CENTROID what is centroid?
The Centroid of any object refers to the point within which the downward force of gravity appears to act.
On any point along a vertical line that passes through the centroid, the object remains balanced. Estimating Location Of CENTROID
Eye estimation: By drawing at least two lines, each of which seems to divide the area into two parts, each of which appears to have the same moment about the line. The intersection of these lines should locate the CENTROID approximately. CONTINUE
PRINCIPLE OF SYMMETRY: If a plane figure has a line or lines of symmetry, its centroid lies on that line or lines. Formulas to Find Out CENTRIOD
Plane Triangle: The point through which all the three medians of a triangle pass is called Centroid of the triangle and it divides each median in the ratio at 2:1.
Centroid=
where , (x1, y1) (x2, y2) (x3, y3)be the coordinates of the vertices of the triangle. Formulas to Find Out CENTRIOD
Rectangle: Centroid of rectangle lies at intersection of two diagonals. Diagonals intersect at width (b/2) from reference x-axis and at height (h/2) from reference y-axis.
Centroid of rectangular area Continue
Examples: Find the centroid of rectangular wall whose height is 12 ft. and base length of wall is 24 ft
SOLUTION : Centroid of rectangular section lies where two diagonals intersect each other. Continue
Hence, centroid from reference Y-axis (X)= b/2 = 24/2 = 12 ft. Centroid from reference X-axis; (y)= h/2 = 12/2 = 6 ft. Formulas to Find Out CENTRIOD
Circle: Centroid of circular section lies at its center i.e., D/2
R= D/2 = 20/2 = 10cm= Answer Formulas to Find Out CENTRIOD
Semicircle: Half circle is known as semi-circle. Centroid of semi-circle is at a distance of 4R/3π from the base of semi-circle. As shown in the figure below: Formulas to Find Out CENTRIOD
Right Circular Cone: Centroid of right circular cone lies at a height h/4 from reference x-axis. As shown in figure below; Formulas to Find Out CENTRIOD COMPOSITE FIGURES
Illustrated example: Centroid of a composite figure: Real life application of Centroid
Centroids indicate the center of mass of a uniform solid. stick a pivot at the centroid and the object will be in perfect balance.
Lots of construction applications and engineering applications to design things so that minimal stress and energy is used to stabilize a component.
In stress and deflection analysis of a beam , the location of centroid is very important. THANK U ALL SimpleMachine – A tool that helps us do work
Machines help us by:
1. Changing the amount of force on an object.
2. Changing the direction of the force.
6 - 1 What is a Simple Machine?
● A simple machine has few or no moving parts. ● Simple machines make work easier. ● Simple machine is a device in which effort is applied at one place and work is done at some other place. ● Simple machines are run manually, not by electric power.
6 - 2 The wrench and screw driver are examples of a wheel and axle, where the screw or bolt is the axle and the handle is the wheel. The tool makes the job easier by changing the amount of the force you exert.
Wheel Axle
6 - 3 All of the simple machines can be used for thousands of jobs from lifting a 500-pound weight to making a boat go. The reason why these machines are so special is because they make difficult tasks much easier.
6 - 4 What is a Compound machine?
● Simple Machines can be put together in different ways to make complex machinery.
● If a machine, consists of many simple machines, it is called compound machine. ● Such machines are run by electric or mechanical power. ● Such machines work at higher speed. ● Using compound machines more work is done at less effort. ● For Ex: scooter, Lathe, crane, grinding machine etc. 6 - 5 What is a Lifting machine ?
Lifting machine is a device in which heavy load can be lifted by less effort. e.g. - simple pulley - simple screw jack - lift - crane. etc.
