Question

An engine of mass m is moving up a slope of inclination θ at a speed v. The coefficient of friction between engine and the rail is μ. If the engine has an efficiency η then the energy spent by engine in time t is

• A. ηmg (sin η + μ cosη) vt
• B. $\frac { \operatorname { mg } ( \mu \cos \theta ) \mathrm { vt } } { \eta }$
• C. $\frac { m g ( \sin \theta + \mu \cos \theta ) \mathrm { vt } } { \eta }$
• D. $\frac { \mathrm { mg } } { 2 } \left( \frac { \sin \theta } { \eta } \right) \mathrm { vt }$

Answer 1

Answer: (C)

$\frac { m g ( \sin \theta + \mu \cos \theta ) \mathrm { vt } } { \eta }$

Answer 2

Force exerted by engine = mg sin θ + μ mg cos θ = mg (sin θ + μ cos θ)

Power exerted by engine = Fv

Work done by engine = Fvt

Efficiency of engine η = $\frac { \text { output } } { \text { input } } = \frac { \text { Fvt } } { \text { input } }$

∴ Input of engine = $\frac { \mathrm { Fvt } } { \eta } = \frac { \operatorname { mg } ( \sin \theta + \mu \cos \theta ) \mathrm { vt } } { \eta }$