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Question

Answers

A. $4$

B. $8$

C. $6$

D. $2$

Answer

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The gear ratio of the wheels is given by formula

\[{\text{Gear ratio}} = \dfrac{{{\text{Radius of driving wheel}}}}{{{\text{Radius of driven wheel}}}}\] …… (1)

The gear ratio of the wheels is also given by formula

\[{\text{Gear ratio}} = \dfrac{{{\text{Number of teeth driving wheel}}}}{{{\text{Number of teeth driven wheel}}}}\] …… (2)

We have given that the radius \[{r_A}\] of the driving wheel is \[2\,{\text{cm}}\] and the radius \[{r_B}\] of the driven wheel is \[20\,{\text{cm}}\].

\[{r_A} = 2\,{\text{cm}}\]

\[ \Rightarrow{r_B} = 20\,{\text{cm}}\]

Let us calculate first the gear ratio.We can determine the gear ratio using equation (1).Substitute \[{r_A}\] for radius of the driving wheel and \[{r_B}\] for radius of the driven wheel in equation (1).

\[{\text{Gear ratio}} = \dfrac{{{r_A}}}{{{r_B}}}\]

Substitute \[2\,{\text{cm}}\] for \[{r_A}\] and \[20\,{\text{cm}}\] for \[{r_B}\] in the above equation.

\[{\text{Gear ratio}} = \dfrac{{2\,{\text{cm}}}}{{20\,{\text{cm}}}}\]

\[ \Rightarrow {\text{Gear ratio}} = \dfrac{1}{{10}}\]

Hence, the gear ratio for the given system of the driving wheel and driven wheel is \[\dfrac{1}{{10}}\].We have given that the number of teeth in the driven wheel is 40.

\[{n_B} = 40\].We have asked to determine the number of teeth \[{n_A}\] in the driving wheel.Substitute \[{n_A}\] for the number of teeth in the driving wheel and \[{n_B}\] for the number of teeth in the driven wheel in equation (2).

\[{\text{Gear ratio}} = \dfrac{{{n_A}}}{{{n_B}}}\]

Rearrange the above equation for \[{n_A}\].

\[{n_A} = {n_B}\left( {{\text{Gear ratio}}} \right)\]

Substitute 40 for \[{n_B}\] and \[\dfrac{1}{{10}}\] for \[{\text{Gear ratio}}\] in the above equation.

\[{n_A} = 40\left( {\dfrac{1}{{10}}} \right)\]

\[ \therefore {n_A} = 4\]

Therefore, the number of teeth in the driving wheel are 4.

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