6 - 6 Technical terms Related to Simple Machines
● Mechanical advantage (MA) : ● The ratio of load lifted (W) and effort required (P) is called Mechanical advantage. Load Lifted W MA MA Effort required P Where,W= Load and P= Effort
● Velocity ratio (VR) : ● The ratio of distance moved by effort and the distance moved by load is called velocity ratio. y VR Distance moved by effort VR Distance moved by load x
6 - 7 ● Input ➢ Input = effort x distance moved by effort ➢ Input = p.y ● Output: ➢ Output = load x distance moved by load ➢ Output = W.x • Efficiency ( ) : ➢ The ratio of work done by the machine and work done on the machine is called efficiency of the machine. output Efficiency 100 % input Output W .x & input P . y W.x W/P η 100 100 P.y y/x MA 100 % VR 6 - 13 ● Ideal machine : ➢ A machine having 100% efficiency is called an ideal machine. ➢ In an Ideal machine friction is zero. ➢ For Ideal machine, Output = input or MA=VR
● Effort lost in friction (Pf): ● In a simple machine, effort required to overcome the friction between various parts of a machine is called effort lost in friction. ● Let, P = effort Po = effort for Ideal machine • effort lost in friction. Pf=P - Po Pf = effort lost in friction ● For Ideal machine,VR = MA VR=W/Po
Po=W/VR
Pf = P-Po P = P-(W/VR) 6 - 14 f ● Reversible machine : ➢ If a machine is capable of doing some work in the reverse direction, after the effort is removed is called reversible machine. ➢ For reversible machine, 50%
● Non-reversible machine or self-locking machine ➢ If a machine is not capable of doing some work in the reverse direction, after the effort is removed, is called non-reversible machine or self-locking machine. ➢ For non-reversible machine, 50% ➢ A car resting on a screw jack does not come down on the removal of the effort. It is an example of non-reversible machine.
6- 15 ● Condition for reversibility of machine : W = load lifted P = effort required x = distance moved by load y = distance moved by effort P.y = input W.x = output ● Machine friction = P.y – W.x ● for a machine to reverse, output > machine friction W.x P.y – W.x 2 W.x p.y W. x 1 P. y 2 Output 0 .5 Input 50%
6 - 16 For a machine to reverse, 50% Law of machine
● The law of machine is given by relation, ● P= mW+C ● Where, P = effort applied W= load lifted m = constant (coefficient of friction) = slope of line AB C= Constant = Machine Friction= OA
● Following observations are made from the graph : ➢ On a machine, if W = 0, effort C is required to run the machine. Hence, effort C is required to overcome machine friction.
➢ If line AB crosses x-x axis. without effort (P), some load call be lifted, which is impossible. Hence, line AB never crosses x-x axis.
➢ If line AB passes through origin, no effort is required to balance friction. Such a graph is for Ideal machine. 6 - 17 Maximum mechanical advantage
W MA P from law of machine P mW C W 1 C MA (Q neglecting ) C mW C m W W 1 Maxi. MA m
Maximum efficiency ( max ) W MA P from law of machine P mW C MA VR
1 1 m (MA MA max ) VR m 1 max 6 - 18 m x VR Relation Between Load Lifted and the Mechanical Advantage
As the load increases, the effort also increases and the M. A. increases The maximum M. A. is equal to 1/m.
Relation Between Load Lifted and the Efficiency
As the load and effort increases, efficiency also increases. The maximum efficiency is equal to 1/(m x VR)
6 - 14 Simple Machine
• Following are the simple machines. ➢ Simple Wheel and Axle ➢ Differential wheel and axle ➢ Worm and Worm Wheel ➢ Single purchase Crab ➢ Double Purchase Crab ➢ Simple Screw Jack ➢ lever ➢Simple Pulley
6 - 15 Simple Wheel and Axle
C WHEEL AND AXLE :A wheel and axle is a modification of a pulley.
C A wheel is fixed to a shaft.
C Large wheel fixed to smaller wheel (or shaft) called an axle
C Both turn together
C Effort usually on larger wheel, moving load of axle 6 - 17 When either the wheel or axle turns, the other part also turns. One full revolution of either part causes one full revolution of the other part.
6 - 18 DIFFERENTIAL WHEEL AND AXLE
•In this machine load axle is made in two parts having two different diameters d1 and d2. •When effort is applied to rotate the assembly at that time string is wound over larger axle (d1) and unwound from the smaller axle (d2). 6 - 19 WORM AND WORM WHEEL
•In worm and worm wheel machine, effort wheel and worm are on the same shaft and rotates in two bearings as shown. •Similarly worm wheel and load drum are also on the same shaft and rotates in two bearings. Two axes are at right angles. 6 - 20 CRAB WINCH
● Winch crabs are lifting machines in which velocity ratio is increased by a gear system. ● If only one set of gears is used, the winch crab is called a single purchase winch crab and if two sets are used it is called double purchase winch crab.
