Finding the Center of Percussion

[bd2015]

Homework Statement

A rod of mass M and length L rests on a frictionless table and is pivoted on a frictionless nail at one end. A blob of putty of mass m approaches with velocity v from the left and strikes the rod a distance d rom the end as shown, sticking to the rod.

1. Find the angular velocity of the system about the nail after the collision (already done)
2. Is there a value of d for which linear momentum is conserved? If there were such a value, it would be called the center of percussion for the rod for this sort of collision.

Homework Equations

ω final = $\frac{mvd}{(1/3)ML^2 + md^2}$

The Attempt at a Solution

So, I know that linear momentum most be conserved. I don't know how to find the linear momentum of the rod

 

[TSny]

The linear momentum of any object can be calculated as the total mass of the object multiplied by the linear velocity of the center of mass of the object.

 

[bd2015]

The linear momentum of any object can be calculated as the total mass of the object multiplied by the linear velocity of the center of mass of the object.

So then do I just need to set P initial to P final, and solve for d, without having to worry about angular momentum?

 

[TSny]

Essentially, yes. But you'll need to use your result from part 1 which came from conservation of angular momentum. Also, don't forget to include the final linear momentum of the putty.

 

[Nickie]

and putty after the collision, but I do know that it must equal the initial linear momentum of the putty. Therefore, I can set up an equation:

(mv)initial = (m+M)(v+ωd)f

Where (m+M)(v+ωd)f is the final linear momentum of the rod and putty combined. From here, I can solve for the value of d that would make the linear momentum conserved. This value of d would be the center of percussion for this type of collision.

However, it is important to note that this is only true for a perfectly elastic collision. In reality, there will be some loss of energy due to the deformation of the putty and the rod. In this case, the center of percussion may not exist or may be slightly different than the calculated value. It is also possible that the collision is not perfectly inelastic, in which case the center of percussion may not exist at all.

Additionally, the center of percussion is a theoretical concept and may not have a practical application in most cases. It is more useful in situations where a rigid body is attached to a pivot and experiences a collision at a certain distance from the pivot. In these cases, the center of percussion can be used to find the point where the impact force will have the least effect on the pivot.

Overall, finding the center of percussion can be a useful tool in certain situations, but it is important to consider the assumptions and limitations of its application.