The center of oscillation of a physical pendulum has this...

The center of oscillation of a physical pendulum has this interesting property: If an impulse (assumed horizontal and in the plane of oscillation) acts at the center of oscillation, no oscillations are felt at the point of support. Baseball players (and players of many other sports) know that unless the ball hits the bat at this point (called the "sweet spot" by athletes), the oscillations due to the impact will sting their hands. To prove this property, let the stick in Fig. $15-13 a$ simulate a baseball bat. Suppose that a horizontal force $\vec{F}$ (due to impact with the ball) acts toward the right at $P,$ the center of oscillation. The batter is assumed to hold the bat at $O,$ the pivot point of the stick. (a) What acceleration does the point $O$ undergo as a result of $\vec{F} ?$ (b) What angular acceleration is produced by $\vec{F}$ about the center of mass of the stick? (c) As a result of the angular acceleration in (b), what linear acceleration does point $O$ undergo? (d) Considering the magnitudes and directions of the accelerations in $(a)$ and $(\mathrm{c}),$ convince yourself that $P$ is indeed the "sweet spot."

The center of oscillation of a physical pendulum has this interesting property: If an impulse (assumed horizontal and in the plane of oscillation) acts at the center of oscillation, no oscillations are felt at the point of support. Baseball players (and players of many other sports) know that unless the ball hits the bat at this point (called the "sweet spot" by athletes), the oscillations due to the impact will sting their hands. To prove this property, let the stick in Fig. $15-13 a$ simulate a baseball bat. Suppose that a horizontal force $\vec{F}$ (due to impact with the ball) acts toward the right
at $P,$ the center of oscillation. The batter is assumed to hold the bat at $O,$ the pivot point of the stick. (a) What acceleration does the point $O$ undergo as a result of $\vec{F} ?$ (b) What angular acceleration is produced by $\vec{F}$ about the center of mass of the stick? (c) As a result of the angular acceleration in (b), what linear acceleration does point $O$ undergo? (d) Considering the magnitudes and directions of the accelerations in $(a)$ and $(\mathrm{c}),$ convince yourself that $P$ is indeed the "sweet spot."

Key Concepts

Physical Pendulum

A physical pendulum is an extended rigid body that oscillates about a fixed pivot point under the influence of gravity. Its motion is characterized by both its translational and rotational dynamics, which depend on the distribution of mass relative to the pivot. Key quantities in its analysis include the center of mass, moment of inertia, and the restoring torque due to gravity.

Center of Oscillation

The center of oscillation is a point on the physical pendulum where an impulse results in pure translational motion of the support, without inducing any rotational or oscillatory effects. This concept is closely related to the center of percussion; striking the pendulum at this point minimizes the reactive forces at the pivot, which in practical applications is often referred to as the 'sweet spot'.

Torque

Torque is the measure of the rotational force applied to a body and is defined as the product of the force and the lever arm, the perpendicular distance from the pivot to the line of action of the force. It is responsible for producing angular acceleration in a rotating system, and its magnitude and direction determine the subsequent rotational motion of the body.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity of a rotating body, resulting from the net torque applied. It is influenced by the moment of inertia, which quantifies the resistance of an object to changes in its rotational motion. The relationship between torque and angular acceleration is encapsulated by Newton’s second law for rotation.

Linear Acceleration

Linear acceleration is the rate at which the velocity of a body changes with time. In the context of a physical pendulum, it arises both from the direct effect of an applied force (translated through the center of mass) and from the tangential acceleration due to the body's rotation about a pivot.

Impulse

Impulse refers to the change in momentum of a body as a result of a force applied over a short period of time. In collision or impact scenarios, the impulse determines the immediate changes in both linear and angular momentum, influencing the subsequent motion of the body.

Transcript

00:01 For this problem, we're asked to think about a physical pendulum in terms of a baseball bat.

00:09 And the oscillations due to impact.

00:14 So we're given us a scenario where there is a physical pendulum.

00:21 Again, even though it's not really drawn as it, but in terms of a stick where o, is a fixed position, and this is where the baseball player would be holding.

00:37 The bat, c is the center of mass, and then p is the center of oscillation.

00:46 The batter holds at o, the pivot point of the stick, and we're asked to figure out what acceleration does the point o undergo as a result of a force of a horizontal force acting towards the right at p on the stick.

01:05 And then also the angular acceleration is produced by the force f about the center mass of the stick.

01:14 And as a result of the angular acceleration in part b, what linear acceleration does point o undergo? and then finally, considering the magnitudes and directions of the accelerations in a and c, we have to prove that p is indeed the sweet spot where if the batter hits the ball on the sweet spot, they will not feel the oscillations through the baseball bat.

01:43 So first, we're thinking about the force.

01:46 There is a force acting towards the right on point p, which is where the stick hits the ball.

01:58 And we're trying to find the acceleration that point o will undergo as a result of this.

02:07 So the net force, the net horizontal force, which is f, since the batter is assumed to exert no horizontal force, on the bat whatsoever.

02:18 So we can actually assume that the acceleration equals just the force divided by the mass.

02:31 So that is going to be the force felt at point o is going to be f divided by m.

02:38 So now for the angular acceleration produced by the force about the center of mass of the stick...

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