6 - 21 SINGLE PURCHASE CRAB WINCH
6 - 22 DOUBLE PURCHASE CRAB WINCH
•In this machine to increase the V.R. one more pair of gears is used in comparison to single purchase crab. •Since there are totally two pairs of gears it is known as Double Purchase Crab Winch. Similarly in Triple Purchase Crab Winch there will be three pairs of gears. • Construction is similar in all the cases
6 - 23 SIMPLE SCREW JACK
● Screw Jack is a simple machine used for lifting heavy loads, through short distances, with the help of small effort applied at its handle. ● The most common application of screw jack is the raising of the front or rear portion of a vehicle for the purpose of changing the wheel or tyre. ● when one rotation is given to the handle. ● distance moved by effort = 2πR ● distance through which load is lifted = p
6 - 24 LEVERS
● The lever is simple machine made with a bar free to move about a fixed point called fulcrum. ● It enables a small effort to overcome a large load.
● VR = dE/dL
6 - 30 ● ME = FL/FE First Kind of lever
● In a first Kind lever the fulcrum is in between of load and effort. ● load and effort is on either side.
6 - 26 Second Kind of lever
● In a second kind lever the fulcrum is at the end, with the load is in between fulcrum and effort.
6 - 27 Third Kind of lever
● In a third kind lever the fulcrum is again at the end, but the effort is in the middle.
6 - 28 Summary of LEVER CLASSES
1st Class 2nd Class 3rd Class Fulcrum is between the load and Load is between fulcrum and effort •Effort is between the fulcrum and effort load. • Mechanical advantage • MA = b/a •MA = b/a • MA = effort arm/load arm • MA is always greater than 1. • MA is always less than 1 MA= b/a •MA can be more than 1, equal to 1 or less than 1. When MA is greater than 1, less Since. MA is always greater than 1. Since, MA is always less effort would be required to lift a lever of second kind is an effort than 1. lever of the third heavy load. Such type of lever is multiplier lever. kind is only a speed multiplier called effort multiplier lever. lever. Such levers cannot lift heavy loads but provide increase in speed of lifting.
6 - 29 Simple Pulley
● PULLEY: A pulley is a simple machine made with a rope, belt or chain wrapped around a grooved wheel.
● A pulley works two ways. It can change the direction of a force or it can change the amount of force.
● A fixed pulley changes the direction of the applied force. ( Ex. Raising the flag ) . ● A movable pulley is attached to the object are moving.
6 - 30 Direction of Effort In Simple Pulley
● Pulley can change the direction of a Effort(force).
6 - 31 TYPES OF PULLEYS
FIXED PULLEY (like flagpole)
● Pulley stays in one position
● Moves LOAD up, down or sideways
● Changes DIRECTION of force
● Does not reduce EFFORT
6 - 32 TYPES OF PULLEYS
MOVABLE PULLEY (for lifting or lowering heavy objects)
● Moves along with LOAD
● Reduces EFFORT
● Increases DISTANCE
6 - 33 System OF PULLEYS
● First system of pulleys
● Second system of pulleys
● Third system of pulleys
6 - 34 First system of pulleys
First system of pulley :VR = 2n Where, n = no. of moving Pulley
6 - 35 Second system of pulleys
Second system of pulley: VR = n Where, n =total no. of Pullies. 6 - 41 Third system of pulleys
Third system of pulley :VR = 2n - 1 Where, n = total no. of Pullies.
