# Is bouncing ball (100% collision) an oscillatory motion/SHM or both or none?

My teacher told me bouncing ball (100% elastic) is oscillatory motion that does not have a stable equilibrium position and restoring force. It is just to and fro motion and thus called oscillatory motion but unlike most other oscillatory motions it does not have mean/stable equilibrium position and restoring force which is proportional to $$-x^(n)$$ where $$n$$ is odd. And in most books it is written that it is a MUST for oscillatory motion to have stable equilibrium position otherwise it is not oscillatory motion. And on the internet it is given it is oscillatory motion and also has mean stable equilibrium position but no proper explanation given as to where it is or how it is. So, I am very much puzzled and totally confused whether it is oscillatory motion or not and if so how and why and if not how and why? If there are any expert/professional in the field please give a answer which is 100% correct according to what true physics is no matter what. Hope someone helps me out early.

• Which books state that oscillatory motion must have a stable equilibrium position? Commented Jan 24 at 19:48
• Related, if not a duplicate. Commented Jan 24 at 22:15
• @Chemomechanics resnick/halliday and physics galaxy. These are very renowned books of my syllabus. Commented Jan 25 at 15:31
• @Chemomechanics or the book might mean to say that if it does not do to nd fro motion about mean/stable equilibrium position then it must not be called TRUE oscillatory motion Commented Jan 25 at 15:54
• If you're referring to Halliday et al. using repetitive back-and-forth motion, harmonic motion, vibration, and oscillation as equivalent descriptors, I'd say there's some imprecision there; their aim is probably only to link broad topics in the student's mind. The overdamped oscillator they analyze, for example, doesn't execute repetitive motion and doesn't even move back and forth; its motion after release is in one direction only. If I were reviewing that text, I'd suggest adding a "generally"-like qualifier to avoid the type of confusion that arose here. Commented Jan 25 at 20:18

The motion of the bouncing ball with elastic collisions is an oscillation since it is periodic. It is not SHM, since the restoring force is not proportional to the displacement of the ball from an equilibrium position.

If we assume that the ball is rigid, so that the collisions with the ground are instantaneous, then there is no equilibrium position in the motion. However, if we assume that collisions with the ground are not instantaneous (equivalently, we assume that the ball is not rigid) then there is an equilibrium position during each collision, at the point where the ball deforms just enough so that the normal force on it is equal to its weight.

• But the ball has to be rigid; otherwise there would be some energy loss during each collision (energy spent in restoring its shape) and the collision wouldn't be fully elastic then. Commented Jan 25 at 8:19
• @StutiGupta Not necessarily. You could, in theory, have a material that deforms and is also perfectly elastic at the same time. This is an idealisation - any real material will lose some energy as it deforms - but a perfectly rigid ball is also an idealisation. Commented Jan 25 at 9:28
• @gandalf61 Sir, please help me understand :- according to 2 renowned physics books of my syllabus(resnick/halliday & physics galaxy) it is said that the strict condition for oscillatory motion is to do to and fro motion [ABOUT] the mean/stable equilibrium position. And periodic motion just means that the body repeats its motion in same intervals of time and (undamped)oscillatory motion is a special case of periodic motion where body does to and fro motion [ABOUT] mean/stable equilibrium position. (physics.stackexchange.com/q/696470) here a person is claiming it can also be shm. Commented Jan 25 at 15:22

As a general rule, "oscillatory" means periodic in time. The bouncing of a ball with 100% elastic collisions is an example of this.

SHM arises from a restoring force equal to the first power of the displacement and is purely sinusoidal for a single frequency. There are all sorts of oscillatory motions that do not fit this description.

I actually am going to take a bit of a different opinion here, and say that this system still has a point of stable equilibrium. The stable equilibrium is the point of contact with the floor. Just as at the natural length of a spring or the bottom of a pendulum, if the ball were to be on the floor with no velocity, it would stay there forever. This is what a stable equilibrium is. The reason the purely elastic ball bounces forever is because every time it reaches the equilibrium point, it has velocity. This is also why a perfect spring or pendulum also oscillate forever.

The equilibrium point is odd compared to others due to the abrupt nature of its behavior. You can see this more clearly by including the dynamics of the bounce, meaning the ball compressing and springing back to shape and off the floor again. The longer and more pronounced that process is (which requires the ball to be less stiff), the more you can see how at the impact with the floor it actually goes "below" the equilibrium point and then rebounds. As the stiffness of the ball goes to infinity and the time of the impact/bounce goes to zero, that motion about the equilibrium becomes infinitesimal. Please let me know if that does not make sense.

The perfectly elastic bouncing ball has a well defined force that is a function of space (constant weight downward always and whatever elastic force upward during the impact). It is a periodic system that will operate forever. You can find its Fourier series. It is a superposition of many SHMs.