A flat-faced follower is driven using a circular eccentric cam rotating at a constant angular velocity ω. At time t = 0, vertical position of follower is y(0) = 0, and the system is in the configuration shown below

Then vertical position of the follower face, y(t) is given by

The correct answer is e(1 - cos ωt).

Step-by-Step Derivation:

1. Understanding the Geometry of the System:
From the given diagram, we have a circular eccentric cam of radius $R$ and eccentricity $e$. The cam rotates about a fixed pivot (axis of rotation) at a constant angular velocity $\omega$. The distance between the center of the circular cam and the fixed pivot is $e$.
The follower is a flat-faced follower that is constrained to move vertically and remains in continuous tangent contact with the top of the cam.

2. Setting up the Coordinates:
Let us set the fixed pivot as the origin $(0,0)$ of our coordinate system.
At time $t=0$, the center of the circular cam is aligned vertically directly below the pivot. This configuration means the initial position of the cam's center $(x_c,y_c)$ is:
$x_c\left(0\right)=0$
$y_c\left(0\right)=-e$

3. Motion of the Cam's Center:
As the cam rotates through an angle $\theta =\omega t$, the center of the circular cam revolves around the fixed origin $(0,0)$ in a circular path of radius $e$.
Therefore, the vertical position of the cam's center $y_c\left(t\right)$ at any time $t$ is given by:
$y_c\left(t\right)=-e\cos \left(\omega t\right)$

4. Position of the Follower Face:
Since the flat face of the follower remains tangent to the top of the circle, the vertical position of the follower face is always at a distance equal to the radius $R$ above the center of the circular cam.
Thus, the vertical position of the follower face at time $t$ is:
$y_{\text{face}}\left(t\right)=y_c\left(t\right)+R=-e\cos \left(\omega t\right)+R$

5. Follower Displacement:
At $t=0$, the initial vertical position of the follower face is:
$y_{\text{face}}\left(0\right)=-e\cos \left(0\right)+R=-e+R$
We are given that the vertical displacement is defined relative to the initial configuration, so that $y\left(0\right)=0$. Therefore, the vertical displacement $y\left(t\right)$ is:
$y\left(t\right)=y_{\text{face}}\left(t\right)-y_{\text{face}}\left(0\right)$
Substituting the expressions:
$y\left(t\right)=(-e\cos (\omega t)+R)-(-e+R)$
Simplifying the terms:
$y\left(t\right)=e-e\cos \left(\omega t\right)$
Factoring out $e$:
$y\left(t\right)=e(1-\cos (\omega t\left)\right)$