Figure P 35.4 shows an apparatus used to measure the speed...

Figure $\mathrm{P} 35.4$ shows an apparatus used to measure the speed distribution of gas molecules. It consists of two slotted rotating disks separated by a distance $d$, with the slots displaced by the angle $\theta .$ Suppose the speed of light is measured by sending a light beam from the left through this apparatus. (a) Show that a light beam will be seen in the detector (that is, will make it through both slots) only if its speed is given by $c=\omega d / \theta,$ where $\omega$ is the angular speed of the disks and $\theta$ is measured in radians. (b) What is the measured speed of light if the distance between the two slotted rotating disks is $2.50 \mathrm{m},$ the slot in the second disk is displaced $1 / 60$ of one degree from the slot in the first disk, and the disks are rotating at 5555 rev/s? (FIGURE CANT COPY)

Key Concepts

Angular Displacement and Unit Conversion

In many practical applications, including this one, angular displacements might be given in degrees while calculations require radians. Converting between these units is essential to correctly apply mathematical relationships such as c = ?d/?, ensuring that the angular displacement is properly used in the underlying laws of rotational dynamics and kinematics.

Time-of-Flight Measurement

Underlying the operation of the apparatus is the time-of-flight concept, where the time taken by the beam (or gas molecules) to travel between the two disks is directly related to its speed. By establishing the condition that the slot must be aligned at the exact moment the beam arrives, the apparatus effectively measures speed as a function of the known rotational and geometric parameters.

Rotational Motion and Angular Speed

This concept deals with how objects rotate around a fixed axis at a rate described by angular speed (often expressed in radians per second). In the context of this apparatus, the angular speed determines the synchronization timing of the slots across the rotating disks, which is crucial for allowing the beam (or gas molecules) to pass through aligned openings.

Geometric Alignment in Rotating Disks

This idea involves the spatial and temporal arrangement of slots in two rotating disks. The detection of the beam depends on the precise positioning (or displacement) of the slots relative to each other. By relating the displacement angle and the disk separation, one can determine the conditions under which the beam passes through both slots, thereby linking geometry with the observed timings.

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