Why does a rod move faster when struck at the center rather than the edge, despite Newton's second law indicating the same acceleration?"

Question

When applying the same force to the edge or the center of a rod, according to Newton's second law, the rod will acquire the same acceleration in both cases. That is, the speed the rod acquires in both cases should be the same. However, in reality, I notice that any object, when I strike it at the center, gains more speed than when I strike it at the edge. When I strike it at the center, it doesn't rotate much or perhaps not at all, and it moves faster. But when I strike it at the edge, it rotates quickly but moves slower.

Could someone explain a perspective that might correct my understanding.

Answer 1

The total momentum that you are exchanging is likely not the same in each case.

1. Strike in the center. Because the force applied is in line with the center of mass, all the mass must be dragged along before any noticeable motion occurs.

2. Strike the edge. The applied force is NOT in line with the center of mass. The requirement that all mass must be dragged along before any noticeable motion happens no longer applies. This makes the object feel "lighter" than what it really is. The fact that it is rotating also eats up energy that could have been used for forward motion.

Depending on material of the rod, and material of the striker, Case 1 almost certainly has more kinetic energy by the end of the strike compared to case 2 assuming the strike mechanism identical. There is slightly less opportunity for momentum to be exchanged because in case 2 the object "gets out of the way" faster.

Although I would be curious on more concrete examples on what exact objects you are talking about and what you are hitting them with.

Answer 2

I think it is the contact time between the force and the rod that causes the real world experience to differ from theory. When struck at its center (of mass), an object is able to stay in touch with the source of the force longer than when struck elsewhere. During the latter, the induced rotation breaks contact more quickly. There are two effects:

• The longer contact time leads to larger momentum transfer ($\Delta \vec p =\int \vec F dt$) and larger energy transfer ($\Delta E=\int \vec F.\vec vdt$)

• usually, an applied force drops in magnitude once contact begins to break. A longer contact time leads to a steady force being applied longer.

In theory, the instantaneous force and torque are most generally modeled as a $\delta$-function linear/angular impulse and wouldn't account for the contact time (unless differently normalized).

This is only a hypothesis and needs data to substantiate. One experimental design choice is to make the contact time independent of or negligibly small compared to the motion of the object. Another is to use non-contact forces.

The induced rotational motion is in addition to the translational motion and has little bearing on why the object doesn't translate as far (in the real world). To repeat, torque is induced in addition to the applied force. No translational energy is "stolen" by the rotation, it is provided in addition by the source (that would be the case if the energy imparted to the rod was the same in both cases - center and edge - which I think is not)

Answer 3

One word explains why your ideal world analysis disagrees with your real world observation - friction.

Ideal world analysis
Linear acceleration of centre of mass of the rod is the same irrespective of where the rod is hit.
Angular acceleration about centre of mass is greater when the applied force applies a larger torque about the centre of mass, ie force is applied at right angles to the rod and further from the centre of mass.

Real world observation
Contradicts theory, as explained above, because of the presence of friction and so much so that many think the theoretical analysis is flawed.
In some ways this is like the old idea that a force needed to be applied to a cart (or any other object) to keep it moving at constant velocity.

A series of YouTube videos made by Veritasium which is similar to the rod experiment with very little friction.

The Bullet Block Experiment

Bullet Block Experiment Result

Bullet Block Explained!

Answer 4

The first observation is that when you strike an object along its center of mass, you use a higher force than when striking it some offset distance from the center of mass. This is why you observe different accelerations.

The reason is that when striking an object you are imparting an impulse and not a force. Impulse can be thought as the product of force and time, or in rough terms

$$ J = F\, \Delta t $$

where $J$ is the impacting impulse and $F$ is the average force applied over a period of time $\Delta t$.

Impulse $J$ has units of momentum and for in impacting ball, for example, it is equal to the momentum of the ball $J = m v$.

In any case, for the sake of this argument, consider the impulse $J$ known, and then the force seen by the object is $$ F = \frac{J}{\Delta t}$$

If the body is rotating due to the impact, then the objects will be in contact for a longer period of time than if the object just bounced off. When $\Delta t$ is larger , then $F$ is smaller.

And the magnitude of $F$ determines the acceleration of the center of mass o the body

$$ a = \frac{F}{M} $$

Answer 5

Let us say that there is a heavy mass with high velocity impacting a rod laid on the ground. Newton's second Law is for a particle and force applied on it when the force is applied at its CG, resulting in translation alone.

Energy conservation principle gives more information about distribution of PE and KE, the potential and kinetic energies. The rod stuck at its end has both PE and KE, and rotational KE slows down the translational speed.