Center of percussion: finding the sweet spot
In this blog post, I explore the concept of the center of percussion, a crucial idea in mechanics with practical applications in sports and engineering. I examine how to determine the optimal impact point on an object, like a bat or stick, to minimize or eliminate reaction forces at its pivot. This “sweet spot” ensures a smooth, efficient transfer of momentum. My analysis uses the impulse-momentum principle, applied to a simplified model of a rigid body impacting another object.
Imagine hitting a ball with a bat. If you hit the ball at a certain point on the bat, you’ll feel minimal vibration in your hands. This point is known as the center of percussion. From a physics perspective, it’s the point where an impact force can be applied to a pivoted rigid body without producing a reaction force at the pivot.
To understand this mathematically, I consider a simple model: a rigid rod pivoted at one end, struck by an impulse force. I use the impulse-momentum principle, which states that the change in momentum of a body is equal to the impulse applied to it. This principle can be applied linearly and rotationally.
I consider the linear impulse-momentum equation in the y-direction (perpendicular to the rod’s length):
m \dot{y}_{C_i} + \int_{t_i}^{t_f} \left( R_y - F \right) \, \mathrm{d}t = m \dot{y}_{C_f}
where:
- m is the mass of the rod,
- \dot{y}_C is the velocity of the center of mass,
- R_y is the reaction force at the pivot,
- F is the impact force.
I also consider the angular impulse-momentum equation about the pivot point:
I_O \omega_i - \int_{t_i}^{t_f} d F \, \mathrm{d}t = I_O \omega_f
where:
- I_O is the moment of inertia about the pivot,
- \omega is the angular velocity,
- d is the distance from the pivot to the impact point.
My goal is to find the value of d for which the reaction force R_y is zero. This simplifies the linear impulse-momentum equation. By combining the linear and angular impulse-momentum equations and setting R_y = 0, I arrive at the following expression for the center of percussion:
d = \frac{I_O}{mL}
For a uniform rod pivoted at one end, I_O = \frac{1}{3}mL^2, which simplifies the equation to:
d = \frac{2}{3}L
Where L is the length of the rod.
This result shows that the center of percussion is located two-thirds of the way down the rod from the pivot point. Hitting an object at this point minimizes or eliminates the reaction force at the pivot, resulting in a smoother impact and more efficient energy transfer.
For more insights into this topic, you can find the details here.