Question

Two wheels are constructed, as shown in Figure, with four spokes. The wheels are mounted one behind the other so that an observer normally sees a total of eight spokes but only four spokes are seen when they happen to align with one another. If one wheel spins at 6 rev/min, while other spins at 8 rev/min in same sense, how often does the observer see only four spokes?

• A. 4 times a minute
• B. 6 times a minute
• C. 8 times a minute
• D. Once in a minute

Answer 1

The correct option is C 8 times a minute

The observer will see only four spokes if the difference of angular

displacements of wheel 1 \& 2 is

$n\frac{2\pi }{4}=n\frac{\pi }{2}$ Difference in angular displacement = (8rev/min - 6rev/min)\times t
$=2\times 2\pi \times t=4\pi \times t\left(tinminutes\right)$ The least time after which observer will see only four spokes
is given by

$4\pi \times t=\left(1\right)\frac{\pi }{2}$ \therefore t= 1/8 min Therefore no of times observer will see only four spokes in a min= 8
We can also think of this in following way: No of times observer will see only four spokes in a time period will be the value of n as it will tell us the number of $\left(\frac{\pi }{2}\right)$ degrees rotations that the wheels have

taken relative to each other. And everytime they takes a $\frac{\pi }{2}$ degree rotation relative to each other

observer will see only four spokes once.

In 1 min $2\times 2\pi =4\pi =8\frac{\pi }{2}$ . Therefore 8 times a min.