Welcome to Transport on Track (TRA)!
In this chapter, we explore the physics behind modern rail systems. It’s not just about getting from A to B; it’s about how we use momentum to keep passengers safe, capacitors to control signals, and electromagnetic induction to sense speed and brake without even touching the wheels! Don’t worry if some of the math looks intimidating—we will break it down step-by-step.
1. Stopping Safely: Momentum and Impulse
Trains are massive. When they move at high speeds, they have a huge amount of momentum. Stopping them safely requires understanding the relationship between force and time.
Momentum and Newton’s Second Law
Momentum (\(p\)) is simply the "oomph" an object has. It is calculated as \(p = mv\).
Newton’s Second Law tells us that Force is the rate of change of momentum: \(F = \frac{\Delta p}{\Delta t}\).
The Impulse Equation
If we rearrange that formula, we get Impulse:
\(Impulse = F\Delta t = \Delta p\)
This means the change in momentum depends on both the force applied and how long it is applied for.
Real-world Example: If a train crashes into a solid wall, the time (\(\Delta t\)) is tiny, so the Force (\(F\)) is huge—that’s a recipe for disaster! If the train hits a "crumple zone" or "crash-buffer," the time (\(\Delta t\)) increases, which makes the Force (\(F\) smaller and the impact safer.
Elastic vs. Inelastic Collisions
• Elastic Collision: Kinetic energy is conserved. The objects "bounce" perfectly.
• Inelastic Collision: Kinetic energy is not conserved. It’s transferred into heat or sound, or used to deform the objects (like a train car crumpling during a crash). In a "perfectly inelastic" collision, the objects stick together.
Quick Review: To reduce the force in a crash, you must increase the time it takes for the momentum to change. This is why "crash-proofing" is all about making impacts last longer.
2. Signalling and Control: Capacitors
Railways use electrical circuits to know where trains are. Capacitors play a huge role in these control systems by storing and releasing electrical charge.
What is Capacitance?
Think of a capacitor as a "bucket" for electrical charge. The capacitance (\(C\)) tells you how much charge (\(Q\)) it can hold per volt (\(V\)) applied:
\(C = \frac{Q}{V}\)
It is measured in Farads (F).
Discharging a Capacitor
When a capacitor discharges through a resistor, it doesn't empty all at once. It empties exponentially. This means it loses a large chunk of its charge quickly at first, then slows down.
The amount of charge (\(Q\)) remaining after time \(t\) is:
\(Q = Q_0 e^{-t/RC}\)
The same pattern applies to current (\(I\)) and potential difference (\(V\)):
\(I = I_0 e^{-t/RC}\)
\(V = V_0 e^{-t/RC}\)
The Time Constant (\(RC\))
The time constant is simply \(R \times C\). It tells us how long the discharge takes.
Memory Aid: After one time constant (\(t = RC\)), the charge has dropped to about 37% of its original value. A bigger resistor or a bigger capacitor means a longer time constant—like a bigger bucket with a smaller hole takes longer to empty!
Using Logs for Graphs
If you see a natural log equation like \(\ln V = \ln V_0 - \frac{t}{RC}\), don't panic! This is just a way to turn a curvy exponential graph into a straight line (\(y = mx + c\)). If you plot \(\ln V\) on the y-axis and \(t\) on the x-axis, the gradient will be \(-\frac{1}{RC}\).
Quick Review: Capacitors store charge. The discharge is exponential and is governed by the time constant \(RC\). Larger \(R\) or \(C\) equals a slower discharge.
3. Sensing and Braking: Electromagnetism
How do we measure a train's speed without mechanical parts wearing out? We use Magnetic Flux.
Magnetic Flux and Linkage
• Magnetic Flux (\(\phi\)): Think of this as the number of magnetic field lines passing through an area. It’s calculated as \(\phi = BA\), where \(B\) is magnetic flux density.
• Flux Linkage (\(N\phi\)): If you have a coil with \(N\) turns of wire, the total flux "linked" to the coil is \(N \times \phi\).
Faraday’s Law and Lenz’s Law
When a train passes over a sensor, the magnetic field through the sensor's coil changes. This induces an electromotive force (e.m.f.).
Faraday's Law: The magnitude of the induced e.m.f. is proportional to the rate of change of flux linkage:
\(\mathcal{E} = -\frac{d(N\phi)}{dt}\)
Lenz's Law: The minus sign in that formula represents Lenz's Law. It means the induced e.m.f. will always try to oppose the change that created it.
Analogy: Lenz’s law is like a stubborn teenager. If you try to push a magnet into a coil, the coil creates a magnetic field to push it back. If you try to pull it away, the coil tries to pull it back in!
Eddy Current Braking
This "stubbornness" is used for braking! If you move a strong magnet over a metal rail, "Eddy currents" are induced in the rail. Because of Lenz’s Law, these currents create a magnetic field that opposes the motion of the train, slowing it down without any friction or touching! This is Eddy Current Braking.
Force on a Wire
If we have a current-carrying wire in a magnetic field, it feels a force:
\(F = BIl \sin\theta\)
We use Fleming’s Left-Hand Rule to find the direction:
• First Finger = Field
• seCond finger = Current
• Thumb = Thrust (Force)
Key Takeaway: Changing magnetic fields induce electricity (Faraday). The induced electricity tries to stop the change (Lenz). We use this for speed sensors and frictionless brakes.
4. Powering the Track: Alternating Current (AC)
Most trains run on AC. But AC is constantly changing, so how do we measure it effectively?
Peak vs. R.M.S. Values
AC voltage looks like a sine wave. The highest point is the Peak Value (\(V_0\)). However, since the voltage spends most of its time below the peak, we use an average called the Root-Mean-Square (r.m.s.) value for power calculations.
To find the r.m.s. value:
\(V_{rms} = \frac{V_0}{\sqrt{2}}\)
\(I_{rms} = \frac{I_0}{\sqrt{2}}\)
Did you know? When we say the UK mains is 230V, that is the r.m.s. value. The actual peak voltage is actually about 325V!
Quick Review: Peak is the maximum height of the wave. R.M.S. is the "effective" value used for power. Use \(\sqrt{2}\) to convert between them.
Common Mistakes to Avoid
• Impulse Units: Remember that Impulse is measured in \(N s\) (Newton-seconds) or \(kg m s^{-1}\). They are the same thing!
• The Exponential "e": When calculating capacitor discharge, make sure you use the \(e^x\) button on your calculator, not "10^x".
• Lenz's Law Direction: Students often forget that the induced field opposes the change. If the flux is increasing, the induced field points the opposite way.
• Degrees vs. Radians: When using \(F = BIl \sin\theta\), check if your calculator is in Degrees!
Don't worry if this seems tricky at first—physics is all about seeing the patterns in how the world moves. Keep practicing those equations, and you'll be on the right track!