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CENTROID What Is Centroid?

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CENTROID What Is Centroid? 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.
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