Question

It has been mentioned that two gear wheels which are meshed together are having a radius of $0.5cm\text{ and }1.5cm$. What will be the number of revolutions made by the smaller one when the larger one goes through 3 revolutions?
\[\begin{align}
  & A.5\text{ revolution} \\
 & B.20\text{ revolution} \\
 & C.1\text{ revolution} \\
 & D.10\text{ revolution} \\
\end{align}\]

Answer 1

Hint: The angular velocity can be found by taking the ratio of the product of the number of turns and circumference of the gear wheel to the time period of the revolution. This angular velocity will be the same in the question. Substitute the conditions in the question and find the answer. This will help you in answering this question.

Complete step by step solution:
The angular velocity can be found by taking the ratio of the product of the number of turns and circumference of the gear wheel to the time period of the revolution. That is we can write that,
\[\omega =\dfrac{n2\pi r}{T}\]
For the first gear wheel we can write that,
\[\omega =\dfrac{{{n}_{1}}2\pi {{r}_{1}}}{T}\]
And for the second gear wheel we can write that,
\[\omega =\dfrac{{{n}_{2}}2\pi {{r}_{2}}}{T}\]
As the angular velocity is same in both the wheels, we can compare the conditions as,
\[\dfrac{{{n}_{1}}2\pi {{r}_{1}}}{T}=\dfrac{{{n}_{2}}2\pi {{r}_{2}}}{T}\]
Let us cancel the common terms in the equation. That is,
\[{{n}_{1}}{{r}_{1}}={{n}_{2}}{{r}_{2}}\]
As we already know, the number of turns of the second wheel is given as,
\[{{n}_{2}}=3\]
The radius of the first wheel will be given as,
\[{{r}_{1}}=0.15cm\]
The radius of the second wheel will be,
\[{{r}_{2}}=0.5cm\]
Substituting the values in the equation will give,
\[{{n}_{1}}\times 0.15=3\times 0.5\]
Rearranging the equations, we can write that,
\[{{n}_{1}}=\dfrac{3\times 0.5}{0.15}=10\text{ revolutions}\]
Therefore the correct answer has been obtained.

It has been mentioned as option D.

Note:
Angular velocity is defined as the measure of an object rotation or revolution with respect to another point. That is it is how fast the angular position or orientation of a body varies with time. There are two kinds of angular velocity. One is the orbital angular velocity and another one is spin angular velocity.