lecture 2 kosygin

The general principle states that the effect of a force acting on a rigid body does not change if the force is moved along its line of action to another point on the body.

In clear, the point of application of a force can be moved anywhere along its line of action without changing the external reaction forces on a rigid body.

Any force that has the same magnitude and direction, and which has a point of application somewhere along the same line of action will cause the same acceleration and will result in the same moment. Therefore, the points of application of forces may be moved along the line of action to simplify the analysis of rigid bodies.

Principle of transmissibility of a force

The main thing to be noted is when analyzing the internal forces (stress) in a rigid body, the exact point of application does matter.

This difference in stresses may also result in changes in geometry which will may in turn affect reaction forces. For this reason, the principle of transmissibility should only be used when examining external forces on bodies that are assumed to be rigid.

Limitation of principle of transmissibility: Principle of transmissibility can be used only for rigid bodies and cannot be used for deformable bodies.

The Principle Of Superposition states that when two waves of the same kind meet at a point in space, the resultant displacement at that point is the vector sum of the displacements that the two waves would separately produce at that point.

Net effect of forces applied in any sequence on a body is given by the algebraic sum of effect of individual forces on the body

Assumptions made in Engineering Mechanics i) All bodies are RIGID.

ii) PARTICLE concept can be used wherever applicable.

iii) PRINCIPLE OF PHYSICAL INDEPENDENCE of forces is valid.

iv) PRINCIPLE OF SUPERPOSITION of forces is valid.

v) PRINCIPLE OF TRANSMISSIBILITY of forces is valid.

SYSTEM OF FORCES A group or set of forces is called system of forces.

Coplanar force system: If all the forces in the system lie in a single plane, it is called coplanar force system.

Concurrent force system: If line of action of all the forces in a system passes through a single point it is called concurrent force system.

Collinear force system: In a system, all the forces parallel to each other, if line of action of all forces lie along a single line then it is called a collinear force system.

Concurrent force system: If the line of action of forces forming the system pass through a common point (point of concurrence) then the system is said to be concurrent.

Non-concurrent force system : f the lines of action of forces forming the system do not pass through a common point, then the system is said to be non-concurrent.

Parallel force system :It is a particular case of non-concurrent force system in which the line of action of forces forming the system are parallel.

COMPOSITION OF FORCESComposition of Forces or compounding is the method to find resultant force of

a number of forces acting simultaneously on a body.

Methods for finding resultant force(FR) are• Parallelogram Law• Triangle Law• Polygon law• Resolution of forces or Method of Resolution

GRAPHICAL METHODIn Graphical method, forces are drawn on graph with some suitable scale and their resultant is measured by Parallelogram law, Triangle Law or Polygon Law. While Parallelogram law is also used as analytical method for resolution of forces. Let’s study the laws used for graphical method for composition of forces.

• Parallelogram Law• Triangle Law• Polygon law

PARALLELOGRAM LAW OF FORCES“If two consecutive sides of a parallelogram represents the magnitude and direction of two coplaner forces acting on a particle, then its diagonal represents the Resultant Force(FR) in magnitude and direction.”

Consider two vectors P⃗ and Q ⃗ as shown in the figure. Vector Q⃗ is displaced parallel to itself till the tail end of both the vectors touch at a point O. The parallelogram is completed as shown in the figure. Applying the law of triangle of vectors, to the triangle OAC, we have,

OA + AC = OC or + = P⃗ Q⃗ R⃗

→ → →

Two forces F1 and F2 of magnitudes 80KN and 50KN respectively acting simultaneously on a particle such that the angle between the two forces is 60o. Considering bigger force as F1.

Taking some suitable scale and drawing a parallelogram ABCD with Sides AB and AD representing the forces F1 and F2 and having an angle (θ) between them.F1 as AB = 80 mm = 8 cm,F2 as AD = 50 mm = 5 cmθ = 60o

According to law of parallelogram diagonal AC is representing the resultant force (FR). Length of the diagonal AC is the magnitude of the resultant force and the angle (α), which diagonal AC is making with the side AB, represents the direction of resultant force (FR).By Measurement

Thus the Resultant FR is equal to 113.5 KN acting at an angle (α) 22.5o with F1.

