*Notes to a video lecture on http://www.unizor.com*

__Equilibrium__

*Statics*is a part of Mechanics that studies forces not as

quantitative measure of a motion they cause (that is a subject of

Dynamics), but from the more fundamental viewpoint of whether these

forces are or are not balanced, that might or might not cause the motion

of objects these forces act upon.

Historically, studies of static aspects of forces precede quantitative

studies of motion in Dynamics. People, first of all, were concerned with

how to build bridges and buildings, so that they stay in place and not

destroyed by gravity, wind or load. And a concept of

*equilibrium*is a central point of Statics, it's goal and purpose.

*Equilibrium*, as it is understood in Statics, is a state of forces, that result in an object acted upon to be in the state of rest.A person standing on a floor is at rest because two main forces that act

upon him, the gravity and the reaction of the floor are equal in

magnitude and opposite in direction, they balance each other, which

results in a state of

*equilibrium*.

A person standing on the weights to check his weight is in a state of

*equilibrium*

because its weight is balanced by elastic force inside the weights that

is equal in magnitude and opposite in direction to the force of

gravity. The equality in magnitude allows to measure the weight by

measuring the elasticity inside the weights.

A building is in a state of

*equilibrium*because its each part's weight is balanced by equal in magnitude and opposite in direction force of reaction.

An airplane, flying horizontally on its route, is in a state of

*vertical equilibrium*

because its weight is balanced by the lifting power of the air under

its wings, where the air pressure is larger because of the wings' shape.

Let's study forces applied to the same object and balance each other causing this object to be in

*equilibrium*.

As we know, force is a vector.

**All forces acting on the same point-object can be added as vectors**, using the rules of the Vector Algebra.

**If the result is a null-vector, we have an**

*equilibrium*.So, if there are

*N*forces

**(where**

*F*_{i}*1 ≤*)

**i**≤ N**condition of equilibrium**is

Σ

**.**

*F*_{i}= 0Let's consider a few examples.

1. An object of weight

**is hanging on three threads as pictured below.**

*W*Thread

**is horizontal, thread**

*a***is at angle**

*b***to horizon.**

*φ*The system is in equilibrium.

What is the magnitude of tension forces

**and**

*T*_{a}**on threads**

*T*_{b}**and**

*a***?**

*b**Solution*

There are three forces acting at the point where all threads are connected:

vertical down - the weight

**of the object;**

*W*tension

**horizontally to the left along the**

*T*_{a}**thread;**

*a*tension

**at angle**

*T*_{b}**to horizon along the**

*φ***thread.**

*b*Since the system is in equilibrium, this point where threads come

together does not move in any direction. In particular, it does not move

in a vertical, nor horizontal direction.

This is sufficient to determine magnitude of tension vectors

**and**

*T*_{a}**.**

*T*_{b}In the horizontal direction the force of gravity is irrelevant, so the

only two forces acting on out object in horizontal direction are tension

**acting to the left and a projection of tension**

*T*_{a}**on the horizontal line acting to the right.**

*T*_{b}That gives the first equation about magnitudes of tension forces:

*T*_{a}= T_{b}·cos(φ)Now consider the vertical direction. The thread

**is irrelevant. So, the only two forces acting vertically, are weight**

*a***pulling down and a vertical component of**

*W***that is equal to**

*T*_{b}**pulling upwards.**

*T*_{b}·sin(φ)This gives the second equation

*T*_{b}·sin(φ) = WFrom these equations we derive:

*T*_{b}= W/sin(φ)After which we can find

**:**

*T*_{a}

*T*_{a}= W·cos(φ)/sin(φ)2. Two objects of mass

**(larger, on an inclined plane) and**

*M***(smaller, hanging freely over the edge of an inclined plane) are connected with a thread that goes over a pulley.**

*m*What is the angle of an inclined plane

**for this system to be in equilibrium?**

*φ*Ignore the friction.

*Solution*

Let's represent the weight of an object on an inclined plane as a sum of two forces:

one is perpendicular to the plane and balanced by a plane's reaction;

another is parallel to the plane and balanced by a tension of a thread.

To be in equilibrium, the force of weight of an object hanging freely

over the edge of an inclined plane must be equal in magnitude to a

tension of a thread (same thread, same tension as before).

Therefore, a component of the weight of an object on a plane that goes

parallel to a plane must be equal in magnitude to a weight of another

object.

*M·g·sin(φ) = m·g*From this we derive

*sin(φ) = m/M*

*φ = arcsin(m/M)*
## No comments:

Post a Comment