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Chapter, (A) Force. Work, Power and Fnergy, Force, SYLLABUS, (1) Turning forces concept; moment of a force; forces in cquilibrium; centre of gravity, (discussions using simple, examples and simple numerical problems)., Scope of syllabus - Elementary introduction of translational and rotational motions; moment (turning effect) of a force, also, called torque and its C.G.S. and S.I. units; common examples, door, steering wheel, bicycle pedal, etc; clockwise and, -, anticlockwise moments; conditions for a body to be in equilibrium (translational and rotational): principle of moments and, its venification using a metre rule suspended by two spring balances with slotted weights hanging from it: simple numerical, problems; centre of gravity (qualitative only) with examples of some regular bodies and irregular lamina., (ii) Unitorm circular motion., Scope of syllabus – As an example of constant speed, though acceleration (force) is present. Differences between centrifugał, and centripetal force., In class IX, we have read that a force when applied on a rigid body can only cause motion in, it, while when applied on a non-rigid body it can cause both a change in its size or shape and, motion as well. In a quantitative way, force applied on a body is defined as the rate of change in, dp, d(mv), its linear momentum i.e., F, or F, = ma (if mass m is constant). Force is a vector, dt, dt, quantity and its S.I. unit is newton (symbol N) or kilogram-force (symbol kgf) where 1 kgf = g N, if g is the acceleration due to gravity at that place (= 9-8 m s average value on the earth's surface)., %3D, %3D, (A) MOMENT OF A FORCE AND EQUILIBRIUM, 1.1, TRANSLATIONAL AND ROTATIONAL motion. For example in Fig. 1.1, on pushing a, MOTIONS, ball lying on a floor, it begins to move., A rigid body when acted upon by a force, can (2) Rotational motion, have two kinds of motion :, If a body is pivoted at a point and a force is, applied on the body at a suitable point, it rotates, the body about the axis passing through, the pivoted point. This is the turning effect of, the force and the motion of the body is, When a force acts on a stationary rigid body called rotational motion. For example, if a, which is free to move, the body starts moving in wheel is pivoted at its centre and a force is, a straight path in the direction of the applied applied tangentially on its rim as shown in, (1) linear or translational motion, and, (2) rotational motion., (1) Linear or translational motion, force. This is called linear or translational Fig. 1.2, the wheel, Ball, Wheel, rotates about an axis, its, Pivot, through, Similarly when a force, is applied normally on, Push, centre., Direction of, motion, Fig. 1.1 Translational motion, the handle of a door, Fig. 1.2 Rotational motion, 1, Scanned by TapScanner
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The unit N m ot moment of force (or torque) is not written joule (J., the door begins to rotate about an axis passing Moment of force about the axis passing throunh, = Force x Perpendicular distance, of force from point O, point O, through the hinges on which the door rests., 1.2 MOMENT (TURNING EFFECT) OF A, FORCE OR TORQUE, = Fx OP, ..(1.1), Consider a body which is, pivoted at point 0. If a force, F is applied horizontally on, the body with its line of, action in the direction AP as, shown in Fig. 1.3. the force is Fig. 1.3 Moment, unable to produce linear, motion of the body in its, direction because the body is not free to move,, but this force turns (or rotates) the body about the, vertical axis passing through the point O, in the, direction shown by the arrow in Fig. 1.3 (1.e., the distance is metre, so the S.I. unit of moment, Note : For producing maximum turning effect on a, body by a given force, the force is applied on the, body at a point for which the perpendicular distance, of the line of action of the force from the axis of, rotation is maximum. In this situation, the given, force provides the maximum torque to turn the body., 90, PIVOT, AF, of a force, Units of moment of force, Unit of moment of force, = unit of force x unit of distance, The S.I. unit of force is newton and that of, force rotates the body anticlockwise)., of force is newton x metre. This is abbreviated, as N m., Factors affecting the turning of a body, The turning effect of a force on a body, depends on the following two factors:, (1) the magnitude of the force applied, and, (2) the perpendicular distance of the line of then the unit of moment of force in S.I. system, action of the force from the axis of rotation is kgf x m and in C.G.S. system, the unit is, (or pivoted point)., Indeed, the turning effect on the body depends, on the product of both the above stated factors., This product is called the moment of force (or, torque). Thus, the body rotates due to the moment, of force (or torque) about the pivoted point. In and 1 gf x cm = 980 dyne cm, The C.G.S. unit of moment of force is, dyne x cm., But if force is measured in gravitational unit,, gf x cm., These units are related as follows :, IN m = 10 dyne x 102 cm, = 107 dyne cm, %3D, }, %3D, ..