Suppose an impulsive force F acts horizontally to the right at point O (the center ofoscillation) in Fig. 17-12. Assume that the pendulum is initially at rest. (a) By combining the effects of translation and rotation, show that the resulting acceleration of a particle at point P is zero. (b) What do you conclude about the force at P that results from the applied force F? Because of this property, the center of oscillation is often called the center of percussion.

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:

$$I_{\text{cm}}\alpha =FR$$

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

$$\alpha =\frac{FR}{I_{\text{cm}}}$$

### 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:

$$a_P=a_{\text{cm}}+\alpha \times l^{\prime }$$

where $l^{\prime }$ is the distance from the center of mass to point $P$.

- Substituting the values:

$$a_P=\frac{F}{M}+\left(\frac{FR}{I_{\text{cm}}}\right)l^{\prime }$$

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

$$\frac{F}{M}=-\frac{FR}{I_{\text{cm}}}{l}^{\prime }$$

- Rearranging:

$$l^{\prime }=\frac{I_{\text{cm}}}{MR}$$

This distance $l^{\prime }$ 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).