6 - 42 . Centre of Gravity & Moment of Inertia
. Concept of Gravity
• Gravity is a physical phenomenon, specifically the mutual attraction between all objects in the universe. • In a gaming setting, gravity determines the relationship between the player and the "ground," preventing the player or game objects from flying off into space, and hopefully acting in a predictable/realistic manner. • Gravity is the weakest of the four fundamental forces, yet it is the dominant force in the universe for shaping the large scale structure of galaxies, stars, etc. • The gravitational force between two masses m1 and m2 is given by the relationship:
Gm m F 1 2 where G 6.6710-11 Nm2 / kg2 gravity r 2
• This is often called the "universal law of gravitation" and G the universal gravitation constant. • It is an example of an inverse square law force. The force is always attractive and acts along the line joining the centers of mass of the two masses. The forces on the two masses are equal in size but opposite in direction, obeying Newton's third law. Gravitational Force
• Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. • This is a general physical law derived from empirical observations by what Newton called induction. • It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica ("the Principia"), first published on 5 July 1687. • In modern language, the law states the following: • Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them: m m F G 1 2 r2 where: • F is the force between the masses, • G is the gravitational constant, • m1 is the first mass, • m2 is the second mass, and • r is the distance between the centers of the masses. Centroid and Center of Gravity
• In general when a rigid body lies in a field of force acts on each particle of the body. We equivalently represent the system of forces by single force acting at a specific point. • This point is known as centre of gravity. • Various such parameters include centre of gravity, moment of inertia, centroid , first and second moment of inertias of a line or a rigid body. These parameters simplify the analysis of structures such as beams. Further we will also study the surface area or volume of revolution of a line or area respectively. Centre of Gravity
• Consider the following lamina. Let’s assume that it has been exposed to gravitational field. • Obviously every single element will experience a gravitational force towards the centre of earth. • Further let’s assume the body has practical dimensions, then we can easily conclude that all elementary forces will be unidirectional and parallel. • Consider G to be the centroid of the irregular lamina. As shown in first figure we can easily represent the net force passing through the single point G. • We can also divide the entire region into let’s say n small elements. Let’s say the coordinates to be (x1,y1), (x2,y2), (x3,y3)……….(xn,yn) as shown in figure. • Let ΔW1, ΔW2, ΔW3,……., ΔWn be the elementary forces acting on the elementary elements • Clearly, W = ΔW1+ ΔW2+ ΔW3 +…………..+ ΔWn • When n tends to infinity ΔW becomes infinitesimally small and can be replaced as dW. Centre of gravity : Cenroids of Areas and Lines
• We have seen one method to find out the centre of gravity, there are other ways too. Let’s consider plate of uniform thickness and a homogenous density. • Now weight of small element is directly proportional to its thickness, area and density as: • ΔW = δt dA. • Where δ is the density per unit volume, t is the thickness , dA is the area of the small element. Centroid for Regular Lamina And Center of Gravity for Regular Solids
• Plumb line method • The centroid of a uniform planar lamina, such as (a) below, may be determined, experimentally, by using a plumb line and a pin to find the center of mass of a thin body of uniform density having the same shape. • The body is held by the pin inserted at a point near the body's perimeter, in such a way that it can freely rotate around the pin; and the plumb line is dropped from the pin. (b). • The position of the plumb line is traced on the body. The experiment is repeated with the pin inserted at a different point of the object. The intersection of the two lines is the centroid of the figure (c). • This method can be extended (in theory) to concave shapes where the centroid lies outside the shape, and to solids (of uniform density), but the positions of the plumb lines need to be recorded by means other than drawing.
1 • Balancing method • For convex two-dimensional shapes, the centroid can be found by balancing the shape on a smaller shape, such as the top of a narrow cylinder. • The centroid occurs somewhere within the range of contact between the two shapes. • In principle, progressively narrower cylinders can be used to find the centroid to arbitrary accuracy. In practice air currents make this unfeasible. • However, by marking the overlap range from multiple balances, one can achieve a considerable level of accuracy. Position of center of gravity of compound bodies and centroid of composition area
• Of an L-shaped object • This is a method of determining the center of mass of an L-shaped object.