TRIANGLE LAW OF FORCES“If two sides of a triangle, taken in order, represents the magnitude and direction of two coplaner forces acting on a partical, then its third side, taken in opposite order represents the Resultant Force(FR) in magnitude and direction.”

Taking some suitable scale (1KN = 1 mm) drawing a triangle ABC with sides AB and BC representing the forces F1 and F2 and having an angle (θ) between BC and extended side ABF1 as AB = 80 mm = 8 cm,F2 as BC = 50 mm = 5 cmθ = 60o

According to triangle law of forces side AC is representing the resultant force (FR). Length of side AC is the magnitude of the resultant force and the angle (α), which side AC is making with side AB represents the direction of resultant force (FR).By Measurement

Triangle and parallelogram law are best suited for system of forces having two forces acting simultaneously on a body. For a system of forces having more than two forces, Polygon law of forces is applied.


POLYGON LAW OF FORCE“If sides of a polygon, taken in order, represents the magnitude and direction of more than two coplaner forces, acting on a partical, then its closing side, taken in opposite order, represents the Resultant Force(FR) in magnitude and direction.”

Let us consider a system of forces F1, F2, F3 & F4. Having magnitude as 25 KN, 35 KN, 30 KN & 20 KN, making an angle of 30o, 45o, 40o and 80o with horizontal

Taking some suitable scale (1KN = 1 mm) drawing the sides of polygon ABCDE with Sides AB, BC, CD & DE representing the forces F1, F2, F3 & F4, having respective angle between them.Joining point A and E to draw the closing side AE.

Joining point A and E to draw the closing side AE.

F1 as AB = 25 mm = 2.5 cm,F2 as BC = 35 mm = 3.5 cmF3 as CD = 30 mm = 3 cm,F4 as DE = 20 mm = 2 cmAngle between F1 and F2 is 105o

Angle between F2 and F3 is 95o

A

BC

DE

F1 as AB = 25 mm = 2.5 cm,F2 as BC = 35 mm = 3.5 cmF3 as CD = 30 mm = 3 cm,F4 as DE = 20 mm = 2 cm

A B

B C

C D

D E

Rotate 330Original

According to polygon law of forces, side AE is representing the resultant force (FR). Length of side AE is the magnitude of the resultant force and the angle (α) which side AE is making with side AB represents the direction of resultant force (FR).


ANALYTICAL METHODAnalytical method is different from graphical method. Let us study analytical method for finding the resultant force for number of forces acting simultaneously on a particle. There are two analytical methods for composition of forces, Parallelogram law and Method of Resolution.

PARALLELOGRAM LAW OF FORCES

“If two consecutive sides of a parallelogram represents the magnitude and direction of two coplaner forces acting on a partical,

Then the Resultant Force (FR) and angle α is obtained by

Solving Illustration 1 by parallelogram law

Resultant Force FR will be

The angle (α) which resultant FR is making with force F1

RESOLUTION OF FORCES FOR RESULTANT FORCEBefore understanding the method of finding resultant force by method of Resolution let us understand what resolution of forces is.

Resolution of ForcesResolution of forces is the process of breaking a force into two components, basically horizontal and vertical components.Suppose force (F) acting on a particle such that, it is making an angle (θ) with horizontal axis (AB).

Horizontal ComponentIt is the product of magnitude of force and the cosine of the angle, it is making with horizontal axis. Direction of the component is considered same as the direction of horizontal axis, with which it is making the angle θ.Vertical ComponentIt is the product of magnitude of force and the sine of the angle, it is making with horizontal axis. Direction of the component is considered same as the direction of vertical axis, normal to the horizontal axis with which it is making the angle θ.

