A power transmission mechanism consists of a belt drive and a gear train as shown in the figure.

Diameters of pulleys of belt drive and number of teeth (T) on the gears 2 to 7 are indicated in the figure. The speed and direction of rotation of gear 7, respectively, are

The correct option is: 255.68 rpm; clockwise

Step-by-Step Derivation and Analysis:

1. Analysis of the Belt Drive:
From the given system layout, we have the driver pulley (motor shaft) with a diameter:
$d_0=150\text{ mm}$
rotating at a speed of:
$N_0=2500\text{ rpm (anticlockwise)}$
This driver pulley is connected via an open belt drive to a driven pulley of diameter:
$d_1=250\text{ mm}$
The rotational speed of the driven pulley ($N_1$) is calculated as:
$N_1=N_0\times \frac{d_0}{d_1}$
Substituting the given values:
$N_1=2500\times \frac{150}{250}=1500\text{ rpm}$
Since this is an open belt drive, the driven pulley rotates in the same direction as the driver pulley (anticlockwise).
Because Gear 2 is mounted on the same shaft as this driven pulley, they share the same speed and direction:
$N_2=1500\text{ rpm (anticlockwise)}$

2. Transmission from Gear 2 to Gear 3:
Gear 2 (having teeth $T_2=18$) meshes externally with Gear 3 (having teeth $T_3=44$).
The rotational speed of Gear 3 ($N_3$) is:
$N_3=N_2\times \frac{T_2}{T_{\mathrm{3<!--\; </msub-->}}}$
Substituting the teeth numbers:

Since Gear 2 and Gear 3 are in external contact, the direction of rotation reverses. Thus, Gear 3 rotates in the clockwise direction.

3. Transmission from Gear 4 to Gear 6:
Gear 4 is keyed to the same shaft as Gear 3, which means:
$N_4=N_3\approx 613.636\text{ rpm (clockwise)}$
Gear 4 (having teeth $T_4=15$) meshes with Gear 5 ($T_5=33$), which in turn meshes with Gear 6 (having teeth $T_6=36$).
Here, Gear 5 behaves as an idler gear, and it does not affect the overall speed ratio between Gear 4 and Gear 6. The speed of Gear 6 is given by:
$N_6=N_4\times \frac{T_4}{T_{\mathrm{6<!--\; </msub-->}}}$
Substituting the parameters:
$N_6=613.636\times \frac{15}{36}\approx 255.68\text{ rpm}$

4. Determining the Final Speed and Direction of Gear 7:
Let us determine the direction of rotation through the gear sequence:
• Gear 4 rotates clockwise.
• Meshing with Gear 5 causes Gear 5 to rotate anticlockwise.
• Meshing with Gear 6 causes Gear 6 to rotate clockwise.
Since Gear 7 is mounted on the same shaft as Gear 6, it rotates at the exact same speed and direction:
$N_7=N_6=255.68\text{ rpm (clockwise)}$