Why is the Touque rxF?

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Objects in the world seem free to move, but there are many constraints that limit the movement of objects.

The ball on the comb always has to roll along and the swing always has to go back and forth.

If there were no constraints, the world would have been a mess.

Imagine a sliding book on a desk can fly anywhere on or under the desk.

And thankfully, these constraints help us a lot when we analyze the motion of an object.

For example, objects that move on the ground allow us to think about three-dimensional space in two dimensions.

 How grateful are you?

Rotation force, called torque in English, is also a concept designed to easily analyze the movement of objects under these constraints.

The spherical coordinate system is a coordinate system that represents the position of an object as r,ta, and quai.

When looking at space as a spherical coordinate system, there is a movement in which the r does not change in the position of the object.

Like the movement of a rock that rotates around the origin of the coordinate system,

 or the movement of a rigid body.

A rigid body is an object whose distance between components does not change.

It's a very hard object.

So the movement that the size of the r coordinate axis doesn't change is called rotary motion.

 In the rotational motion of an object, there is no need to think about a change in the direction of r.

 It was unnecessary to apply the same concepts that we used to use in the rotation movement.

The relationship between force and acceleration is very important in analyzing the motion of an object.

But in rotational motion, angular acceleration, which is the rotational acceleration, is a more useful amount for analyzing motion.

So what we're going to do is to explain the relationship between the forces that work and the angular acceleration, 

which we're going to find out today, the turning force in English.

In space, there is a particle in the position of the rvector.

This particle has neither mass nor volume, but is held at its origin by any existing string.

So this particle can't change in the direction of the r, it can only rotate.

A force F acts on this particle.

This force accelerates the object.

However, particles cannot be accelerated in the direction of the r cap, 

so dividing the force acting on the particles by the components in the direction of the r vector and the vertical direction of the r vector can only accelerate the particles.

When the angle between position vector and force vector F is A, the vertical force of the r vector becomes Fsin(A).

Consider the θcoordinates of the plane in which the particles rotate.

As particles move, the size of the θ changes.

The distance S and θ at which the particles move have the following relationships:

S=rθ

The change in travel distance over time is speed.

(△S / △T ) = Speed V = r△θ / △t

That's the conclusion that.

△θ / △t is called angular velocity, which is the speed at which the angle changes.

It uses the symbol ω (Omega)
Therefore, the speed V is rω.
In the same logic, the acceleration a is rα.
α is the angular acceleration which means the change of ω over time.

In rotational motion, the velocity and acceleration of particles are very difficult to analyze because the direction of the movement continues to change depending on the position.

So it's easy to use angular velocity and angular acceleration, which are the components of the spherical coordinate system.

When a force is applied to a particle from the outside, the force produces the acceleration of the particle, which means that the particle has angular acceleration.

Therefore, there is a relationship between force F and angular acceleration alpha.

We have to find it.

The r-cap vertical component of the force, Fsin(A), produces acceleration a.

In accordance with Newton's second law, Fsin(A) in minutes is a.

This acceleration a shows that angular acceleration alpha has the following relationship. 

a=rα

Orient the angular acceleration alpha to create angular acceleration.

Set this direction to vectors perpendicular to the plane in which the particles accelerate.

So the unit vector that represents this direction is as shown in the figure.

Because the r cap and f cap are unit vectors of R vector and F vector in size 1 and point out the direction of the R vector and F vector respectively.

So the vector product of the two is SinA in size, and the direction is perpendicular to the plane that the two vectors make.

This is the direction of angular acceleration that we set. 

If you divide it into sinA and change the size to 1, it becomes a unit vector that tells you the direction. 

Let's call this unit vector alpha cap.

A is the relation of scalar, which is still the size of the rα

I will multiply this by the side of the unit vector α cap that indicates the direction.

And now this scalar has a breath of direction. 

ααcap is alpha in size and angular acceleration vector alpha in direction.

We need to clean up the left side a little because we need to change it to a relationship with power.

The acceleration a is produced by the tangent component of the rotational orbit of force F. 

It was FsinA / m.

So organize a (rcapxFcap)/sinA as shown.

The FF cap is a force vector F.

So we found out the relationship between the force vector F and the angular acceleration vector alpha, as we initially aimed for.

This relationship can be arranged more simply by multiplying both sides of the relationship by the straight line r from the origin of the coordinate system to the particle.

rxF=mr²α

If you look at this equation about rotation, it looks like Newton's Second Law.

Newton's second law, F, means that the mass is the property of the material that determines the magnitude of the acceleration when the force is applied to the material.

 

Similarly, when the rcapXF vector is applied to a substance, it can be thought of as a force to produce angular acceleration for rotational motion.

And the mass that determines the relationship between this force and angular acceleration is called mr2.

So we call this force rvector x F vector turning, torque in English, and mr2 is the rotational inertia of an object.

We can think of some of the principles of rotational motion in this law.

1. When a force is applied to an object with a constraint that requires rotational motion, the greater the distance from the axis of rotation, the greater the angular acceleration.

2. When the same force is applied at the same point as the axis of rotation, the greatest turning force is obtained when the direction of the force is parallel to the direction of rotation, i.e. when it is applied perpendicular to the position vector r.

3. The larger the rotational inertia, the smaller the angular acceleration.

This means that the greater the rotational inertia, the stronger the tendency to maintain the current state of rotation.

This rotational inertia is the greater the mass,

 and the greater the mass present, the greater the position of the mass from the axis of rotation.

to sum up,

Rotation force (torque) is a significant concept for analyzing rotational motion, which has the following relationship with angular acceleration alpha:

τ=Iα

Here I is the amount of rotational inertia that tries to maintain the current rotation state, which depends on the size and location of the mass.