Question

Cant \(C\) on a Broad Gauge railway track is calculated for an equilibrium speed \(V\) (in km/h), dynamic gauge \(G\) (in mm) and radius of curve \(R\) (in m) using formula/formulae:

• A. \(C = \dfrac{GV^2}{127R}\)
• B. \(C = \dfrac{13.76\,V^2}{R}\)
• C. \(C = \dfrac{13.20\,V^2}{R}\)
• D. \(C = \dfrac{8.33\,V^2}{R}\)

Hint:

Start from \(C=GV^2/127R\) and substitute the Broad Gauge dynamic gauge \(G \approx 1750\) mm; check which options this matches.

Answer 1

The Correct Option is A, B

Step 1: Start from the physical requirement for cant.
On a curved track, a vehicle moving at speed $V$ experiences an outward centrifugal force. Cant (superelevation) raises the outer rail so that, at the chosen equilibrium speed, the resultant of gravity and the centrifugal force lies along the normal to the rail plane, giving passengers a comfortable, wear-free ride.

Step 2: Set up the force triangle.
For a small cant angle $\theta$, $\tan\theta \approx C/G$, where $C$ is the cant and $G$ is the (dynamic) gauge, both in the same length unit. Equilibrium of forces along the incline gives:
$$\tan\theta = \frac{V^2}{gR}$$
with $V$ in m/s, $R$ in m and $g = 9.81\,\text{m/s}^2$. Equating the two expressions for $\tan\theta$:
$$\frac{C}{G} = \frac{V^2}{gR}, \quad \text{so} \quad C = \frac{G V^2}{gR}$$

Step 3: Convert units to match the given formula.
Converting $V$ from km/h to m/s introduces a factor of $(1000/3600)^2 = 1/12.96$, so:
$$C = \frac{G}{g \times 12.96} V^2 = \frac{G}{127} V^2 \quad \text{(with $C$, $G$ in mm, $V$ in km/h, $R$ in m)}$$
This reproduces option (A) exactly: $C = GV^2/127R$, so option (A) is a correct general formula, valid for any gauge as long as the matching value of $G$ is used.

Step 4: Plug in the Broad Gauge dynamic gauge value.
For BG track, $G \approx 1750$ mm, so:
$$C = \frac{1750}{127}\frac{V^2}{R} \approx 13.78\frac{V^2}{R} \approx 13.76\frac{V^2}{R}$$
matching option (B). This confirms option (B) is just the BG-specific case of option (A), and so it is also correct.

Step 5: Rule out options (C) and (D).
Option (C), $13.20V^2/R$, and option (D), $8.33V^2/R$, come from substituting smaller dynamic-gauge values, appropriate to Metre Gauge or Narrow Gauge track, not to Broad Gauge; since $G_{BG} \approx 1750$ mm is fixed by the question, these two constants do not apply here.
$$\boxed{\text{The correct formulae for BG cant are options (A) and (B).}}$$