2 • 1. Divide the shape into two rectangles. Find the center of masses of these two rectangles by drawing the diagonals. Draw a line joining the centers of mass. The center of mass of the shape must lie on this line AB. • 2. Divide the shape into two other rectangles, as shown in fig 3. Find the centers of mass of these two rectangles by drawing the diagonals. Draw a line joining the centers of mass. The center of mass of the L-shape must lie on this line CD. • 3. As the center of mass of the shape must lie along AB and also along CD, it is obvious that it is at the intersection of these two lines, at O. (The point O may or may not lie inside the L-shaped object.) • Of a composite shape • This method is useful when one wishes to find the location of the centroid or center of mass of an object that is easily divided into elementary shapes, whose centers of mass are easy to find (see List of centroid). • Here the center of mass will only be found in the x direction. The same procedure may be followed to locate the center of mass in the y direction. Centroids of Composite Areas • We can end up in situations where the given plate can be broken up into various segments. In such cases we can replace the separate sections by their centre of gravity. • One centroid takes care of the entire weight of the section. Further overall centre of gravity can be found out using the same concept we studied before. • Xc (W1 + W2 + W3+…..+Wn) = xc1W1 + xc2W2 + xc3W3+…….……..+xcnWn • Yc (W1 + W2 + W3+…..+Wn) = yc1W1 + yc2W2 + yc3W3+…….……..+ycnWn • Once again if the plate is homogenous and of uniform thickness, centre of gravity turns out to be equal to the centroid of the area. In a similar way we can also define centroid of this composite area by: • Xc (A1 + A2 + A3+…..+An) = xc1A1 + xc2A2 + xc3A3+…….……..+xcnAn • Yc (A1 + A2 + A3+…..+An) = yc1A1 + yc2A2 + yc3A3+…….……..+ycnAn • We can also introduce the concept of negative area. It simply denotes the region where any area is left vacant. We will see its usage in the coming problems. CG of Bodies with Portions Removed • Rigid body is composed of very large numbers of particles. Mass of rigid body is distributed closely. • Thus, the distribution of mass can be treated as continuous. The mathematical expression for rigid body, therefore, is modified involving integration. The integral expressions of the components of position of COM in three mutually perpendicular directions are : • Note that the term in the numerator of the expression is nothing but the product of the mass of particle like small volumetric element and its distance from the origin along the axis. Evidently, this terms when integrated is equal to sum of all such products of mass elements constituting the rigid body. • Symmetry and COM of rigid body • Evaluation of the integrals for determining COM is very difficult for irregularly shaped bodies. • On the other hand, symmetry plays important role in determining COM of a regularly shaped rigid body. There are certain simplifying facts about symmetry and COM : 1.If symmetry is about a point, then COM lies on that point. For example, COM of a spherical ball of uniform density is its center. 2.If symmetry is about a line, then COM lies on that line. For example, COM of a cone of uniform density lies on cone axis. 3.If symmetry is about a plane, then COM lies on that plane. For example, COM of a cricket bat lies on the central plane. Moment of Inertia
• What is a Moment of Inertia?
• It is a measure of an object’s resistance to changes to its rotation. • Also defined as the capacity of a cross-section to resist bending. • It must be specified with respect to a chosen axis of rotation. • It is usually quantified in m4 or kgm2. • Perpendicular Axis Theorem
• The moment of inertia (MI) of a plane area about an axis normal to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane and passing through the given axis.
• That means the Moment of Inertia Iz = Ix+Iy. • Parallel Axis Theorem
• The moment of area of an object about any axis parallel to the centroidal axis is the sum of MI about its centroidal axis and the prodcut of area with the square of distance of from the reference axis.
