Normal Force & SHM: Answers to Peter G.

[Peter G.]

Hi,

As I was going through some SHM problems I came across a doubt regarding normal forces.

Let's use as an example a book standing on a table: There are two forces acting on the book: It's weight acting downwards and a normal reaction force from the table acting upwards. My doubt: Is there a reaction force for the Normal Force acting on the box on the table? Or not even a reaction force, just a Normal force since the table is in contact with the book as much as the book is in contact in the table... I don't know if I am conveying my doubt clearly but, in case there was a Normal Force on the table as well, a free body diagram of it would show two forces pointing downwards, the normal force and the weight, no?

Thanks,
Peter G.

 

[Doc Al]

Let's use as an example a book standing on a table: There are two forces acting on the book: It's weight acting downwards and a normal reaction force from the table acting upwards.

OK. Note that both forces act on the book; they are not action/reaction pairs.

My doubt: Is there a reaction force for the Normal Force acting on the box on the table? Or not even a reaction force, just a Normal force since the table is in contact with the book as much as the book is in contact in the table...

Sure. The book presses down on the table and the table presses up on the book. Both of those contact forces are really two sides of the same interaction. They are third law pairs.

I don't know if I am conveying my doubt clearly but, in case there was a Normal Force on the table as well, a free body diagram of it would show two forces pointing downwards, the normal force and the weight, no?

A free body diagram of what? The table? There would be at least two forces on it: The downward normal force from the book and the downward force of the table's weight. Of course, there will also be an upward normal force of the floor on the table.

 

[Peter G.]

Ok, cool! Got it! Now, regarding the SHM part:

1. State what is meant by an oscillation and Simple Harmonic Motion
2. A Ball goes back and forth along a horizontal floor bouncing off two vertical walls. Is the motion an example of an oscillation? If yes, is the oscillation simple harmonic?
3. A ball bounces vertically off the floor. Is the motion of the ball an example of an oscillation? If yes, is the oscillation simple harmonic?

1. An oscillation is a periodic movement about an equilibrium position. In Simple Harmonic Motion, the acceleration of the body in oscillatory motion must be proportional and opposite to the displacement. (A general oscillation must have a constant period or no?)

2, 3: For two and three, I presume that, to determine whether they are or not Simple Harmonic I must consider whether the displacement is proportional and opposite to the acceleration. Firstly, I believe both of them are oscillations - they are being displaced periodically from an equilibrium position.

Let's deal with the Questions 2 scenario first. As the ball hits the wall and moves back to the midpoint between the two walls (equilibrium position), it's displacement will be at its maximum and so will it's acceleration. As it moves towards the center, the rate of change in speed will decrease as the distance from the equilibrium position decreases, therefore, the displacement and acceleration are proportional. As it moves towards the center, the force acting on the ball will be friction, which, decreases as the displacement decreases (because the velocity is decreasing (still proportional)). If the ball is moving to the right, the frictional force will be to the left and, since the acceleration is always in the same direction as the resultant force, the acceleration would also be to the left. Since we considered the ball was moving back to its equilibrium position towards the right, that means it was displaced to the left relative to the equilibrium position, hence, the displacement is in the same direction as the acceleration, the motion is not SHM.

In Question Number 3, I think we can already consider the motion not to be SHM because the acceleration by gravity is constant throughout, in other words, it is not dependent upon the displacement.

I hope I didn't go crazy and my line of thought is possible to follow. Excuse me if I made erroneous assumptions but this was my attempt!

Thanks,
Peter

 

[Doc Al]

1. An oscillation is a periodic movement about an equilibrium position. In Simple Harmonic Motion, the acceleration of the body in oscillatory motion must be proportional and opposite to the displacement. (A general oscillation must have a constant period or no?)

Sounds good to me. Depending upon your textbook, an "oscillation" can refer to any periodic motion, which by definition has a constant period.

2, 3: For two and three, I presume that, to determine whether they are or not Simple Harmonic I must consider whether the displacement is proportional and opposite to the acceleration. Firstly, I believe both of them are oscillations - they are being displaced periodically from an equilibrium position.

Sounds good.

Let's deal with the Questions 2 scenario first. As the ball hits the wall and moves back to the midpoint between the two walls (equilibrium position), it's displacement will be at its maximum and so will it's acceleration. As it moves towards the center, the rate of change in speed will decrease as the distance from the equilibrium position decreases, therefore, the displacement and acceleration are proportional. As it moves towards the center, the force acting on the ball will be friction, which, decreases as the displacement decreases (because the velocity is decreasing (still proportional)). If the ball is moving to the right, the frictional force will be to the left and, since the acceleration is always in the same direction as the resultant force, the acceleration would also be to the left. Since we considered the ball was moving back to its equilibrium position towards the right, that means it was displaced to the left relative to the equilibrium position, hence, the displacement is in the same direction as the acceleration, the motion is not SHM.

Yes, not SHM. Forget about friction to make it even more obvious that it's not SHM. As the ball moves between walls there is no acceleration; when it bounces off the wall there is a short-lived but large acceleration as it reverses direction.

In Question Number 3, I think we can already consider the motion not to be SHM because the acceleration by gravity is constant throughout, in other words, it is not dependent upon the displacement.

OK, but again don't forget to also consider the relatively large force exerted by the floor on the ball as it bounces. In any case, not SHM.