As the parameter t changes, the component moves along the rail, and the engineer needs to know how the slope of the path changes at different stages. The task is to determine how y changes with respect to x, and to interpret the result as part of the design check for smooth motion.

Given $x = t - \sin t$.

$$\frac{dx}{dt} = \frac{d}{dt}(t - \sin t) = 1 - \cos t$$

Given $y = 1 + \cos t$.

$$\frac{dy}{dt} = \frac{d}{dt}(1 + \cos t) = -\sin t$$

Using the chain rule for parametric equations, $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.

$$\frac{dy}{dx} = \frac{-\sin t}{1 - \cos t}$$

The slope of the path is given by $\frac{dy}{dx} = \frac{-\sin t}{1 - \cos t}$. For smooth motion, the slope should change continuously. Sudden changes in slope occur when $\frac{dy}{dx}$ is undefined or changes abruptly.
The expression for $\frac{dy}{dx}$ becomes undefined when the denominator $1 - \cos t = 0$.
This occurs when $\cos t = 1$.
The values of $t$ for which $\cos t = 1$ are $t = 2n\pi$, where $n$ is an integer.
At these points, $\frac{dx}{dt} = 1 - \cos(2n\pi) = 1 - 1 = 0$, and $\frac{dy}{dt} = -\sin(2n\pi) = 0$.
When both $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are zero, the curve has a singular point, which for this specific parametric curve (a cycloid) is a cusp. A cusp represents a sharp point where the direction of the tangent changes instantaneously, leading to a sudden and infinite change in slope. This would cause increased friction, vibration, and wear on the sliding component, violating the requirement for smooth motion. Therefore, the design engineer must ensure that these values of $t$ are avoided or the design is modified to eliminate these cusps.

The rate of change of $y$ with respect to $x$ is:

$$\boxed{\frac{dy}{dx} = \frac{-\sin t}{1 - \cos t}}$$

The interpretation is that the motion will not be smooth at points where $t = 2n\pi$ (for integer $n$), as these correspond to cusps in the path where the slope is undefined, leading to sudden changes in direction.