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 Engineering Formulae
Machines

 About Machines Inclined Plane Efficiency of Machines The Lever Principle of Work The Wheel and Axle Uses of a Machine Screw Types of Simple Machine Wedge Pulley

Machines:

A machine is a device by which a force applied to some point in certain direction is made available at some point and in some other direction. The force P applied on the machine is called the effort or the power while the resistance W overcome by the machine is called weight or load.

The ratio, load/effort is called mechanical advantage of a machine. The mechanical advantage should be greater than one. If, in a machine, the ratio is less than one, it would be more accurate to call is mechanical disadvantage. • Velocity Ratio:

The ratio is called the velocity ratio of a machine. The two distances are moved in the same interval of time, so they are proportional to the velocities of the effort and load.

P�d1 = W�d2   in a perfect machine.

or where d1 and are d2 are the distances moved by effort P and load W respectively. Hence, in a perfect machine the mechanical advantage is equal to velocity ratio.

Efficiency of Machine:

In all machines some work is always wasted friction. The result of it is that the work done by the effort in a given time, called total work or work input (= p � d1) is always greater than the work done on the load (= W � d2) called useful work or work output. The difference of the latter from the former = lost work (Pd1-Wd2).

The ratio is called efficiency of machine. It is also defined as the ratio In any actual machine the efficiency is always less than one but in a perfect or ideal machine in where there is no friction at all the efficiency is equal to unity. Or Mechanical advantage = Efficiency � velocity ratio

i.e. M.A. =Velocity Ratio � η

The Principle of Work:

In any actual machine, the useful work obtained is always less than total work done by the effort. This is because (i) work has to be done in lifting its parts which have weight, and (ii) because there is always some internal friction which has to be overcome. A perfect or ideal machine is one which has no weight and the efficiency of the machine is unity. Principle of the work is the principle of conservation of energy i.e., the work done by a machine is equal to the work done on an ideal or perfect machine. According we have If W increase, then d2 decrease in the same ratio. Hence in a machine, whatever is gained in power is lost in speed or distance.

Uses of a machine:

(i) This enables one to lift weight or overcome resistances much greater than one could do unaided as in the case of a pulley-system, a wheel and axle, a crow bar, a simple screw jack, etc.

(ii) This enable one to convert a slow motion at some point into a more rapid motion at some other desired point, viz., a bicycle,. A sewing machine etc. An opposite effect may also be arranged in practice when necessary

(iii) This enable one to use a force acting at a point to be applied at a more convenient point, as in the case of a poker for stirring up fire, or to use a force acting at a point in a more convenient manner, e.g., lifting of a mortar bucket to the top floor by means of a rope passing over a pulley fixed at the top of the building, the other end of the rope being pulled down by an agent remaining on the ground.

(iv) This enables one to convert a rotatory motion into a linear motion or vice versa, as in the case of a rack and pinion, etc

(v) This enables one to convert a to and fro motion into a rotatory motion or vice versa, e.g., a crank used in the heat engine.

Types of Simply Machines:

The following six simple machines represent the types of principles used in making practical machines.

(1)   Pulley.

(2) Inclined plane.

(3) Lever.

(4) Wheel and axle.

(5) Screw and.

(6) Wedge.

(1) Pulley:

A pulley is a simple machine consists of a grooved wheel, called sheave, over which a string can pass. The wheel is capable of turning freely about an axle passing through its centre. The axle is fixed to a frame work, called the block. The pulley is fixed or movable accordingly as its block is fixed or movable.

The Single Fixed Pulley: Shows a fixed pulley in which the block of the pulley is fixed to a rigid support. The load W is attached to one end of the string passing round the groove of the pulley and effort P is applied at the other of the pulley and effort P is applied at the other end. With a perfectly smooth pulley and a weightless string, the tension of the string will be the same throughout. Hence the distance through which the load is raised, is equal to the distance through which the effort descends. For equilibrium, the moments of P and W about O, the centre of the wheel, must be equal and opposite.

Or        P � AO = W � OB AO = OB Mechanical advantage In practice, pulleys are not perfectly frictions, and W is always less than P, that is, the mechanical advantage is always less than 1. But in spite of this, the arrangement is useful as the operator can use the weight of this body for raising the land. It is generally used for raising weights, drawing curtains etc.

Single Movable pulley: Here one end of the string passes round the pulley A and is attached to a fixed support M as shown in. The effort P is applied at the other end of the rope passing over a fixed pulley B. The load w to be raised is attached to the block of movable pulley A. The fixed pulley is used only to apply the power in the downward direction. It is assumed that the pulleys are frictionless and the tension in the string is the same at all points and is equal to P. When the strings are vertical the total upward force is 2d and neglecting the weight of the pulley, the downward force is W.