Ixx IG Ad • Essentially, 2 • A is the cross-sectional area. d is the perpendicular distance between the centroidal axis and the parallel axis. • Parallel Axis Theorem – Derivation • Consider the moment of inertia Ix of an area A with respect to an axis AA’. Denote by y, the distance from an element of area dA to AA’. I y dA x 2
3 • Derivation • Consider an axis BB’ parallel to AA’ through the centroid C of the area, known as the centroidal axis. The equation of the moment inertia becomes: I y dA ( y'd) dA x 2 2
y'2 dA 2 y' dA d 2 dA
4 Derivation
• Modify the equation obtained with the parallel axis theorem: Radius of Gyration of an Area
• The radius of gyration of an area A with respect to the x axis is defined as the distance kx, where Ix = kx A. With similar definitions for the radii of gyration of A with respect to the y axis and with respect to O, we have
5 IMAGE REFERENCES
Sr. No. Source/Links
1 http://www.wonderwhizkids.com/resources/content/images/3201.jpg
2 http://nrich.maths.org/content/id/2742/cog3.gif http://images.tutorvista.com/content/rigid-body/parallel-axes- 3 theorem.gif
4 http://images.tutorvista.com/content/rigid-body/parallel-axes- theorem.gif http://www.transtutors.com/Uploadfile/CMS_Images/21994_M 5 oment%20of%20inertia%20of%20a%20circular%20section.JPG CONTENT REFERENCES
A TEXT BOOK OF ENGINEERING MECHANICS , R.S.KHURMI , S.CHAND & COMPANY PVT. LTD. A TEXT BOOK OF ENGINEERING MECHANICS , Dr. R.K.BANSAL , LAXMI PUBLICATION Any Question?? Thank You Mechanics Question: The diagram shows a horizontal force of magnitude 30 N acting on a block of mass 2 kg, which is at rest on a plane inclined at 50⁰ to the horizontal. Find the magnitude and direction of the frictional force on the block. 30 N (Assume g = 10 ms-2)
50⁰ Answer: A force of 30 cos 50⁰ N (= 19.28 N) is acting along the inclined plane in the upward direction. A force 20 sin 50⁰ N (= 15.32 N) is acting along 30 N the inclined plane in the downward direction.
50⁰ Therefore, there is a resultant force of 3.96 N 20 cos N 50⁰ 50⁰ acting in the upward direction, which will tend to move 50⁰ W = mg = (2 ⨯ 10) N = 20 N the block upwards. m = 2 kg , g = 10 ms-2 Since, the frictional force will oppose motion, it will act in the downward direction and will have a magnitude of 3.96 N (since the block is at rest). SIMPLE & COMPOUND MACHINES .::DEFINITIONS::.
MACHINE : A machine is a tool consisting of one or more parts that is constructed to achieve a particular goal. SIMPLE MACHINE :: A simple machine is a mechanical device that changes the direction or magnitude of a force. COMPOUND MACHINE : Combination of two or more simple machine is called Compound Machine “The 6 Simple Machines”
Inclined Plane Screw Wedge
Pulley Wheel and Axle Lever Inclined Planes • An inclined plane is a flat surface that is higher on one end • Inclined planes make the work of moving things easier Screws
A screw is a mechanism that converts rotational motion to linear motion, and a torque (rotational force) to a linear force. It is one of the six classical simple machines. The most common form consists of a cylindrical shaft with helical grooves or ridges called threads around the outside Screw
The mechanical advantage of an screw can be calculated by dividing the circumference by the pitch of the screw. Pitch equals 1/ number of turns per inch. Wedges • A wedge is a triangular shaped round tool, a compound and portable inclined plane, and one of the six classical simple machines. It can be used to separate two objects or portions of an object, lift an object, or hold an object in place. Levers
A lever is a machine consisting of a beam or rigid rod pivoted at a fixed hinge, or fulcrum. .::Types Of Levers::. • In a first class lever the fulcrum is in the middle and the load 1) Levers-First Class and effort is on either side • Think of a see-saw
PLIER • In a second class lever 2)Levers-Second Class the fulcrum is at the end, with the load in the middle. 3)Levers-Third Class • In a third class lever the fulcrum is again at the end, but the effort is in the middle. Pulleys
•A pulley is a wheel on an axle that is designed to support movement of a cable or belt along its circumference. •Pulleys are used in a variety of ways to lift loads, apply forces, and to transmit power. WHEEL AND AXEL
The wheel and axle is generally considered to be a wheel attached to an axle so that these two parts rotate together in which a force is transferred from one to the other. Some other applications of Wheels & Axles
RAILWAY WHEELS Turning a doorknob rotates the spindle which moves the latch. COMPOUND MACHINES
1. Combination of two or more simple machine is called Compound Machine. 2. Simple Machines can be put together in different ways to make complex machinery.
TANKS LATHE MACHINE
CYCLE