| 3.Stopping and Starting |
The product between mass and velocity is momentum:
momentum (kg m s-1) = mass (kg) × velocity (m s-1)
In Physics Code:
p = mv
Momentum is a vector quantity with units of kilogram metres per second (kg m s-1). Unlike most physics quantities, you can't feel it or see it. However it is very useful in explaining collisions.
We are familiar with the relationship that sums up Newton's Second Law:
F = ma
If we consider the make up of this statement we can write:
Force = mass × acceleration
Force = mass × change in velocity
time interval
It is very reasonable to write that the product of mass × change in velocity is the change in momentum. We can therefore write:
Dp = mDv
So we can rewrite Newton II as:
F = Dp
Dt
We can rearrange the equation to give:
Dp = FDt
The term Dp is the change in momentum or impulse. We can also consider the impulse to the product between force and time interval. Therefore impulse can have the units newton seconds (N s) which is the same as kilogram metres per second (kg m s-1).
A big force applied for a short time interval will have the same impulse as a small force applied for a long time interval.
The area under this graph is the impulse. You can see that the area of both rectangles is the same.
You will be familiar with the concept of work and energy. You will be able to recall the expression:
DW = FDs
When the brakes are applied, a force opposing the forward movement of the train results in negative acceleration until the train stops. The moving train has kinetic energy which is transferred to heat by the brakes. When trains had metal brake shoes, you could see them glowing red hot and sparks flying from them.
We can describe the energy transfer as:
Energy transfer = kinetic energy at the end - kinetic energy at the start
DE = (1/2 mv2) - 1/2(mu2)
In terms of force we can also write:
DE = -FDs
The minus sign tells us that the force is in the opposite direction to the movement. Since the train comes to a halt, the kinetic energy at the end is zero. so we can write:
-FDs = - 1/2(mu2)
The minus signs cancel out and we can rearrange the equation to give:
Ds = mu2
2F
From this relationship we can easily see that if the speed of the train is doubled, the stopping distance goes up four times.
Think about a locomotive that has stopped at a signal on a hill. The slope on the hill as been considerably exaggerated to show the idea. The angle of the hill is q.
The forces acting on the locomotive are:
The weight that always acts vertically downwards. It is the resultant of:
The normal force, a force acting at 90 o to the rail;
The downhill force acting parallel to the rail.
We know that weight is given by:
W = mg
So we can resolve the other two vectors as:
Normal force
Fx = mg cos q
Downhill force
Fy = mg sin q
When the locomotive starts we may have to take into account the drag force of the train, Fd.
So the force produced by the traction motors will have to be:
Ftot = Fd + mg sin q
If we put a wire in a magnetic field and pass a current through it, we see that the wire moves. It has a force acting on it. It does not matter whether the wire is made of a magnetic or non-magnetic material. The direction of the movement is governed by Fleming's Left Hand Rule.
You will remember that:
Thumb gives the movement;
First Finger gives the field;
Second Finger gives the current.
The three components are at 90 o to each other.
The force that acts on the wire is governed by three factors:
The current flowing in the wire;
The strength of the magnetic field;
The length of wire in the magnetic field.
These can be combined to give a simple formula:
F = BIl
The term B is the magnetic field strength, which is measured in Tesla (T).
1 T = 1 N A-1 m-1
The magnetic field strength can be thought of as the density of the field lines in a given area. The closer the lines are together, the higher the magnetic field strength.
If the wire is at an angle q to the magnetic field lines, the formula takes this into account by being modified to:
F = BIl sin q
The electric motor uses this idea to convert the force on the wire into rotary motion. The moving part is called the armature. The picture here shows the armature from a locomotive's traction motor.
The drain pipe gives an idea of the scale. In the foreground you can see the commutator which consists of many segments. Current is brought to the commutator by brushes. The whole thing rotates in a magnetic field produced by field coils. This motor would take a current of about 500 A at 750 V.
If we can get a wire to move by passing a current through a magnetic field, it is reasonable to suppose that if we move the wire in the magnetic field, we will get a current. This is the case, provided the wire is connected to an outside circuit. If the wire is not connected, we still get charge separation that leads to a voltage, called the electromotive force (emf). It's not a force really. The emf is given the physics code E (Curly E). [You can get this on your computer using a font called Kunstler Script.]
We find that the emf is proportional to:
Magnetic Field Strength;
How fast the wire is moved;
Number of turns in the coil.
It does NOT matter whether we move the wire or move the magnet; we still get the same emf.
We don't even need to move the wire; if we make an electromagnet and mount a second coil on the electromagnet, we get an emf when we turn the current on or off. If we use an alternating current, whose value and direction is constantly changing, we induce an alternating emf.
In other words we induce an emf by changing the magnetic flux.
We need to write down some definitions before we see how they are related:
Magnetic flux - the total number of field lines in a magnetic field.
Magnetic flux density - how many field lines there are in a given area.
Magnetic flux linkage - the product between the magnetic flux and the number of turns of wire.
