Home | TRA Notes

MomentumThe 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 = mv/t = ma

Also, from equation [1] above .....

     Ft = p = change in momentum

The quantity Ft is called the impulse on the body.

Thus .....

     impulse = change in momentum

Since impulse = Ft, 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 BrakingRecall 2 basic principles from HFS .....

     work done = energy transferred     [1]

     work done = force x distance moved in direction of force = Fs     [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 .....

     1/2mu2 - 1/2mv2 = Fs

..... where F is the braking force, and s is the braking distance.

Usually, the final velocity v after braking = 0. In this case .....

     1/2mu2 = Fs

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 EffectA 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 FluxWe 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 InductionConsider 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.

TransformersA 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)     (x Ns)

The following formula can be used .....

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 ControlA 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.

CapacitorsA 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 CapacitorsConsider 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 = 1F 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 1-second 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 MomentumThe 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     1/2m1u12 + 1/2m2u2

2 = 1/2m1v12 + 1/2m2v2

2

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     taking velocities to the right as +ve

     mv = MV