So, for equilibrium W=2P and mechanical advantages Thus a given effort can raise twice its weight.

If the weight of the pulley is w, then W+w=2P

Or Mechanical advantages = The single movable pulley is much used in cranes. Sails in boats and flags are raised and lowered with the help of movable pulleys.

Combination of Pulleys: A combination of pulleys is very often used to secure a mechanical advantages greater than two. Different system�s having different mechanical advantages are used for different purposes, but the most important combination, which is in general use, is given here.

Pulley Block:

This system consists of two blocks, each containing 2 or 3 pulleys. The upper block is fixed to a support and the lower one is movable to which the load W is attached. The string is attached to the upper or to the lower block, and is then passed round a movable and a fixed pulley in turn finally passing over a fixed pulley, the effort p being applied at the free end. It should be noted that when the string is attached to the upper block, the numbers of wheels in the two blocks must be the same, but when it is attached to the lower block, the number of wheels in it will be one less than that in the upper one.

The tension everywhere round the string is the same and is equal to the effort P, the pulleys being assumed frictionless. If the number of position of the string in the lower block be n, the total upward force on it is nP and this must be equal to the load W supported. Thus, we have

nP = W + w, when w, is the weight of the lower block. (2) The Inclined Plane:

An inclined plane is a smooth rigid flat surface inclined at an angle to the horizontal. It is used to facilitate the rising of a heavy body to a certain height by the application of a force which is less than the weight of the body. The fair-case and the roads on the hills are typical examples.

Let AB be a plane inclined at an angle θ to the horizontal line AC and BC is the height of the plane. The body placed on the plane is acted upon by three forces (i) W, its weight acting vertically downwards (W=mg), (ii) p, the force or effort, and (iii) R, the reaction of the plane.

Case. I. Let the force P act upward along the plane.

In order that the body may be in equilibrium, the sum of the resolved parts of the forces parallel parts of the forces parallel and perpendicular to the plane AB are separately equal to zero. Resolving W=mg parallel and perpendicular to AB, we have mg sin θ perpendicular to AB.

Hence

mg sin θ � P = 0;

mg cos θ � R = 0

P = mg sin θ:

R = mg cos θ. that is, a body of weight W = mg can be supported by a force P = W/2 acting up the plane.

(3) The Lever: A lever is a simple machine and consists the rest of the lever can turn. This fixed point is called the fulcrum. The perpendicular distance between the fulcrum and the point of application of the power is called the power arm the perpendicular distance between the fulcrum and the point of application of the weight arm of the lever.

Let AB be a lever with the power P is applied at A and weight W to be lifted is suspended at B. AF and Bf are the power and weight arms respectively. The forces act perpendicular to the arms and keep the lever in equilibrium.

Then, by the principle of moments, we have

P � AF = W � BF This is known as the principle of lever.

The lever is used for pulling weights or overcoming resistance by the application of force at a suitable point. From the above relation it is clear that by increasing the length of the power arm, we can lift a greater weight with the lever.

The Straight Levers: a) b) c)

When the lever is straight and the effort and the weights act perpendicular to the lever, the following three distinct classes of levers are found in practice according to the relative positions of A, B and F- the points of application of the effort, the weight and the fulcrum respectively. In each of these cases, three parallel forces are acting on the body. They, therefore, are in the equilibrium condition. The reaction R at the fulcrum must be equal and opposite to the resultant of P and W acting at A and B.

A lever of the first order:

In this case the effort P and the weight W act on the opposite sides of the fulcrum F. Here reaction R is the middle force, therefore, P and W act in the same direction and R in the opposite direction. To lift the weight, the effort must be applied downwards, and the reaction acts upwards so that lever presses downwards on the fulcrum. Taking moments about F, we have   AF may be either greater, equal or less than BF, so that mechanical advantages is either or equal or less than unity.

The examples of this type of lever are a poker used to raise the coal in a grate, handle of water pump, a balance, see-saw, a spade used in the digging of earth and a bicycle brake. A pair of scissors and a pair of pincers are examples of double levers of this class.

Second Order:

In this case, the weight is placed between the effort and the fulcrum. Hence W is the middle force. Therefore, P and R act in the same direction and W in the opposite direction. To lift the weight, the effort must be applied in the direction. The reaction of the fulcrum also acts upwards. Taking moments about the fulcrum F, we have,  But in this class of levers AE is greater than Bf, there force, mechanical advantage W/P is greater than unity.

Also                  W = P + R

Or                     R = W- P

The examples of this class are a crow bar, a wheel barrow, a tin opener, foot bellows and lifting of a lid of box. A pair of ordinary nut-crackers and cork squeezer are examples of double levers of this case.

