Classwork Series and Exercises {Physics- SS2}: Equilibrium of Forces

Concept of Equilibrium: Dynamic and Static Equilibrium

An object is in equilibrium when it is not accelerated, that is, there is no net force acting on it in any direction. For such a body in equilibrium, the forces acting on it are so related in magnitude and direction that no acceleration results. Thus the body may either be at rest or may be moving with constant velocity. Such bodies at rest are said to be in static equilibrium. When the body is moving with a constant velocity in a straight line, or when it is rotating with a constant angular velocity about a fixed axis through its center of mass, the equilibrium is said to be kinetic or dynamic equilibrium. This means that the forces that set the body in motion balance the forces that resist the motion. For example a car running at a constant speed for a considerable length of time is in dynamic equilibrium.

A body is said to be in equilibrium when:

(i) the body as a whole either remains at rest or moves with a straight with constant speed and;

(ii) the body is either not rotating at all or is rotating at a constant angular velocity.

When an object is in equilibrium the sum of all the forces is zero. The object is at rest or moving at a constant speed in a straight line.

A single force that will place an object in equilibrium is called the equilibrant. The equilibrant is equal in magnitude to the resultant and opposite in direction.

To determine the equilibrant find the resultant and then reverse the direction.

Resultant and Equilibrant Forces

Students encounter the equation f=ma soon after they begin to study physics. If an object experiences a net force, it will experience a corresponding acceleration proportional to the magnitude of that net force. When multiple forces simultaneously act upon the object, you’ll need to add the different forces together to get the resultant force. A force identical in magnitude but in the opposite direction is the equilibrant force

The resultant force is that single force which acting alone will have the same effect in magnitude and direction as to or more forces acting together.

The equilibrant of two or more forces is that single force which will balance all the other forces taken together. It is equal in magnitude but opposite in direction to the resultant force.

If three forces F₁, F­­₂ and F₃ acting at a point are in equilibrium, the resultant of any two of the forces is equal but opposite in direction to the third force. Anyone of these forces is said to be the equilibrant of the other two. The equilibrant of F­₁ and F₂ is F₃, the equilibrant of F₂ and F₃ is F₁ and the equilibrant of F₁ and F₃ is F­₂.

Also the line of action of the three forces keeping a body in equilibrium must all pass through one point. Otherwise the resultant of two of the forces cannot be counterbalanced by the third force.

Conditions of Equilibrium under the action of Parallel Coplanar Forces

Coplanar forces are forces that lie in the same plane. Parallel forces are forces whose lines of action are parallel to each other.

A body acted upon by several forces is s aid to be in equilibrium if it does not move or rotate. Under this equilibrium condition, the sum of the forces acting in one direction (e.g. upwards) must be equal to the sum of the forces acting in opposite direction (e.g. downwards). Thus the total forces acting upwards must balance the total forces acting downwards.

Also the body can also remain in equilibrium if the moment of the forces about any point act in such a way as to cancel each other. That is, the total clockwise moments of all the forces about any point of the object must be exactly counterbalanced by the by the total anticlockwise moment about the same point.

Hence the two conditions for equilibrium of parallel coplanar forces can be stated as follows:

1. Forces

The algebraic sum of the forces acting on the body in any given direction must be zero. That is, the sum of the upward forces must equal the sum of the downward forces or the sum of the forces acting in one direction must be equal to the sum of the forces acting in opposite direction.

2. Moments

The algebraic sum of the moments of all the forces about any point on the body must be zero, or the total clockwise moments of the forces about any point on the body must be equal to the total anticlockwise moments of the forces about the same point. The second condition above is known as the

Principle of Moments

 The Principle of Moments states that if a body is in equilibrium, then the sum of the clockwise moments about any point on the body is equal to the sum of the anticlockwise moments about the same point.

