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Centroid and Centre of Gravity Centroid and Centre of Gravity By: Amit Kumar Centroid and Centre of Gravity Centroid Center of Gravity • It is defined as a point about • It is defined as a point about which the entire line, area or which the entire weight of the volume is assumed to be body is assumed to be concentrated. concentrated. • It is related to distribution of • Center of mass. length, area and volume. • It is related to distribution of mass. Examples Examples Examples Examples Axis of Reference Center of Gravity Consider system of n particles fixed within a region of space. The weights of the particles can be replaced by a single (equivalent) resultant weight having defined point G of application Center of Gravity Center of mass Center of Volume / Volume centroid Center of Area / Area centroid Center of Line / Line centroid Axis of symmetry Axis of symmetry Thank you APPLIED MECHANICS E-NOTES FOR ------ MACHINE BY – Er Sachin (Lectr in Civil Engg Deptt) A machine (or mechanical device) is a mechanical structure that uses power to apply forces and control movement to perform an intended action. Machines can be driven by animals and people, by natural forces such as wind and water, and by chemical, thermal, or electrical power, and include a system of mechanisms that shape the actuator input to achieve a specific application of output forces and movement. They can also include computers and sensors that monitor performance and plan movement, often called mechanical systems. Simple machine vs compound machine Simple Machines A simple machine is the simplest device that performs work. A simple machine applies a single force to a single load and generally acts as a building block in developing more complex, compound machines. Simple machines usually fall into one of six categories, including: lever, wheel and axle, pulley, inclined plane, wedge and the screw. Simple machines do not contain an energy source so cannot perform more work than is applied to them in their use. Types of Simple Machines A lever is an object that acts as a pivot point that multiplies the force that can be applied to another object. The wheel and axle is a rod attached to a wheel which can multiply an applied force. A pulley consists of a wheel on an axle with a rope running over the wheel. Pulleys are used to change the direction of an applied force. The inclined plane is a flat surface with ends at different heights. Inclined planes reduce the amount of force required to move an object. Wedges are triangular-shaped and used to separate, hold or lift an object. Screws are cylindrical shafts with grooves that pass through or move other objects through rotational force. Compound Machines Compound machines are a collection of simple machines working together. Compound machines are the most common type of machine and do more complex work than individual simple machines. They perform more work and therefore offer a greater advantage than simple machines alone. Compound machines may consist of various and innumerable combinations of simple machines Types of Compound Machines There is a huge variety of compound machines available to assist with a large number of tasks. Several examples of compound machines include: bicycles, pliers, wheelbarrows and scissors. In bicycles, the pedals and wheels form cooperating wheel and axle systems, the brakes are levers and the parts are held together with multiple screws. Pliers are constructed with multiple levers. A wheelbarrow is a combination of a lever and wheel and axle. Scissors consist of two pivoting levers. Load: This is the resistance to be overcome by the machine. Effort : This is the force required to overcome the resistance to get the work done by the machine.(p) Mechanical advantage: This is the ratio of load lifted to effort applied. Thus, if W is the load and P is the corresponding effort, then Velocity Ratio: This is the ratio of the distance moved by the effort (D) to the distance moved by the load ( d) in the same interval of time. Thus, Input : The work done by the effort is known as input. Input = PD Output: It is defined as useful work got out of the machine, i.e. the work done by the load. Output = Wd Efficiency : This is defined as the ratio of output to the input. Efficiency of an ideal machine is 1. In that case, Wd=PD, which gives W/P= D/d. Thus, for an ideal machine, mechanical adavntage is equal to velocity ratio. Law of Machines The relationship between the load lifted and the effort required in a machine is called the law of machine. This is generally a straight line which does not pass through the origin. This is found by conducting experiments in which efforts required for lifting different loads are determined and then load versus effort graph as shown in the following figure is plotted: Reversibility of a Machine If the removal of effort while lifting results in lowering of the load, the machine is said to be reversible. The machine is said to be self-locking if the load is not lowered on removal of the effort. A lifting machine is reversible if its efficiency is greater than 50 percent and self locking if its efficiency is less than 50 percent. This can be proved as follows. Suppose a load W has been lifted to a height h. The potential energy of the load is Wh. Let us say the frictional energy loss is Ef. Now, in order load can come back to its previous position, its potential energy must be more than friction loss. In the limiting case, Wh=Ef. In that case, the efficiency will be 50 %. Or Lifting machine It is a device, which enables us to lift a heavy load W, by a comparatively small effort P. The following terms are commonly used in lifting machines 1. Mechanical advantage (M.A.). It is the ratio of load lifted (W) to the effort applied (P). 2. Velocity ratio (V.R.). It is the ratio of the distance moved by the effort (y) to the distance moved by the load (x). 3. Input of the machine. It is the workdone on the machine. It is equal to the product of effort and the distance through which it moves (i.e. P x y). 4. Output of the machine. It is the workdone by the machine. It is equal to the product of load lifted and the distance through which it has been lifted (i.e. W x X). 5. Efficiency of the machine. It is ratio of output to the input of the machine. Mathematically, efficiency of the machine, 6. Ideal machine. If the efficiency of the machine is 100 %, i.e. if output is equal to input, then the machine is said to be a perfect or ideal machine. 7. Reversible machine. If a machine is capable of doing some work in the reversed direction, after the effort is removed, then the machine is known as reversible machine. The condition for a machine to be reversible is that its efficiency should be more than 50 %. 8. Non- reversible or self locking machine. If a machine is not capable of doing some work in the reversed direction, after the effort is removed, then the machine is known as non-reversible or self locking machine. The condition for a machine to be non-reversible or self locking is that its efficiency should be less than 50 %. 9. Law of the machine. It is the relationship between the load lifted (W) and the effort applied (P). It is given by the equation, P = m.W+C where m = A constant (called coefficient of friction) which is equal to the slope of the line AB as shown in Fig. 1.23, and C = Another constant, which represents the machine friction. 10. Maximum mechanical advantage. The maximum mechanical advantage of a lifting machine is given by Max. M.A. = 1 / m 11. Maximum efficiency. The maximum efficiency of a lifting machine is given by Relation between Mechanical Advantage (MA), Velocity ratio (VR), Efficiency(η) - Explained Relation between Mechanical Advantage (MA), Velocity ratio (VR), Efficiency(η) - Explained What is Mechanical Advantage? Mechanical Advantage is defined as the ratio between Load( L) and Effort(E) MA = L / E What is Velocity ratio? Velocity ratio is defined as the ratio between distance moved by effort (x) and distance moved by load (y) VR = x / y What is Efficiency? Efficiency is defined as the ratio between Output and Input η = Output / Input Relation: Work done by effort = Input = E x x Work done by load = Output = L x y η = Output / Input = L x y / E x x = ( L / E) / ( x / y) = L / E is MA and ( x / y) is VR = MA / VR η = MA / VR--------------------ans What is Systems of Pulleys? Differentiate first, second and third Systems of Pulleys. The following three systems of pulleys are commonly used : 1. First system of pulleys. For such a system, velocity ratio is: where n = Number of pulleys 2. Second system of pulleys. For this system, velocity ratio: 3. Third system of pulleys. For such a system, velocity ratio: .
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