Question

The wheel of a railway carriage is 3 feet in diameter and makes 3 revolutions in a second; how fast is the train going? \[\]

Answer 1

Hint: We find the radius $r$ of the wheel by taking of the diameter given in the question. We find its circumference $C=2\pi r$. We use the fact that the wheel has to pass its circumference to complete 1 revolution and find the distance covered per second by multiplying number of revolutions to $C$. We find how fast is the train by using the definition speed that is distance covered per second. \[\]

Complete step by step answer:
Let us denote the radius of the railway carriage as $r$. We know that the wheel is in circular shape and all the wheels are congruent. We are given the question that the diameter of the wheel of the railway carriage is 3 feet. We know that radius is half of the diameter. So we have
\[r=\dfrac{3}{2}=1.5\text{feet}\]
Let us denote the circumference of the wheel of the railway carriage as $C$. We know that the circumference of a circle is given by $C=2\pi r$. So circumference of the wheel is
\[C=2\times \pi \times r=2\times \pi \times 1.5=3\pi \text{feet}\]
We observe that a wheel when it rotates by 1 revolution it covers a distance of its circumference that is $C$. \[\]
We observe it from the above diagram where the point P returns back to its original position to make 1 revolution. It has covered the circumference of the wheel to complete one revolution. \[\]
We are given the question that makes 3 revolutions in a second. It means the wheel covers three times its circumference in a second. So the amount of distance say $v$ the wheel covers in feet per second is
\[v=3\times C=3\times 3\pi =9\pi \]
We are asked the question how fast the train is going, which means the speed of the train. The speed of the train is equal to the speed of each wheel in the carriage. Speed is defined as the amount of distance an object covers per unit of time. Here the unit time is second and per second the wheel covers $9\pi $feet.\[\]
We take $\pi =3.14$ and find the speed of the train in metre (1feet=.3048mtre) per second as
\[v=9\times \pi =9 \times 0.3048\times 3.14=8.62\]

So, the correct answer is “8.62m/s”.

Note: The amount of distance covered per unit of time is called linear velocity and is conventionally measured in metre per second. The angle covered by the wheel per unit of time is called angular velocity $\omega $ and is conventionally measured in radian per second. The number of revolutions per second is called frequency and is given by $f=\dfrac{1}{\omega }$.