# Solving Classical Mechanics Forces on Stick: F_3a=F_2(a+b) Explained

• Grand
In summary, the conversation discusses a problem in classical mechanics involving forces acting on a horizontal stick. The author proves that F_3a=F_2(a+b) by considering the relative positions of the forces and introducing the concept of torque balance. This is based on the intuitive result that when one force is doubled, the other must also double, maintaining their relative positions. The conversation also touches on the interesting symmetry in mechanical problems and the rule of position in statics.
Grand

## Homework Statement

I'm reading Classical mechanics by Morin and I can't understand the following reasoning. There is a horizontal stick at the ends of which 2 forces $$F_1$$ and $$F_2$$ are applied perpendicular and upward to the stick, and another force $$F_3$$ applied downward and again perpendicular to the stick. The distances from the point where $$F_3$$ is applied to the ends of the stick are $$a$$ and $$b$$. Now the author proves that $$F_3a=F_2(a+b)$$ by saying that since the force is linear, then the following must hold:

$$F_3f(a)=F_2f(a+b)$$
where f is a function to be determined. I can't fully understand this reasoning, so could anyone give me some more detail. I'll be very grateful.

Greetings,
Grand

## The Attempt at a Solution

I luckily have Morin's book, so I can interpret his idea in the way I understand. At the point of this problem, Morin doesn't even introduce the formulas of torque and angular momentum. What I understand is this: For this particular problem, from the condition of force balance that the author introduces before, we have F3 = F1 + F2. But that's not enough and not so intuitive, since F3 must be exerted at a certain point on the stick. There must be some other condition (and most of us know that is torque balance; but let's pretend that we don't know it). And the author's aim is, after finding that condition, he will introduce the concept of torque balance.

So his argument is this: Because this condition has something to do with both magnitude of the force and the position where the force is exerted, it can be expressed in terms of the force (F3, for example) and the position (a, or even a+some length, for example). But he chooses a, as the position is relative. That is, he chooses to consider the relative positions of the forces to the point where F1 is exerted, and therefore F1 is no longer considered. There is a real case that matches this choice: suppose that at the end where F1 is exerted, the stick is fixed to a pivot. So F1 is the force due to the pivot. We know that if a force F2 is exerted at the free end, then F3 must be exerted with one certain magnitude and one certain position: F2 determines F3, regardless of F1. Therefore the equation should contain F3, accompanied by a (relative position of F3 to F1), and F2, accompanied by a+b (relative position of F2 to F1).

Intuitively when F3 is doubled, F2 must be doubled, provided that their positions maintain the same. Therefore the equation must show the linearity of the force. Thus, the most likely equation is: $$F_3f(a) = F_2f'(a+b)$$ (the two function f and f' may be different!).

However there is interesting symmetry: F2 and F3 have the same role. If we choose to consider F2 as the applied force, F3 must be the force which is here to balance out F2. On the other hand, if F3 is the applied force, F2 must be the balancing force.
Out of topic: Such symmetry actually comes from an intuitive and interesting characteristic of mechanical problems: A mechanical problem should have one and only one solution (or show one and only one resultant phenomenon) corresponding to certain initial conditions. Here, F2 and F3 (and their positions) are initial conditions, and these initial conditions lead to that the stick stays at rest. To put it another way, if the stick stays at rest, then with one F2 (one certain magnitude and one certain position), we have one and only one F3. Notice: The last sentence "To put it another way..." is consistent with the above characteristic only under certain circumstances!

So due to that F2 and F3 shares the same role, f and f' must be the same, or the rule of the position of each force in statics is the same. Later on, Morin shows that this rule is the linearity of position (f(a) = a), an intuitive result.

Just my 2 cents

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## What is classical mechanics?

Classical mechanics is a branch of physics that deals with the motion and forces of objects in a macroscopic scale. It is based on Newton's laws of motion and the principles of conservation of energy and momentum.

## What is the force equation F_3a=F_2(a+b)?

The force equation F_3a=F_2(a+b) is a mathematical representation of the third law of motion proposed by Sir Isaac Newton. It states that for every action, there is an equal and opposite reaction. In this equation, F_3a represents the action force, while F_2(a+b) represents the reaction force.

## How does this equation relate to forces on a stick?

This equation can be used to explain the forces acting on a stick when a force is applied to one end. The force applied (F_3a) will result in an equal and opposite force (F_2) on the other end of the stick, causing it to move or rotate.

## Can this equation be applied to other objects besides a stick?

Yes, this equation can be applied to any object as long as there is an external force acting on it. It follows the principle of action and reaction, which is a fundamental law of classical mechanics.

## Why is understanding classical mechanics important?

Understanding classical mechanics is important because it helps us explain and predict the behavior of objects in our everyday lives. It is the foundation of many other branches of physics, such as thermodynamics, electromagnetism, and quantum mechanics.

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