Question

A wheel completes 2000 revolutions to cover the 9.5 km distance, then the diameter of the wheel is:
(A) 1.5 m
(B) 1.5 cm
(C) 7.5 cm
(D) 7.5 m

Answer 1

Hint: In making one revolution, the wheel covers an angular distance of 2$\pi$. The circumference of the circle is exactly the distance covered in one revolution. The total distance can be known by this and can be equated with the given value of 9.5 km.

Complete answer:
We are given that the wheel makes 2000 revolutions. This multiplied by the circumference of the circle is the distance the wheel is supposed to cover;
${\rm{distance = 2000}} \times {\rm{(2}}\pi {\rm{r)}}$
If we just replace the 2r above with d for diameter, we get:
${\rm{distance = 2000}} \times {\rm{(}}\pi d{\rm{)}}$.
Keeping the value of distance as 9.5 km, we get:
$ \dfrac{9500m}{2000\pi }$ = $d$
This will give the diameter as:
d = 1.512 m approximately.

Therefore the correct option is (A).

Additional Information:
In rolling without slipping, a body translates the same distance that the outer circumference of the body covers. If the body moves such that a point on the rim appears back at the same point where it started, we say that body has translated by the distance equal to its circumference.

Note:
Two of the options given are in cm and two are in m. So, one should carefully replace the km with 1000 m while solving else the order obtained will be very different. The concept used here is in general very useful in understanding rotational motion. The distance the wheel covers in translation is entirely given by the number of times the whole circumference of the wheel rotates to cover that distance. If rolling was missed anywhere or it had undergone skidding anywhere, the revolution would have not been a true measure of the distance covered.