Why Must P Be Perpendicular to R for Minimum Force on an Inclined Plane?

[masterflex]

Homework Statement

This is a conceptual/logic question I have. I don't understand why for minimum P: P must be perpendicular to R; angle beta = psi_s. (please see attached picture). Once I can get my arms around that statement, I can solve the problem. Thanks for any logic help. Or if there's another way to approach this, that's cool too.

Homework Equations

The Attempt at a Solution

Why is the angle Beta not = 0? That is the most puzzling thing conceptually. It would seem that if angle Beta=0, you would get the most mechanical efficiency because the direction of force P would then be perpendicular to the normal N.

 

[anathema]

There are a few key concepts to understand in order to answer this question. First, we need to understand the definition of mechanical efficiency. Mechanical efficiency is a measure of how well a machine converts input energy into useful output energy. It is calculated by dividing the useful output energy by the input energy.

In this scenario, the input energy is the force P and the output energy is the normal force N. In order to maximize mechanical efficiency, we want to minimize the input energy (force P) while still maintaining the same output energy (normal force N).

Now, let's consider the angle beta. This angle represents the direction of the force P relative to the normal N. If beta is equal to 0, then the force P is parallel to the normal N. In this case, the normal force N would be equal to the force P, resulting in a higher input energy and therefore a lower mechanical efficiency.

On the other hand, if beta is equal to psi_s (the angle between P and R), then the force P is perpendicular to the normal N. In this case, the normal force N would be equal to the component of P that is perpendicular to N, resulting in a lower input energy and therefore a higher mechanical efficiency.

In summary, the angle beta must be equal to psi_s in order to maximize mechanical efficiency. This is because a perpendicular force P results in a lower input energy and therefore a higher mechanical efficiency compared to a parallel force P.

 

[m2003]

I would explain this concept using the principles of friction and inclined planes.

Firstly, let's define what friction and inclined planes are. Friction is a force that resists the motion of two surfaces in contact with each other. Inclined planes are surfaces that are at an angle to the horizontal plane.

Now, let's look at the situation in the attached picture. The inclined plane is at an angle, which means that the force of gravity acting on the object is not directly perpendicular to the surface. This results in a component of the force acting down the inclined plane.

When we add the force P, it is important to remember that it also has components in different directions. The force P is acting perpendicular to the surface, and it also has a component that is parallel to the surface.

Now, let's consider the different angles. If we make angle Beta = 0, that means that the force P is acting parallel to the surface. In this case, there is no component of the force acting perpendicular to the surface, which means that there is no force to counteract the force of gravity acting down the inclined plane. This would result in the object sliding down the plane without any opposition.

On the other hand, if we make angle Beta = psi_s, which is the angle of friction, the force P is acting perpendicular to the surface. This means that there is a component of the force acting perpendicular to the surface, which will counteract the force of gravity acting down the inclined plane. This will result in a more stable equilibrium, where the object will not slide down the plane.

Therefore, in order to achieve the most mechanical efficiency, we need to make angle Beta = psi_s, so that the force P can counteract the force of gravity and keep the object in a stable equilibrium.

I hope this explanation helps to clarify the concept for you. If you have any further questions, please let me know.