Solved Problem on Electric Current

A 100-ton electric locomotive is driven by four electric engines operating with a potential difference of 1000 volts and moves at a speed of 72 km/h. Assuming the friction coefficient between the wheels of the locomotive and the rails equal to 0.5 and the free-fall acceleration g=10 m/s², determine:
a) The electric current circulating on each engine;
b) If in each engine we have a current equal to 4000 amperes, how many railroad cars, 15 tons each, will the locomotive be able to pull?

Problem data:

Mass of the locomotive: M = 100 t;
Mass of a railroad car: M_R = 15 t;
Speed of locomotive: v = 72 km/h;
Coefficient of friction: μ = 0.5;
Voltage line: U = 1000 V;
Free-fall acceleration: g = 10 m/s².

Problem diagram:

Figure 1

Solution

First, we convert the unit of mass given in tons (t) to kilograms (kg), and the speed is given in kilometers per hour (km/h) to meters per second (m/s) used in the International System of Units (S.I.).

\[ \begin{gather} M=100\;\cancel{\text{t}}\times \frac{1000\;\text{kg}}{1\;\cancel{\text{t}}}=100\times 1000\;\text{kg}=100000\;\text{kg}\\[10pt] M_{R}=15\;\cancel{\text{t}}\times \frac{1000\;\text{kg}}{1\;\cancel{\text{t}}}=15\times 1000\;\text{kg}=15000\;\text{kg}\\[10pt] v=\frac{72\;\cancel{\text{km}}}{1\;\cancel{\text{h}}}\times \frac{1000\;\text{m}}{1\;\cancel{\text{km}}}\times \frac{1\;\cancel{\text{h}}}{3600\;\text{s}}=\frac{720\;\text{m}}{36\;\text{s}}=20\;\text{m/s} \end{gather} \]

a) The power generated by one of the engines will be given by

\[ \begin{gather} \bbox[#99CCFF,10px] {\mathscr{P}=Ui} \tag{I} \end{gather} \]

The total power generated by the four engines will be

\[ \begin{gather} \mathscr{P}_{T}=4\mathscr{P} \tag{II} \end{gather} \]

substituting the expression (I) into (II), we have

\[ \begin{gather} \mathscr{P}_{T}=4Ui \tag{III} \end{gather} \]

From Classical Mechanics, we have the total power required to make the locomotive move itself

\[ \begin{gather} \bbox[#99CCFF,10px] {\mathscr{P}_{T}=Fv} \tag{IV} \end{gather} \]

equating expressions (III) and (IV)

\[ \begin{gather} 4Ui=Fv \tag{V} \end{gather} \]

So the locomotive move, the engines should overcome the force of friction. In Figure 1, the train wheels push the rails back with the force \( \vec{F} \), the rails react to the wheels with the force of friction making the train move itself (Newton's Third Law)

\[ \begin{gather} F=F_{f}=\mu N \tag{VI} \end{gather} \]

substituting the expression (VI) into (V), we obtain

\[ \begin{gather} 4Ui=\mu Nv \tag{VII} \end{gather} \]

Figure 2

The normal \( \vec{N} \) and the weight \( \vec{W} \) of the locomotive cancel out (Figure 2)

\[ \begin{gather} N=W \tag{VIII} \end{gather} \]

substituting the expression (VIII) into (VII)

\[ \begin{gather} 4Ui=\mu Wv \tag{IX} \end{gather} \]

the weight is given by

\[ \begin{gather} \bbox[#99CCFF,10px] {W=Mg} \tag{X} \end{gather} \]

substituting the expression (X) into (IX), we have

\[ \begin{gather} 4Ui=\mu Mgv\\ i=\frac{\mu Mgv}{4U} \tag{XI} \end{gather} \]

substituting the values given in the problem, we obtain

\[ \begin{gather} i=\frac{0.5\times 100000\times 10\times 20}{4\times 1000}\\ i=\frac{10000000}{4000} \end{gather} \]

\[ \bbox[#FFCCCC,10px] {i=2500\;\text{A}} \]

b) Applying the expression (XI) to the current and using the data of this item, we have the total mass M_T that can be driven by the engines

\[ \begin{gather} M_{T}=\frac{4Ui}{\mu gv}\\M_{T}=\frac{5\times 1000\times 4000}{0.5\times 10\times 20}\\ M_{T}=\frac{16000000}{100}\\ M_{T}=160000\;\text{kg} \end{gather} \]

Figure 3

Of this total mass, we have 100 000 kg represent the mass of the locomotive itself, then leftover for the railroad cars \( 1600000-100000=60000\;\text{kg} \),

\[ 1600000-100000=60000\;\text{kg,} \]

as each railroad car has a mass of 15 000 kg, the number of railroad cars will be

\[ n=\frac{60000\;\cancel{\text{kg}}}{15000\frac{\cancel{\text{kg}}}{\text{railroad car}}} \]

\[ \bbox[#FFCCCC,10px] {n=4\;\text{railroad cars}} \]

Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .