To solve this problem, let's analyze the motion of the pendulum when an impulsive force \( F \) acts at the center of oscillation (point \( O \)).

### Step 1: Understanding the Motion of the Pendulum
- The pendulum consists of a rigid body pivoted at a fixed point.
- The applied force \( F \) at \( O \) will cause both translational and rotational motion.
- We need to determine the acceleration at point \( P \), which is the center of percussion.

### Step 2: Translational Motion
- The entire pendulum experiences a linear acceleration due to the force \( F \).
- The equation of motion for translation is given by:

\[
a_{\text{cm}} = \frac{F}{M}
\]

where \( M \) is the total mass of the pendulum and \( a_{\text{cm}} \) is the acceleration of the center of mass.

### Step 3: Rotational Motion
- The applied force \( F \) at \( O \) also creates a torque about the center of mass, causing angular acceleration \( \alpha \).
- The torque about the center of mass is:

\[
\tau = F \cdot R
\]

where \( R \) is the distance from \( O \) to the center of mass.

- Using the rotational equation of motion:

\[
I_{\text{cm}} \alpha = F R
\]

where \( I_{\text{cm}} \) is the moment of inertia about the center of mass.

\[
\alpha = \frac{F R}{I_{\text{cm}}}
\]

### Step 4: Acceleration at Point \( P \)
- The acceleration of a point \( P \) at a distance \( l \) from the pivot (measured along the pendulum) is given by:

\[
a_P = a_{\text{cm}} + \alpha \times l'
\]

where \( l' \) is the distance from the center of mass to point \( P \).

- Substituting the values:

\[
a_P = \frac{F}{M} + \left(\frac{F R}{I_{\text{cm}}} \right) l'
\]

- The center of oscillation (percussion) is defined such that the net acceleration at \( P \) is zero. That happens when:

\[
\frac{F}{M} = -\frac{F R}{I_{\text{cm}}} l'
\]

- Rearranging:

\[
l' = \frac{I_{\text{cm}}}{M R}
\]

This distance \( l' \) corresponds to the center of percussion.

### Step 5: Conclusion
- Since the acceleration at \( P \) is zero, we conclude that the applied force at \( O \) does not cause any reaction force at \( P \).
- This property makes the center of oscillation the **center of percussion**, where an impulsive force produces no reactive force at the pivot.
- This concept is important in sports (e.g., hitting a baseball with a bat) and engineering applications (e.g., designing hammers).