PRINCIPLE OF FINDING RESULTANT FORCE BY THE METHOD OF RESOLUTION OF FORCES

“The resolved parts of the forces in a particular direction will be equal to the resolved parts of resultant force in same direction.”

Method of resolution of forces starts with summation of Horizontal Components in a system of forces is represented By ∑H while summation of Vertical Components is represented by ∑V.

∑H = Sum of Horizontal Components in a system of forces. = F1cos θ1 + F2cos θ2 + F3cos θ3 +_ _ _+ Fn cos θn

∑V = Sum of Vertical Components in a system of forces. = F1sin θ1 + F2sin θ2 + F3sin θ3 +_ _ _+ Fn sin θn

Direction of resultant in terms of angle(α) is obtained by

Magnitude of resultant force is obtained by

Solving Illustration 2 with method of resolution

Evaluating ∑H and ∑V (Using sign convention)

∑H = 25 cos 30o + 20 cos 80o + (-30 cos 40o) + (-35 cos 45o) = 25 cos 30o + 20 cos 80o – 30 cos 40o – 35 cos 45o

= – 22.60∑H = – 22.60 (-ve)Similarly∑V = 25 sin 30o + 35 sin 45o + (- 30 sin 40o) + (- 20 sin 80o) = 25 sin 30o + 35 sin 45o – 30 sin 40o – 20 sin 80o

= – 1.73∑V = – 1.73 (-ve)Resultant Force FR

Resultant Force FR

The angle (α), which resultant (FR) making with horizontal axis.

In Solution of illustration 2 by polygon law, α is the angle which resultant FR is making with force F1. Here α is the angle, which resultant FR is making with horizontal axis in negative direction.Thus angle between FR and F1 will be

Thus sum of both horizontal and vertical Component of forces is negative, the resultant will lie in 3rd quadrant.

Two forces 5 N and 20 N are acting at an angle of 120 degree between them . Find the resultant force in magnitude and direction.

When vectors act as maximum and minimum?Solution : It is to be noted that the magnitude of the resultant of two vectors is maximum , when the vectors act in the same direction and is minimum when they act in opposite directions.

Two forces 10 N and 14 N are acting upon the a body . What can be the maximum and minimum resultant force on the body ?

Two forces 5 N and 20 N are acting at an angle of 120 degree between them . Find the resultant force in magnitude and direction.

Two forces 10 N and 14 N are acting upon the a body . What can be the maximum and minimum resultant force on the body ?

Drawing Free-Body DiagramsFree-body diagrams are diagrams used to show the relative magnitude and direction of all forces acting upon an object in a given situation.

1. The size of the arrow in a free-body diagram reflects the magnitude of the force.

2. The direction of the arrow shows the direction that the force is acting.

3. Each force arrow in the diagram is labelled to indicate the exact type of force.

It is generally customary in a free-body diagram to represent the object by a box and to draw the force arrow from the centre of the box outward in the direction that the force is acting

Force Symbol Magnitude Direction

Gravity (weight) F(G) = mobject*g (9.8 m/s2) Downward

Normal (surface) F(N) any (up to breaking load) perpendicular to surface

Tension F(T) any (up to breaking load) along string/rope/chain

Friction F(fr)not slipping: between zero and ms*F(N) slipping: = mk*F(N)

Direction opposing relative motion.

•A book is at rest on a tabletop. Diagram the forces acting on the book.

•A girl is suspended motionless from the ceiling by two ropes. Diagram the forces acting on the combination of girl and bar..

http://www.physicsclassroom.com/Class/newtlaws/U2L2c.cfm#2

A book on a table In this example, there are two forces acting on a book at rest on a table:1) The weight W exerted by the earth on the book2) The normal force N exerted by the table on the book.

A suspended blockIn this example, there are two forces acting on the suspended block at rest:1) The weight W exerted by the earth on the block2) The tension force T exerted by the string (or rope) on the block.

A block on a floor with an acting force Fa to pull the block In this example, the block is being pulled and therefore a force of friction acts on the block. So there are four forces acting on the block.

