Equilibrium of Forces (PDF)

EQUILIBRIUM OF FORCES Equilibrium is a state of stability. An object is said to be in equilibrium when the body as a whole remains at rest or moves in a straight line at a constant speed. Such a body or object is said to be in a state of equilibrium. An object can also be said to be in equilibrium w...

EQUILIBRIUM OF FORCES Equilibrium is a state of stability. An object is said to be in equilibrium when the body as a whole remains at rest or moves in a straight line at a constant speed. Such a body or object is said to be in a state of equilibrium. An object can also be said to be in equilibrium when the object is either not rotating at all or rotating at a constant angular velocity. Such a body is said to be in dynamic equilibrium. RESULTANT FORCE: The resultant force is that single force which acts alone and will have the same effect in magnitude and direction as the two or more forces acting together. The resultant force can also be defined as a single force that can be used to represent or replace a system of forces. The resultant force is obtained by the parallelogram law of vectors. EQUILIBRANT FORCES: Equilibrant of two or more forces is that single force which will balance all other forces together. An equilibrant can also be defined as a single force that is needed to keep a body in a state of equilibrium. It is equal in magnitude but opposite in direction to the resultant forces. B C Q R O P A E The resultant of two forces P and Q is counter balanced by the equilibrant force E as shown in the diagram above. The three forces P,Q and E keeps the point O in equilibrium. The forces that keep the body in equilibrium can be represented in magnitude and direction by the three sides of the triangle. Consider three forces F1, F2, and F3 acting at a point. They are in equilibrium as shown in the diagram below: F2 F1 F3 F1 is the equilibrant of F2 and F3 F2 is the equilibrant of F1 and F3 F3 is the equilibrant of F1 and F2 PARALLEL FORCES: These are forces attached with an inextensible string on a horizontal platform or surface. Moment of a Force The moment of a force about a point is defined as the product of the force and the perpendicular distance from the point to the line of action of the force. It is the turning effect of a force. Mathematically; Moment = Force x perpendicular distance = F X d = Fd Thus for the diagram below, the moment of the force, F about O = f x h Case 1 h F Case 2 In the figure below the moment of force, F about O = F x d Sin θ Case 3 F1 F2 X1 x2 X4 x3  F4 F3 A.C.W C.W Where there are several forces acting on a body, the resultant moment is the algebraic sum of the moments. If we take clockwise moments as positive then the anticlockwise moment should be negative. A.C.W = Anticlockwise moment C.W = clockwise moment EXAMPLE 1: A meter rule is pivoted at its midpoint C with a vertical force of 10N hanging from the distance 30 cm from C. at what distance must a 15N force hang to balance the ruler horizontally. Solution: 30cm y 10N 15N For the ruler to balance horizontally, we have Clockwise moment = anticlockwise moment 10 x 30 = 15 x y Y= = 20 cm EXAMPLE 2: A uniform meter rule balances on knife edge at the55cm mark when a mass of 40g is hung from the 95cm mark. Find the weight of the ruler. Solution: 50cm 5cm 4cm W 0.4N W x 5cm = 0.4N x 40cm W X 0.05m = 0.4N X 0.4m W= =3.2N Mass of ruler = 0.32kg Examples: A beam A B C D E is shown resting on two points at B and D, The weight of the beam is 450N. 55N and 70N are attached at A and E at the distances shown below. Find the reaction at the pivots. COUPLE This is a pair of equal forces acting on a body in different direction. When two equal and parallel but opposite forces acts on a body they constitute a couple or torque. The turning moment of a couple is called TORQUE. Examples of couple are: turning of water tap, screw driver, etc. the resultant force of a couple is zero, but the resultant moment is “Fd”, the product of one of the forces and the perpendicular distance between them. F  F Resultant force of a couple Centre of Gravity The centre of gravity of a body is defined as the point through which its resultant weight appears to act. The centre of gravity of a uniform symmetry body is at the centre of the body. e.g. 1. The centre of gravity of a uniform meter rule is at 50cm mark. 2. The centre of gravity of a circular object is at the centre. 3. The centre of gravity of a square, rectangle and parallelogram lies at the point of intersection of their diagonals. 4. The centre of gravity of a triangle lies at the median points. Principle of Moment The principle of moment states that if a body is in equilibrium, then the sum of the clockwise moment about any point is equal to the sum of the anticlockwise moment about the same point. GENERAL CONDITION FOR EQUILIBRIUM UNDER THE ACTION OF PARALLEL COPLANAR FORCES A body is in equilibrium if it does not move or rotate under the action of a system of forces, i.e. if the resultant force and moment are each zero. From this we deduce the three conditions for equilibrium of a body. This is otherwise known as condition necessary for equilibrium. 1. The sum of the clockwise moment about a point must be equal to the sum of anticlockwise turning moment at that same point. 2. The sum of forces acting in one direction must be equal to the sum of forces acting in opposite direction, i.e. upward forces equals downward forces. In summary the algebraic sum of moment about a point is zero. 3. The total component in a particular direction equals the total component in opposite direction. STABILITY Types of Stability 1. Stable Equilibrium: A body is said to be in a point of stable equilibrium, if when slightly displaced or tilted, it returns to its original position, e.g. A cone resting on its base. 2. Unstable Equilibrium: A body is said to be in unstable equilibrium if when tilted or slightly displaced moves further away from its original position e.g. a cone resting on its apex end. 3. Neutral Equilibrium: A body is said to be at a point of neutral equilibrium if when tilted it rest in another position which is different from its original position e.g. A cone resting on its slant side. 4. Static Equilibrium: This is the kind of equilibrium possessed by stationary bodies. All bodies at rest are in a static equilibrium. 5. Dynamic Equilibrium: This is the type of equilibrium possessed by bodies moving at a constant speed or uniform velocity Stable equilibrium Unstable equilibrium Neutral equilibrium