 

Turning Effects: When a force is applied a distance away from the centre of mass (COM) then we produce a turning effect or “moment”. If you have a situation where an object is in equilibrium then you can solve a problem by equating all the turning effects about any point in the situation…

Clockwise moments = anticlockwise moments

Moments: When a force is applied it has to be perpendicular to the distance to be classed at a moment. To solve problems you sum up all the moments applied “perpendicular” to a point…

Clockwise moments = anticlockwise moments

See-Saw: When we have simple see-saw we can sum up by just balancing from the middle of the see-saw and add any Wd from the pivot point..

W₁d₁ = W₂d₂

or if you have two people on one side…

W₁d₁ = W₂d₂ + W₃d₃

Complex Moments: Now what happens if the see-saw is not equal on each side. Now the weight of the see-saw must be taken into account when taking moments. This acts from the COM but looks strange at first…

W₁d₁ = W₂d₂ + W₃d₃

Conditions of Equilibrium under the action of non-parallel coplanar forces

Sometimes non parallel forces acts on a body. Such forces can also keep the body in equilibrium. The non parallel forces can easily be resolved into horizontal and vertical components giving rise to two sets of parallel forces at right angle to each other. The problem can then be treated as in Conditions of Equilibrium under the action of Parallel Coplanar Forces and its method can be applied to both the vertical and the horizontal components separately. The two conditions of equilibrium under the action of non parallel forces then become:

1. Forces

The vector sum of all the forces acting on the body must be zero. In order words the algebraic sum of all the forces or components of the forces acting on the body in any direction must be zero. Thus, (i) the algebraic sum of the horizontal components of the forces must be zero, i.e. SF_x = 0, (ii) the algebraic of the vertical components of the forces must be zero, i.e., SF_y = 0.

2. Moments

The algebraic sum of the moments of all the forces about any axis perpendicular to the plane of the forces must be zero. In other words the sum of the clockwise moments about any such axis equals the sum of the anticlockwise, moments about the same axis.

Equilibrium under action of three non-parallel forces

(i) The three forces must lie in a plane.

(ii) Their lines of action must intersect in a common point.

(iii) The vector representing the three forces can be arranged to form a closed triangle with sides respectively parallel to the directions and proportional in length to the magnitude of the forces.

Example 1: A 10 kg sign is being held up by two wires that each make a 30° angle with the ground. Determine the tension (force) in each of the wires.

F_g = mg
= (10) (9.81)
F_g = 98.1 N

sinq = opp / hyp
sinq = 1/2 F_g / F_T
F_T = (1/2 F_g) / sinq
= (49.05) / sin 30° = 98.1 N

EXERCISES

Lets see how much you’ve learnt, attach the following answers to the comment below

Two forces are pushing an object along the ground. One force is 10 N [fusion_builder_container hundred_percent=”yes” overflow=”visible”][fusion_builder_row][fusion_builder_column type=”1_1″ background_position=”left top” background_color=”” border_size=”” border_color=”” border_style=”solid” spacing=”yes” background_image=”” background_repeat=”no-repeat” padding=”” margin_top=”0px” margin_bottom=”0px” class=”” id=”” animation_type=”” animation_speed=”0.3″ animation_direction=”left” hide_on_mobile=”no” center_content=”no” min_height=”none”][W] and the other is 8.0 N [S]. The diagram below shows the equilibrant of these two forces.

1. Determine the equilibrant. A. 14 N B. 13 N C. 16 N  D. 17
2. Calculate the angle where the 8.0 N force and the equilibrant touch A. 51⁰ B. 54.5⁰  C. 56⁰  D. 59⁰
3. Which of these is not correct about equilibrium? A. An object is in equilibrium when it is not accelerated B. An object is in equilibrium when there is no net force acting on it in any direction C. A body is in equilibrium when there is no torque acting on it D. A body is in equilibrium when it is vibrating.
4. Coplanar forces are forces that lie in A. the same plane B. different planes C. a higher plane  D. a lower plane
5. A car running at a constant speed for a considerable length of time is in A. Static equilibrium B. dynamic equilibrium C. Elastic equilibrium D. Inelastic equilibrium

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