(1.2), 1 kgf x m =, 9.8 N m, %3D, other words., Clockwise and anticlockwise moments:, The turning effect on the body about an axis is, due to the moment of force (or torque) applied, on the body., Conventionally, if the effect on the body is to turn, it anticlockwise, the moment of force is called, anticlockwise moment and it is taken positive., Measurement of moment of force (or torque) while if the effect on the body is to turn it, clockwise, the moment of force is called, clockwise moment and it is taken negative., The moment of a force (or torque) is equal to, the product of the magnitude of the force and, the perpendicular distance of the line of action, of the force from the axis of rotation., The moment of force is a vector quantity. The, direction of anticlockwise moment is along the, axis of rotation outwards, while that of clockwise, moment is along the axis of rotation inwards., In Fig. 1.3, the line of action of force F is, shown by the dotted line AP and the, perpendicular drawn from the pivoted point 0 on, the line of action of force is OP. Therefore,, * The unit N m of moment of force (or torque is not written joule 131., However, the unt N m for work or energy is written joule (J) because, torque is a vector, while work or energy is a scalar quantily., 2, Scanned by TapScanner
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On applying a force on a pivoted body, its, direction of rotation depends not only on the, direction of force but also on the point of, application of the force. Thus the direction of, rotation of a body can be changed by two ways :, A, B, FRAME, A, HINGE, (fixed in wall), R, HANDLE, HINGE, F, (1) By changing the point of application of, force – Fig. 1.4(a) shows the anticlockwise, and clockwise moments produced in a disc, pivoted at its centre by changing the point, of application of the force F from point A to, point B., D, Fig. 1.5 Opening of a door, (2) The upper circular stone of a hand flour, grinder is provided with a handle near its rim, (i.e., at the maximum distance from the, centre) so that it can easily be rotated about, the iron pivot at its centre by applying a, в, DISC, PIVOT-, small force at the handle., DISC, F., (3) For turning a steering wheel, a force is, applied tangentially on the rim of the wheel, (Fig. 1.6). The sense of rotation of the wheel, is changed by changing the point of, application of force without changing the, direction of force. In Fig. 1.6 (a), when, force F is applied at point A of the wheel,, the wheel rotates anticlockwise; while in, A, A, ANTICLOCKWISE, CLOCKWISE, (NEGATIVE), (a) By changing the point of application of force, (POSITIVE), F, AXLE, F, AXLE, ANTICLOCKWISE, (POSITIVE), CLOCKWISE, (NEGATIVE), (b) By changing the direction of force, Fig. 1.6 (b), the wheel rotates clockwise, when force F is applied in the same direction, at point B of the wheel., Fig. 1.4 Anticlockwise and clockwise moments, (2) By changing the direction of force, Fig. 1.4(b) shows the anticlockwise and, clockwise moments produced on a pivoted, axle by changing the direction of force F at, B., F, A, F, the free end of the axle., (a) ANTICLOCKWISE, ROTATION, (b) CLOCKWISE, ROTATION, Common examples of moment of force, (1) To open or shut a door, we apply a force (push, or pull) F normal to the door at its handle P, which is provided at the maximum distance (4) In a bicycle, to turn the rear wheel, from the hinges as shown in Fig. 1.5., We can notice that if we apply the force at a, point Q (near the hinge R), much greater force, is required to open the door and if the force, is applied at the hinge R, we will not be able, to open the door howsoever large the force, may be (because for the force at R, torque, will be zero). Thus, the handle P is provided, near the free end of the door so that a smaller, Fig. 1.6 Sense of rotation changed by the change, of point of application of force, anticlockwise, a small force is applied on the, foot pedal of a toothed wheel of size bigger, than the rear wheel so that the perpendicular, distance of the point of application of force, FOOT, PEDAL, force at a larger perpendicular distance from, the hinges produces the maximum moment of, force required to open or shut the door., AXLE, AXLE, REAR WHEEL, Fig. 1.7 Turning of toothed wheel of a bicycle, 3, Scanned by TapScanner
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from the axle of the wheel is large (Fig. 1.7). forces, not acting along the same line, form a, has a long handle to produce a large moment consisting of two forces : (i) the force which we, exert at the handle of the door, and (ii) an equal, from the axle of the wheel is large (Fig. 1.7). forces, not acting along the same line, form, The toothed wheel is joined to the rear wheel couple. A couple is always needed to produce, rotation. For example, when we open a door, the, a, by a chain passing over their teeth., rotation of the door is produced by a, (5) A spanner, used to tighten or loosen a nut,, has a long handle to produce a large moment consisting of two forces : (i) the force which Ple, of force by a small force applied normally at, the end of its handle as shown in Fig. 1.8. and opposite force of reaction at the hinge, The spanner is turned anticlockwise to loosen, the nut by applying the force in the direction effect, then two forces, equal in magnitude and, shown in Fig. 1.8, while it is turned clockwise opposite in direction, are applied on the bodu, to tighten the nut by applying the force in a explicitly such that both the forces turn the bod, direction opposite to that shown in Fig. 1.8., couple, exert at the handle of the door, and (11) an equal, Sometimes when we require a larger turning, in the same direction., Example : To open the nut of a car wheel., we apply equal forces, each F, at the two ends, of a wrench's arm in opposite directions as shown, in Fig. 1.9., HANDLE, NUT, FORCE, Fig. 1.8 Spanner (wrench) used to loosen a nut, (6) A jack screw used to lift a heavy load such, as a vehicle, has a long arm so that less effort, is needed to rotate it so as to raise or lower, F, F, NUT, the load table., HOLDING, A NUT, Conclusion : From the above examples, we, conclude that the turning of a body about an axis, depends not only on the magnitude of force, but, it also depends on the perpendicular distance of, the line of action of the applied force from the tightening the cap of an inkpot (Fig. 1.11), turning, axis of rotation. Larger the perpendicular distance, the key in the hole of a lock (Fig. 1.12), winding, less is the force needed to produce the same, turning effect and vice-versa., (a) Car wrench, (b) Wrench, Fig. 1.9 Opening the nut of a car wheel by a wrench, Similarly, while turning a water tap (Fig. 1.10),, a clock (or a watch) with a key, turning the, steering wheel of a car (Fig. 1.13), pushing, 1.3 COUPLE, A single force alone does not cause rotation of, a pivoted body. Actually rotation is always produced, by a pair of forces. In the above examples, rotation, occurs due to the force externally applied and the, force of reaction produced at the pivoted point. The, force of reaction at the pivot is equal in magnitude,, but opposite in direction to the applied force. The, moment of the force of reaction about the pivot is, Fig. 1.10 Turning a, Fig. 1.11 Tightening, the cap, water tap, zero because its distance from the axis of rotation, is zero, so the force of reaction at the fixed point, (or pivot) is not explicitly shown in Fig. 1.3 to, Fig. 1.8. The pair of forces formed by the, external force and the force of reaction is called, F, a couple. Thus, two equal and opposite parallel, Fig. 1.12 Turning a key, Fig. 1.13 Turning a, steering wheel, in a lock, 4, Scanned by TapScanner
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Moment of couple : Fig. 1.14 illustrates the moments of all the forces about the fixed point, effect produced by a couple. AB is a bar which is zero, so they do not change the rotational state, the pedals of a bicycle, etc., a pair of forces do not change the state of rest or of linear motion, (couple) is applied for rotation., of the body, and (ii) the algebraic sum of, Moment of couple, effect produced by a couple. AB is a bar which, is pivoted at a point O. At the ends A and B, two, 1.14 illustrates the moments of all the forces about the fixed point, Is zero, so they do not change the rotational state, of the body. then the body is said to be in, equal and opposite forces, each of magnitude F. equilibrium. Thus., are applied. The perpendicular distance between, the two forces is AB (= d) which is called the, couple arm. The two forces cannot produce, translational motion as their resultant sum in any, When a number of forces acting on a body, produce no change in its state of rest or of, linear or rotational motion, the body is said to, be in a state of equilibrium., direction is zero, but each force is capable of, producing a turning effect on the bar in the same Kinds of equilibrium, direction. Thus, the two forces together form a, couple which rotates the bar about the point O., Equilibrium is of two kinds : (1) static, m Fig. 1.14, the two forces rotate the bar in equilibrium, and (2) dynamic equilibrium., anticlockwise direction., (1) Static equilibrium : When a body remains, in a state of rest under the influence of several, forces, the body is in static equilibrium., Examples : (i) In Fig. 1.15, if a body lying, on a table top is pulled by a force F to its left, and by an equal force F' to its right (along the, same line), the body does not move. The reason, is that the applied forces are equal and opposite, (anticlockwise) along the same line, so they balance each other, A, 'B, Fig. 1.14 Couple action, Moment of force F at end A, = F x OA, Moment of force F at end B, (i.e., there is no net horizontal force on the body)., = F x OB, (anticlockwise) Hence, the body remains at rest (i.e., in static, Total moment of couple (i.e., moment of both equilibrium)., the forces) = F x OA + Fx OB, %3D, = Fx (OA + OB) = F × AB, %3D, F', = Fx d, (anticlockwise), Thus,, Fig. 1.15 Abody in static equilibrium, = Either force x perpendicular, (ii) If a book is lying on a table, the weight, distance between the two forces of the book exerted on the table vertically, .(1.3) downwards is balanced by an equal and opposite, Moment of, %3D, couple, (or couple arm), force of reaction exerted by the table on the book, 1.4 EQUILIBRIUM OF BODIES, vertically upwards. Thus, the book is in static, We have read that when a single force acts on equilibrium., a body, it can produce translational motion if the, body is free to move or can produce rotational balanced in horizontal position, the clockwise, motion if the body is pivoted or fixed at a point. moment of force due to the object on its right pan, But in certain circumstances it is possible to balances the anticlockwise moment of force due to, anply a number of forces (two or more) such that the weights on its left pan and the beam has no, i the resultant of all the forces is zero, so they rotational motion i.e., it is in static equilibrium., (iii) In a beam balance, when the beam is, 5, Scanned by TapScanner, F.
