A gear train with five gears shown in figure. The idler gear in this gear train is; (N₁, N₂ … N₅ are number of teeth on respective gear).

Correct Answer: Gear 4

Detailed Explanation:

An idler gear is an intermediate gear in a gear train that does not affect the overall velocity ratio (speed ratio) of the train. Its primary purposes are to change the direction of rotation of the driven gear and to bridge the distance between shafts without requiring extremely large gears.

Let us analyze the gear train layout shown in the diagram:
1. Gear 1 is the driver gear with $N_1$ teeth.
2. Gear 2 (with $N_2$ teeth) and Gear 3 (with $N_3$ teeth) share a common center (indicated by the single "+" symbol). This means they are rigidly mounted on the same shaft, forming a compound gear. Therefore, their rotational speeds are equal:
${\omega }_3={\omega }_2$
3. Gear 4 (with $N_4$ teeth) meshes directly with both Gear 3 and Gear 5.
4. Gear 5 is the driven gear with $N_5$ teeth.

Now, let us derive the velocity ratio of the gear train step-by-step. Let ${\omega }_i$ represent the angular velocity of gear $i$:

For the meshing pair Gear 1 and Gear 2:

$$\frac{{\omega }_2}{{\omega }_1}=\frac{N_1}{N_2}$$

For the meshing pair Gear 3 and Gear 4:

$$\frac{{\omega }_4}{{\omega }_3}=\frac{N_3}{N_4}$$

For the meshing pair Gear 4 and Gear 5:

$$\frac{{\omega }_5}{{\omega }_4}=\frac{N_4}{N_5}$$

The overall velocity ratio of the driven gear (Gear 5) to the driver gear (Gear 1) is given by:

$$\frac{{\omega }_5}{{\omega }_1}=\frac{{\omega }_2}{{\omega }_1}\times \frac{{\omega }_4}{{\omega }_3}\times \frac{{\omega }_5}{{\omega }_4}$$

Substituting the teeth relationships (and using ${\omega }_3={\omega }_2$):

$$\frac{{\omega }_5}{{\omega }_1}=\frac{N_1}{N_2}\times \frac{N_3}{N_4}\times \frac{N_4}{N_5}$$

Simplifying the product:

$$\frac{{\omega }_5}{{\omega }_1}=\frac{N_1\times N_3}{N_2\times N_5}$$

Since the number of teeth of Gear 4 ($N_4$) cancels out completely, it does not affect the overall velocity ratio of the gear train. Therefore, Gear 4 acts purely as an idler gear in this configuration.