Chapter Two Notes:Mechanical Equilibrium

A force is a push or a pull:◦ A force is necessary to cause a change in the state

of motion of an object.◦ Net Force– The net force is the sum of all forces

acting on an object. You pull horizontally on an object with a force of 10

pounds. A friend pulls on the same object with a force of 5

pounds If the friend pulls in the same direction, the net force is

15 pounds. If the friend pulls in the opposite direction, the net

force is 5 pounds in the direction you are pulling.

Hold an object in your hand, and the net force is zero. The earth pulls down on the object, while you push up to hold it there.

◦ Tension and Weight– Suspend an object from a rope, and the force the

rope exerts on the object to keep it from falling is called tension. The pull of earth on the object is called weight.

The net force on the object is zero…◦ Force Vectors-

Forces are represented by arrows, in the direction of the force. The length of the arrow represents the amount (magnitude) of the force.

A vector is the name we give to these arrows. A vector quantity is a quantity that needs both a magnitude and a direction to be complete.

A scalar quantity, on the other hand can be described by only a magnitude, and has no direction.

We will come back to vectors in chapter 5.

Mechanical Equilibrium is a state where no physical changes occur. In other words, the net force is zero.

Equilibrium Rule - You can express the equilibrium rule mathematically!

ΣF = 0The symbol Σ stands for “sum of” and F stands for “Force”.

(This represents the fact that all the forces acting on an object add vectorially to zero.)

If you have an object at rest on a desk, one force is gravity pulling down on the object. Since the object is in equilibrium, the net force must be zero, and there must be another force acting on the object to produce this state. That force must be opposite the force of gravity, which is acting down.

That force must be upward, and must be provided by the desk. We call this the support force.

For an object at rest on a horizontal surface, the support force must equal the object’s weight.

When an object isn’t moving it’s in equilibrium. The forces on it add up to zero. But the state of rest is only one for of equilibrium. An object moving in a straight line path at a constant speed is also in a state of equilibrium.

Once in motion, if there is no net force to change the state of motion, it’s in equilibrium.

Object at rest are said to be in static equilibrium; objects moving at constant speed in a straight-line path are said to be in dynamic equilibrium.

There are several ways to find the sum of two or more vectors.

(Head-Tail, Components, Parallelogram Method)

Addition of vectors by Head to Tail method (Graphical Method)

(not in our book!)   Head to Tail method or graphical method is one of the

easiest method used to find the resultant vector of two of more than two vectors.

DETAILS OF METHOD  

Consider two vectors and acting in the directions shown below:

In order to get their resultant vector by head to tail method we must follow the following steps:

STEP # 1   Choose a suitable scale for the vectors so

that they can be plotted on the paper.

STEP # 2

  Draw representative line of vector . Draw representative line of vector such that the tail of coincides with the head of vector .

STEP # 3   Join 'O' and 'B'.

represents resultant vector of given vectors and i.e.

STEP # 4   Measure the length of line segment and multiply it

with the scale chosen initially to get the magnitude of resultant vector .

STEP # 5   The direction of the resultant vector is directed

from the tail of vector to the head of vector . -----------------////-----------------

NEXT METHOD: (The Books method)

Parallelogram Method to Calculate Resultant

Before tackling the parallelogram method for solving resultant vectors, you should be comfortable with the following topics:

SOHCAHTOA (basic sine, cosine, tangent ) law of cosines law of sines the following properties of parallelograms

◦ opposite sides of parallelograms are congruent◦ opposite angles of parallelograms are congruent

To best understand how the parallelogram method works, lets examine the two vectors below. The vectors have magnitudes of 17 and 28 and the angle between them is 66°. Our goal is to use the parallelogram method to determine the magnitude of the resultant.

Step 1) Draw a parallelogram based on the two vectors that you already have. These vectors will be two sides of the parallelogram (not the opposite sides since they have the angle between them)

Step 2) We now have a parallelogram and know two angles (opposite angles of parallelograms are congruent). We can also figure out the other pair of angles since the other pair are congruent and all four angles must add up to 360. ???

Determine other Angles Step 4) Draw the parallelograms diagonal. This diagonal is

the resultant vector

Step 1) Draw a parallelogram based on the two vectors that you already have. These vectors will be two sides of the parallelogram (not the opposite sides since they have the angle between them)

Step 2) We now have a parallelogram and know two angles (opposite angles of parallelograms are congruent). We can also figure out the other pair of angles since the other pair are congruent and all four angles must add up to 360. ???

Determine other Angles Step 4) Draw the parallelograms diagonal. This diagonal is

the resultant vector

Step 1) Draw a parallelogram based on the two vectors that you already have. These vectors will be two sides of the parallelogram (not the opposite sides since they have the angle between them)

Step 2) We now have a parallelogram and know two angles (opposite angles of parallelograms are congruent). We can also figure out the other pair of angles since the other pair are congruent and all four angles must add up to 360. ???

Step 3) Determine other Angles Step 4) Draw the parallelograms diagonal. This diagonal is

the resultant vector

Step 1) Draw a parallelogram based on the two vectors that you already have. These vectors will be two sides of the parallelogram (not the opposite sides since they have the angle between them)

Step 2) We now have a parallelogram and know two angles (opposite angles of parallelograms are congruent). We can also figure out the other pair of angles since the other pair are congruent and all four angles must add up to 360. ???

Step 3) Determine other Angles Step 4) Draw the parallelograms diagonal. This diagonal is

the resultant vector

To Find the Length of Diagonal Use the law of cosines to determine the

length of the resultant