Question

A railway train is travelling on a circular curve of $1500$ metres radius at the rate of $66km/hr$. Through what angle has it turned in $10$ seconds?

Answer 1

We have, In circular track,

$OA=OB=r=1500m$

$\angle AOB=\theta$ = angle the train turns in 10 seconds

Speed of train $=66km/hr$

$=\frac{66\times 1000}{60\times 60}m/sec$ [Since, $1hr=60\times 60$ seconds]

$=\frac{110}{6}m/sec$

Since, $\text{speed}=\frac{\text{Distance}}{\text{Time}}$
So, distance travelled in $10$ sec $=\text{Speed}\times \text{Time}$
$\therefore$ Train will travel in 10 sec
$=\frac{110}{6}\times 10=\frac{1100}{6}m$
As train travelled in circular motion, so the distance becomes arc length of the circle.
$\therefore \text{arc~A}=\frac{1100}{6}m$

Thus, $\theta =\frac{\text{arc}}{\text{radius}}=\frac{1100}{6\times 1500}=\frac{11}{90}\text{radian}$

$\therefore$ The train will turn by $\left(\frac{11}{90}\right)$ radians angle in 10 sec.