1) The weight W exerted by the earth on the block 2) The normal force N exerted by floor on the block. 3) The acting force Fa to pull the block. 4) The force of friction Ff exerted by floor on the block in the direction opposite the motion due to Fa.

http://www.problemsphysics.com/forces/force_friction.html

A falling object In this example, there is only one force action on the falling object. 1) The weight W exerted by the earth on the falling object.

A falling object with Air Frictionhere only two force action on the falling object. 1) The weight W exerted by the earth on the falling object.2) The Friction FAir

An box on an incline plane (with no frictions) In this example, we assume that the inclined plane is frictionless; two forces act on the box: 1) The weight W exerted by the earth on the box. 2) The normal force N exerted by the inclined plane on the box. N is normal to the inclined plane.

An box on an incline plane with an acting force and friction considered. In this example, a force Fa pulls the box upward and frictions are not negligible. Four forces act on the box: 1) The weight W exerted by the earth on the box. 2) The normal force N exerted by the inclined plane on the box. N is normal to the inclined plane. 3) The acting force Fa. 4) The force of friction Fs exerted by the inclined plane on the box in the direction opposite the motion due to Fa.

A block suspended to the ceiling using three strings.

A) free body diagram for the block; two forces (lower part of figure below) 1) The weight W exerted by the earth on the box. 2) The tension force T '3 exerted by the string on the block.

B) free body diagram of point P; three forces (upper part of figure below) 1) Tension T1 2) Tension T2 3) Tension T3

A system with two blocks, an inclined plane and a pulley.

A) free body diagram for block m1 (left of figure below) 1) The weight W1 exerted by the earth on the box. 2) The normal force N 3) The force of friction Fk 4) The tension force T exerted by the string on the block m1. B) free body diagram of block m2 (right of figure below) 1) The weight of the block W2 2) Tension T '.

Determining the Net Force

In the statement of Newton's first law, the unbalanced force refers to that force that does not become completely balanced (or canceled) by the other individual forces. If either all the vertical forces (up and down) do not cancel each other and/or all horizontal forces do not cancel each other, then an unbalanced force exists. The existence of an unbalanced force for a given situation can be quickly realized by looking at the free-body diagram for that situation. Free-body diagrams for three situations are shown below. Note that the actual magnitudes of the individual forces are indicated on the diagram.

Free-body diagrams for four situations are shown below. The net force is known for each situation. However, the magnitudes of a few of the individual forces are not known. Analyze each situation individually and determine the magnitude of the unknown forces.

The force F is producing, simultaneous x displacement and y-displacement. The part of the force F which is producing x displacement is called x component or horizontal component of the force F(Fx). The part of the force F which produces y – displacement is called y component of the force or vertical component of force F(Fy).

The technique of finding a component of a force along any direction is called resolution of force. The component of a force along any direction is called the resolved component. The components of a force determined along two mutually perpendicular direction are called rectangular components.

RESOLUTION OF A FORCE

In the diagram the force F is resolved into two perpendicular component forces that is the Fy and Fx components (using parallelogram method).

To calculate the magnitude of the vertical (Fy) and horizontal (Fx) forces, we can use simple trigonometry.

Fx = F cos θ , Since cos θ = (Fx/F)

Fy = F sin θ, Since sin θ = (Fy/ F)

To calculate the magnitude of the vertical (Fy) and horizontal (Fx) forces, we can use simple trigonometry.

Fx = F cos θ , Since cos θ = (Fx/F)

Fy = F sin θ, Since sin θ = (Fy/ F)

Note: 1. Sign convention for the direction of components.

The horizontal component or ‘X’ component of a force acting along x direction is the force itself. Whereas, its vertical component or y-component is zero.

2.

‘x’ component of a force acting along Y direction is zero. Whereas, its ‘y’ component is equal to itself.

4. If a force is inclined at 45o

w.r.t. x axis or y axis then its x component will be equal to y component (Fx= Fy).