The picture shows the idea of magnetic flux density
The field lines pass through a loop of area A. The total flux is given by:
magnetic flux (Wb) = magnetic flux density (T) × area (m2)
In Physics code:
F = BA
The symbol F is "Phi" a Greek capital letter "Ph" (or "F").
The units for magnetic flux are Weber (Wb).
1 Wb = 1 T m2 = 1 N A-1 m
If the loop is tilted at an angle, we get:
The relationship becomes:
F = BA cos q
The induced emf is proportional to the rate of change in flux. We can write this as:
E µ DF
Dt
If a coil contains N loops of wire this becomes:
E µ NDF
Dt
The product NF is the magnetic flux linkage. The units are Weber (or Weber-turns).
Lenz's Law states:
The direction of the current induced in a conductor by moving it relative to a magnetic field is such that its own field opposes the motion.
In other words, then a load is applied to a coil, the coil acts as a motor in the opposite direction. The bigger the load, the harder the coil is to move.
This is shown by a minus sign to our relationship:
E = -D(NF)
Dt
Consider a wire of length l, moving through a magnetic field of strength B:
In this picture the magnetic field lines are going vertically into the page (or screen). The wire is traveling from left to right at a constant speed of v m/s. In a time of Dt seconds the wire moves through a distance of vDt metres.
So the area covered per second is:
DA = lvDt
An emf will be induced according to Lenz's Law:
E = -D(NF)
Dt
Since F = BA, we can rewrite this as:
E = -D(NBA)
Dt
And we can substitute for A:
E = -D(NBlvDt)
Dt
The Dt terms cancel out and N = 1, so we can write:
E = Blv
In this derivation, don't get too hung up on the delta notation, nor the minus signs. The equation above is what you want to achieve.
The direction of the current in the complete circuit is determined by Fleming's Right Hand rule:
As before:
Thumb gives the movement;
First Finger gives the field;
Second Finger gives the current.
The three components are at 90 o to each other.
If a piece of metal is moved past a magnetic fields, localised currents are induced in the metal itself. These are called eddy currents as they resemble eddies in a flow of water. The potential difference is very small, but the values of the currents are quite high and will cause significant heating.
In the school physics laboratory, eddy currents can be demonstrated by:
dropping a magnet through a copper pipe, and comparing it with a plastic pipe of the same diameter.
swinging a metal sheet as a pendulum between the poles of a large magnet. Compare that with a plastic sheet of the same mass, and a metal sheet with slots cut in it.
In transport, electric retarders on buses work by large metal disk running between the poles of a powerful electromagnet which is switches on as the driver applies the brakes. As the speed reduces, the effect reduces, and normal brakes are used to stop the vehicle. However this machine reduces quite considerably the wear and tear on the brakes.
Electric trains use their traction motors as generators while coming down a hill, or braking to a halt.
In regenerative braking the current from the motors is fed back to the overhead wires and can be used to provide energy for trains going uphill. This is easy to do on direct current lines, but not so easy (but not impossible) on alternating current lines.
In rheostatic braking, the current is fed to a very large variable resistor. The energy is dissipated as heat.
In both these cases, the braking effect is reduced as the speed gets less. Then the normal brakes are applied.
Electric locomotives take their power from a variety of systems. In most modern electrified railways the power is supplied to the locomotive by a 25 000 V overhead line at 50 Hz ac. However the motors work at about 1500 V, so there needs to be a way of reducing the voltage. This could be done using a resistor, but it would be extremely wasteful. So it's done with a transformer.
The transformer is a machine that is simplicity itself. It consists of:
A primary coil connected to the alternating power source. This provides the changing magnetic field.
A secondary coil connected to the load (in this case, the rectifier and control equipment in the locomotive).
A laminated soft iron core.
The two coils are electrically completely different circuits. Either of the coils can act as a primary. The laminated core is made up of layers of soft iron separated by an insulating layer of varnish or glue. This reduces losses from eddy currents. Soft iron is NOT soft like putty; it is heavy and hard. However "soft" means that it loses its magnetism immediately the current is turned off. Therefore the magnetic field can change forwards to backwards as the current changes.
The ratio of the input voltage to the output voltage is the same as the ratio of the number of turns on the primary to the number of turns on the secondary. We can write this as:
Number of turns on the primary = Primary Voltage
Number of turns on the secondary Secondary voltage
In Physics code:
N1 = V1
N2 V2
If N1 is greater than N2, we have a step-down transformer, because the voltage is reduced. A step-up transformer increases the voltage.
If a transformer is 100 % efficient (and it nearly is) we can say that:
power in = power out
V1I1 = V2I2
Therefore we can say that when the voltage is lower, the current is bigger. We can rewrite the transformer equation in terms of current to give us:
N1 = I2
N2 I1
In practice, the transformer is about 97 % efficient. When a large transformer is transferring a lot of energy, even 3 % losses produce a fair amount of heat. Therefore the transformer is cooled with oil which is pumped to heat exchangers.
Transformers can only work with alternating current; they cannot work with direct current. If a locomotive passes from 25 000 V ac to 3000 V dc a different solution is needed to step down the voltage. The 1500 V motors are put in series.
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