Third Order:

Here the effort P is placed between the weight and the fulcrum as shown in. Since is the middle force, therefore, W and R act in the same direction and P in the opposite direction. Also

P = R +W

Or                    R = P - W

Thus to lift weight the effort must be applied upwards, the reaction of fulcrum acts downwards or lever presses fulcrum upward. Taking moments about the fulcrum F, we have,  But in this lever AF is less than Bf, therefore, mechanical advantage is less than unity. This shows a mechanical disadvantage. This arrangement gives W a large movements for a small movement of the effort P, a fact which is just opposite to what happens in the other two types of levers.

The examples of this type of lever are the forearm, treadle of a sewing machine in kicking a football and in using a cricket bat. A fire tongs, a pair of forceps used in a weight box, the upper and lower jaws of the mouth are examples of double levers of this class.

(4) The Wheel and Axle: The wheel and axle is a modification of the lever. It consists of two cylinders of different diameters capable of turning about a common fixed axis. The larger cylinder is called the wheel and the smaller the axle. The load W to be raised is attached to a rope coiled around the wheel in the opposite direction, so that when the rope round the wheel is un-coiled, the rope round the axle is coiled up and thereby the weight is raised. Shows a section where OB is the radius r of the axle OA the radius R of the wheel. Taking moments about O,

P � OA = W � Ob Application:

The windlass by which water is drawn from a well is the same as the wheel and axle, the crank-handle of which serves the purpose of the wheel. The capstan used for lifting an anchor in ships, a coffee grinder, a spanner used to wind a nut, the steering of a motor car, bicycle pedal etc. are all applications of the wheel and axle.

(5) Screw:

The screw is a rod usually of some hard metal on the surface of which is cut a spiral groove. The successive turns of these grooves are separated by a spiral ridge known as thread. The screw works in a collar or nut through which a hole is bored, having a groove to fit the thread to fit groove of the screw. A screw is generally provided with an arm by means of which it can be rotated. When the screw turns in a fixed nut, it moves forward or backward in the direction of its length. In each turn of the screw, the distance moved is equal to the distance between two connective threads. This distance is called the pith. Hence, by turning the screw round, it may be used to raise weights or overcome resistance applied to its ends. To screw press, screw jack, bolt and nut are some examples of screw.

To find the pitch of screw, count the number of grooves or threads in one centimeter and thus calculate pitch as  when the screw works without friction.

Screw Jack: Screw jack as shown in is used for lifting heavy loads like an automobile in garages and workshops for repair purpose etc.

It consists of a strong hollow metallic tapering cylinder A having at the top a hole in the form of a nut N with grooves cut on its inner side. A square threaded screw moves in this nut and upper end. There is a loose collar C put on d to avoid rotation of the load W placed over it when the screw moves. The screw is turned round by a lever arm of length or by a system of cog wheels. The power p is applied at right angles to the arm. When the lever makes one complete rotation. The screw is raised vertically through a distance equal to the pitch of screw. By using a system of cog wheel, the load is raised through a distance equal to the pitch of the screw when lever is given n rotations, then Winch of Lifting Crab:

This machine is used for lifting comparatively heavy loads by means of a small effort such as can be exerted by hand. It is used when greater velocity ratio and also mechanical advantage is required than can be conveniently obtained with a wheel and axle, in which convenient length of handle is limited by the range of a man�s arm and the diameter of the drum by the consideration of holding all the rope necessary for a given lift.

Single Purchase Crab:

It consists of two stands A1 and A2. Connected rigidly together by the three stays B1, B2 and B3 and having bearings for the spindle B and the drum D. On the spindle C is keyed the pinion E, which gears with and drives the larger spur wheel F on the drum D. The ends of C are squared to receive handles H, on the ends of which the effort is applied. The load is lifted by a rope which is coiled round the drum D.

Velocity Ratio. Let,

R=radius or a length of handle,

T1=number of teeth in pinion,

T3=number of teeth in wheel,

Now in one revolution of the of handle we have-

Motion of effort = 2πR

Motion of pitch circle of pinion

= 1 revolution

=T1p (P=pitch) which is the product of the velocity ratio of wheel and axle and a pair of spur wheels of given numbers of teeth.

(6) Wedge:

A wedge is a simple machine consisting of a solid block of metal or wood shaped as an inclined plane. A small dividing force P applied to the wedge results in a much larger splitting or separating force. It is commonly used for raising heavy bodies for widening a gap, for breaking strong cover joints etc. A double wedge is shown in, being used for widening a gap. The separating forces generated product equal reaction W, W at the edges of the gap. The forces, P, W, W can be represented by a triangle shown in, neglecting friction.

Hence              P = 2w sin θ/2

And                  M.A. = The action of an axe, or a nail knife may to be treated as that of a combination of two wedges.  