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2.9 Equilibrium

Forces in Equilibrium

The principle of the forces in equilibrium states,

“ When forces act upon an object , the object is said to be in a state of equilibrium when the resulting force acting on the object is zero ( no net force acting upon it) ”

When the equilibrium is reached, then the object is in two states, that is

(i)      remains stationary  (if the object is stationary)

(ii)     moves at a constant velocity ( if the object is moving)

Based on   , F = ma  atau  a = F

                                                                 m

When the equilibrium of forces is achieved, then, F = 0  ,  hence  a =0

Thus a = 0  , it means the object remains stationary or moves at a constant velocity.

 

Newton’s Third Law of Motion

Newton’s third law of motion states , “ To every action there is an equal but opposite direction”

                                                               

Examples Forces in Equilibrium

(a)

clip_image002

                                Weight = Normal reaction

(b)          

                  clip_image004

                                Weight = Tension

(c)          

                     clip_image006

                                            Buoyant force = Weight

 

(d)

clip_image008

 

                                     Weight = Normal reaction

(e)          

clip_image010

                               

Weight = Normal reaction

Pulling force = Frictional force

(f)

clip_image012

 

                                Weight = Lifting force

                                Driving force = Dragging force

(g)

clip_image014

Weight = Normal reaction

Engine thrust = Air resistance + Frictional force

(h)

clip_image016

 

                Buoyant force = Weight of load + Weight of helium gas

 

Two Forces in Equilibrium

clip_image018

 

                                                                          P  +  Q = 0

                              We can rewritten into  P  = – Q

 

Example 1

Figure shows a stationary wooden block of mass 2 kg resting on a table.

clip_image020

Calculate

(a)           the weight of the wooden block

(b)           the normal reaction

 

Solution

Three Forces in Equilibrium

 

clip_image022 

P  +  Q  + R = 0

 

When three forces in equilibrium the  triangle of forces in one direction (in order)

Example 2

 

The following figure shows a steel sphere of mass 12 kg suspended from a length of rope which is pulled to the side by a horizontal force of  M. The tension of another rope is N.

                 clip_image024

(a)           Draw a  triangle of forces.

(b)           Calculate the value of

                (i)            M

                (ii)           N

 

Solution

 

 

 

Resultant force

 

Force is a  vector quantity  and hence it has magnitude and direction.

Two or more forces  which act on an object can be combined into a single force called the resultant force.

 

If  two forces are in same line, vector addition is easy. We simply add the forces if both pull or push together;

subtract  them if one is in the opposite direction.

 

If they are at an angle, the resultant force can be determined by the triangle method and the parallelogram method.

 

Parallelogram method :

 

clip_image025

 

In this method the tail of the first vector is joined  to the tail of second vector and then draw a parallelogram.

The diagonal represents the resultant force.

 

Triangle method:

 

clip_image026

 

In  this method the tip of the first vector is joined  to the tail of second vector and then draw a  line to complete the triangle.

The third side represents the resultant force.

 

Example 3

 

Find the resultant force for the following figure:-

 

clip_image028

Solution:

 

 

 

 

 

 

Example  4

clip_image030

The figure shows a trolley is pulled by two forces

 

What is the magnitude and the direction of the resultant force acting on the trolley.

Solution

 

 

                   

Example  5

 

Figure shows a boat is pulled by  two forces.

Calculate the magnitude of the resultant force acting on the trolley.

clip_image032

Solution:

 

 

 

 

 

 

 

 

Resolution of  forces

 

A force can be resolved into two components, that is,

 (i)           the horizontal component, Fx and

(ii)           the vertical component , Fy

clip_image034

                                Fx  = F cos q

                                Fy  = F sin q

 

·         q  is an angle between the force F to the horizontal line

·         the sign of the force depend on the quadrant where the force , F is placed

 

For an object on a inclined plane, the weight,W of the object can be resolved into two components ;

(i)            parallel to inclined plane, A

(ii)           perpendicular to inclined plane,B

 

clip_image036

 

 

A = W sin q

 B = W kos q

 

Example  6

 

Find the values of Px and Py for the following figures.

clip_image037

Solution:

 

 

 

 

 

 

 

 

 

Example 7

clip_image039

Figure shows a stationary wooden block of mass 50 g which is placed on a inclined plane that is at an angle of 40o to the horizontal. 

What is the magnitude of the weight parallel to the inclined plane.

 

 Solution

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