A CNC machine has one of its linear positioning axes as shown in the figure, consisting of a motor rotating a lead screw, which in turn moves a nut horizontally on which a table is mounted. The motor moves in discrete rotational steps of 50 steps per revolution. The pitch of the screw is 5 mm and the total horizontal traverse length of the table is 100 mm. What is the total number of controllable locations at which the table can be positioned on this axis?

The correct option is 1000.

Understanding the Diagram and System Mechanics:
As shown in the diagram, the system consists of a motor that rotates in discrete angular steps. This motor is connected to a lead screw, which translates the rotational motion into the linear movement of a nut. A table is mounted on this nut, moving horizontally as the screw rotates. Since the motor moves in discrete steps, the table also moves in discrete, controllable linear increments.

To find the total number of controllable locations, we first calculate the resolution of the linear positioning axis (the distance the table moves per single motor step).

Step 1: Calculate the Step Resolution
The pitch of the screw is the linear distance the nut (and table) moves in one full revolution of the screw. Here, one revolution corresponds to a linear movement of 5 mm. The motor requires 50 steps to complete one full revolution.
Therefore, the movement of the table per step (step resolution) is calculated as:

$$\text{Step Resolution}=\frac{\text{Pitch of the Screw}}{\text{Steps per Revolution}}$$

Substituting the given values:

$$\text{Step Resolution}=\frac{5\text{mm}}{50}=0.1\text{mm/step}$$

This means the table can be positioned at discrete intervals of 0.1 mm along the axis.

Step 2: Calculate the Total Number of Controllable Locations
The total horizontal traverse length (travel distance) of the table is given as 100 mm. The total number of controllable discrete positions along this length is:

$$\text{Total Controllable Locations}=\frac{\text{Total Traverse Length}}{\text{Step Resolution}}$$

Substituting the values:

$$\text{Total Controllable Locations}=\frac{100\text{mm}}{0.1\text{mm/step}}=1000$$

Thus, there are exactly 1000 controllable locations at which the table can be positioned along this axis.