/sheets/foils separated by an insulator which may simply be a layer of air, but could be a sheet of plastic, paper, etc.
The symbol is .....
Suppose a capacitor has a p.d. V applied across it, as shown below .....
Electrons flow onto one plate - which  has a charge -Q on it. Electrons flow off the
other plate - which has an equal and opposite charge +Q on it. The charge stored in
the capacitor is said to be Q.
Its capacitance C is defined by the formula .....
C = Q/V the unit of C is the farad F
REMEMBER!
Charging and Discharging Capacitors
Consider the circuit shown below .....
When the 2-way switch is to the left, the battery is connected to the capacitor.
Current flows in the direction shown by the arrow, and the capacitor charges. The
charging current I has to pass through the resistor R. If R is made larger, I will be
smaller, and the capacitor will take longer to charge. Also, if the capacitor has a
larger capacitance, it will take longer to charge.
When the 2-way switch is to the right the capacitor discharges. The discharge
current is in the opposite direction to the charging current. The discharge current I
also has to pass through the resistor R. So again, if R is made larger, I will be
smaller, and the capacitor will take longer to discharge. Also, if the capacitor has a
larger capacitance, it will take longer to discharge.
Graphs (against time) of the charge Q on the capacitor, the voltage V across it, and
the charging/discharging current are shown below .....
The following points should be noted .....
 4 of the above 6 curves are exponential decay curves. (The Q and V charging
curves are of the form: "a constant minus an exponential".)
 Since Q = CV, it follows that Q is proportional to V. Therefore the shapes of the Q
and V curves are the same.
 The current I is initially high, irrespective of whether the capacitor is charging or
discharging. It falls towards zero as the capacitor charges (or discharges).
 Q0, V0 and I0 are the starting (or maximum) values of charge, voltage and current
respectively.
The equation for the charge against time curve (for discharge) is as follows .....
Q = Q0e-t/RC
The other exponential decay curves have similar formulae .....
V = V0e-t/RC
and
I = I0e-t/RC
The 2 "odd ones out" (the curves for charge and voltage for a charging capacitor)
have the following equations .....
Q = Q0(1 - e-t/RC)
and
V = V0(1 - e-t/RC)
The quantity RC is known as the time constant of the circuit. It is equal to the time
taken for the charge/voltage/current to fall to 1/e (1/2.718) x its initial value.
(Provided R is in  and C is in F, RC will be in seconds. e.g. R = 1 M and C = 1F
gives RC = 1 s.) This is illustrated by the graph below .....
Thus the importance of the time constant is that it gives us some idea of long it will
take the capacitor to discharge (or charge) - or, at any rate, how long it will take the
charge/voltage/current to drop to about 1/3 of its initial value. (It will take an infinite
time (in theory) for the capacitor to discharge completely.)
(For the Q/t and V/t charging curves the time constant tells us how long it will take for
Q or V to reach about 2/3 of Q0 or V0.)
Recall that in the "digital signals, feedback and control" section we required a 1second pulse. It is in fact a fairly simple matter to change the V/t curve we get from
an R-C circuit .....
to .....
..... using other electronic components.
Conservation of Momentum
The principle of conservation of linear momentum states .....
"The total momentum of a system remains constant, provided no external forces act
on the system."
The diagram below shows how this applies to a collision between 2 vehicles (say)
.....
u1 and u2 are the velocities of the 2 vehicles before the collision - and v1 and v2 are
their velocities after.
According to the principle of conservation of linear momentum, the total momentum
of the system (i.e. the 2 vehicles) remains constant.

total momentum before collision = total momentum after

m1u1 + m2u2 = m1v1 + m2v2
A collision is said to be elastic if kinetic energy is also conserved.
i.e. if
1/
2
2m1u1
+ 1/2m2u22 = 1/2m1v12 + 1/2m2v22
If some k.e. is lost (converted to heat, sound etc) the collision is said to be inelastic.
(In this case, the lhs of the above equation would be greater than the rhs.)
(A real collision is unlikely to be perfectly elastic.)
The diagram below shows how conservation of momentum can be applied to a
canon firing a shell .....
V is the recoil velocity of the canon. v is the velocity of the shell. (The velocity of the
system before firing = 0.)
total momentum before firing = total momentum after

0 = -MV + mv

mv = MV
taking velocities to the right as +ve