Consider the mechanism shown in the figure. There is rolling contact without slip between the disc and ground.

Select the correct statement about instantaneous centers in the mechanism.

The correct option is: All points P, Q, R, S, T and U are instantaneous centers of mechanism

1. Identification of Links and Joint Locations:
From the given mechanism diagram, we can identify four distinct links:
• Link 1: The ground/frame (represented by the horizontal dash-dot line passing through pivot P and the hatched ground surface).
• Link 2: The crank link pivoted to the ground at point P.
• Link 3: The connecting rod joining point Q and the center of the disc at point S.
• Link 4: The circular disc of radius ST rolling on the ground.

2. Calculating the Total Number of Instantaneous Centers:
For a planar mechanism with N links, the total number of instantaneous centers ($I_{\text{centers}}$) is given by the formula:

$$I_{\text{centers}}=\frac{N(N-1)}{2}$$

Substituting $N=4$ into the formula:

$$I_{\text{centers}}=\frac{4\times 3}{2}=6$$

Thus, there are exactly 6 instantaneous centers for this mechanism: $I_{12}$, $I_{23}$, $I_{34}$, $I_{14}$, $I_{13}$, and $I_{24}$.

3. Determining the Location of Each Instantaneous Center:
We can locate all 6 instantaneous centers using direct observation and the Aronhold-Kennedy Theorem of Three Centers (which states that if three links move relatively to each other, their three relative instantaneous centers must lie on a straight line):

• Center $I_{12}$ (Point P): This is the permanent pin joint connecting the crank (Link 2) to the fixed ground (Link 1). Therefore, point P is the instantaneous center $I_{12}$.
• Center $I_{23}$ (Point Q): This is the pin joint connecting the crank (Link 2) and the connecting rod (Link 3). Therefore, point Q is the instantaneous center $I_{23}$.
• Center $I_{34}$ (Point S): This is the pin joint connecting the connecting rod (Link 3) to the center of the disc (Link 4). Therefore, point S is the instantaneous center $I_{34}$.
• Center $I_{14}$ (Point T): Since the disc (Link 4) rolls without slipping on the ground (Link 1), the point of contact T has zero instantaneous velocity relative to the ground. Thus, point T is the instantaneous center $I_{14}$.
• Center $I_{13}$ (Point U): According to Kennedy's Theorem, the three centers $I_{12}$, $I_{23}$, and $I_{13}$ must lie on a straight line (line PQ). Similarly, the three centers $I_{14}$, $I_{34}$, and $I_{13}$ must lie on a straight line (vertical line TS through the disc's center). The intersection of the extension of line PQ and the vertical line TS locates the instantaneous center $I_{13}$, which is labeled as point U.
• Center $I_{24}$ (Point R): By Kennedy's Theorem, the three centers $I_{12}$, $I_{14}$, and $I_{24}$ must lie on a straight line (the horizontal line of centers/ground line PT). Additionally, the three centers $I_{23}$, $I_{34}$, and $I_{24}$ must lie on a straight line (line QS along Link 3). The intersection of the horizontal line PT and the link axis line QS defines the instantaneous center $I_{24}$, which is labeled as point R.

Conclusion:
Since all six points P, Q, R, S, T, and U correspond exactly to the six instantaneous centers of this four-link mechanism, all of these points are instantaneous centers.