A railway train is travelling on a circular curve of \(1500\) metres radius at the rate of \(66\) km/hr. Through what angle has it turned in \(10\) seconds?
Solution:
Radius of the curve:
\[ r=1500 \text{ m} \]
Speed of the train:
\[ 66 \text{ km/hr} \]
Convert speed into m/s:
\[ 66\times\frac{1000}{3600} \]
\[ \frac{55}{3} \text{ m/s} \]
Distance travelled in \(10\) seconds:
\[ s=\frac{55}{3}\times10 \]
\[ s=\frac{550}{3} \text{ m} \]
Using,
\[ s=r\theta \]
\[ \theta=\frac{s}{r} \]
\[ \theta=\frac{550/3}{1500} \]
\[ \theta=\frac{11}{90} \]
Therefore, the angle turned by the train in \(10\) seconds is:
\[ \boxed{\frac{11}{90} \text{ radians}} \]