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ICSE Class X Notes 2026 : Physics (Baldwin Co - Education Extension High School, Bangalore)

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Gitaali U Naidu
Baldwin Co - Education Extension High School, Bangalore
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Force (A) TORQUE, EQUILIBRIUM AND CENTRE OF GRAVITY In our everyday life, we use the term force to walk, to lift objects, to push or pull objects or even to change the shape of objects. For example, force is applied for pulling a rope, stretching a rubber band, squeezing a lemon, etc. In physics, force is the cause of the change of state of rest or state of motion in a body, or it is used to denote an action that deforms a body. Force is an external influence that changes, or tends to change the state of rest or motion of a body or deforms a body; that is, the force changes its shape and size. Force is a vector quantity. Its SI unit is Newton. 1 .1 TRANSLATIONAL AND ROTATIONAL MOTION A body that does not get deformed under the action of a force is a rigid body. Ideally, no real body is truly rigid; however, wood, metals, stone, glass etc. can be regarded as rigid bodies. A rigid body when acted upon by a force, can have two types of motion: Translational motion In translational motion, the body moves along a straight-line path in the direction of force applied. It is also known as linear motion. In this type of motion, the rigid body is not pivoted or fixed in any way, and every particle of the body has the same displacement. Examples: Motion of a rectangular wooden block down the inclined plane, a car moving on a straight road, a ball rolling on the floor in a straight line path (Fig. 1.1). Rotational motion If a body is pivoted at a point and a force is exerted on the body at a suitable point, it starts rotating the body about the axis passing through the pivoted point. This is known as the turning effect of the force and the motion of the body is called rotational motion. For example, when a wheel is pivoted at its centre and a force is applied tangentially on its rim as shown in Fig. 1.2, the wheel rotates about its centre. Merry-go-round Fig. 1.2 Rotational motion Examples: Rotation of a merry-go-round, a spinning top, a potter's wheel, movement of a door around the hinges, etc. In general, the motion of a rigid body is a combination of translational and rotational motion. Axiof 1.2 MOMENT OR TURNING EFFECT OF A FORCE (TORQUE) The turning effect of force is known as moment of force. It is the product of the force multiplied by the perpendicular distance from the line of action of the force to the pivot or point where the object will turn. the S.I unit is newton metre (Nm). Factors Affecting the Moment of a Force The turning mentioned effect of two a force factors:on a body depends on th following 1. The magnitude of the force (F) exerted. 2. The perpendicular distance of the line of action the force (XY) from the point of action or axis ofthe rotation (AOB). Moment of force about the point O = Force x Perpendicular distance of force from the point O Unit of moment of force = Unit of force x Unit of distance = Nm Clockwise and anticlockwise moments Conventionally, when the effect on the body turns it anticlockwise, the moment of force is called anticlockwise moment and it is taken positive, while if the effect on the body turns it clockwise, the moment of force is called clockwise moment and it is taken negative. Ihe direction of anticlockwise moment is along the axis of rotation outwards, while that of clockwise moment is along the axis of rotation inwards. When we apply a force on a pivoted body, its direction of rotation depends both on the direction of force and the point of application of the force. That is why, we can change the direction of rotation of a body in two ways: 1. By changing the direction of force: It is clear from the Fig. 1.4(a) that the anticlockwise and clockwise moments are produced on a pivoted axle by changing the direction of force F at the free end of the axle. 2. By changing the point of application of force: It is clear from the Fig. 1.4(b) that the anticlockwise and clockwise moments are produced in a disc pivoted at its centre by changing the point of application of the force F from point A to point B. ANTICLOCKWISE CLOCKWISE (POSITIVE) (a) ANTICLOCKWISE (POSITIVE) (NEGATIVE) By changing the direction of force CLOCKWISE (NEGATIVE) (b) By changing the point of application of force Fig. 1.4 Anticlockwise and clockwise moments Some examples of turning effect of force 1. Turning a steering wheel: We need to apply a force tangentially on the rim of the wheel for turning a steering. A line passing through the centre of the wheel will form its axis of rotation. By changing the point of application of force on the wheel, its rotation can be changed without changing the direction of the force. When we apply a force at point A on the steering wheel (i.e. at the bottom from the left direction), the wheel rotates in an anticlockwise direction, whereas if a force is applied at point B on the wheel (i.e. at the top), in the tangential direction, then the wheel rotates clockwise (Fig. 1.5). (a) Anticlockwise rotation (b) Clockwise rotation Fig. 1.5 Point of application of force for turning a steering wheel 2. Opening and shutting a door: A door is attached on one side to the hinges. A vertical line drawn through the hinges forms its axis of rotation. To open or close a door, we apply a force on its handle (i.e. normal to the door). The handle is provided at the other free end of the door (at point A), so that the distance from the hinges is more, as shown in Fig. 1.6. If we apply a force at a point somewhere in the middle of the door (point B) or near to the hinges (point C), a much greater force is required to open or close the door, and if we apply a force on the hinges of the door, the door will not open or close even if the magnitude of force is large, Fig. 1.6 Opening and shutting a door because the line of action is passing through the point of rotation. The handles being at a large perpendicular distance from the hinges of the door, comparatively a much smaller force is required to apply on the handles to open or close the door. A smaller force on the handle at the free end will produce the same turning effect as a larger force on the handle if it is located nearer to the hinges. It is due to this reason the handle is provided near the free end (or farther from the hinges) of the door. 3. Rotation of a bicycle wheel: In a bicycle, the axle of the wheel is the axis of rotation. lhe pedal is kept at a distance from the axle of the wheel so that the perpendicular distance from the line of action to the axis of rotation is large (Fig. 1.7). Due to this, a small force applied on the pedals can rotate the wheels. Foot Axle Fig. 1.7 Turning of a bicycle wheel 4. Turning a spanner: A spanner or a wrench is a tool used for tightening or loosening a nut. It has a long handle so that a large torque is produced when a force is applied normally at its end. As the perpendicular distance from the point of application of force being large, only a small force is needed to turn a nut. When the end of a spanner is held by hand such that the force is applied at the end of the handle in an upward direction, then the spanner turns anticlockwise and loosens a nut. If the direction of force is in a downward direction, then the spanner turns clockwise and tightens a nut (Fig. 1.8). Loosening a nut with a spanner Tightening a nut with a spanner We conclude from the above examples, that the turning ofa body about an axis depends on both the magnitude of force, and the perpendicular distance of the line of action of the applied force from the axis of rotation. Larger the perpendicular distance, less is the force required to produce the same torque. 1.3 COUPLE A single force alone can never produce a turning effect It is always produced by a pair offorces. In all the examples discussed earlier, a pair of forces required to produce a turning effect. One of the forces is the force applied externally and the other is the reaction force at the fixed or pivoted point, which is equal in magnitude to the applied force but opposite in direction. This pair of forces acting together to turn the body in the same direction constitutes a couple. A pair of equal and opposite parallel forces whose lines of action are not the same, constitute a couple. A couple is always required can be to balanced produce by a a turning equal effect or rotation. A couple and parallel forces (i.e. an equal couple but acting in the opposite direction). Examples: 1. While turning a steering wheel, the left hand pulls with force on the wheel, while the right hand pushes with the same force, as shown in Fig. 1.9. The two forces make the wheel turn in an anticlockwise directi011 If the wheel is to turn in clockwise direction, the left hand pushes and the right hand pulls the steering wheel. Fig. 1.9 Turning a steering wheel 1.3.1 Moment of Couple The moment of a couple is equal to the product of either of the two forces and the perpendicular distance between the line of action of both the forces. Fig. 1.14 illustrates the effect produced by a couple. Consider a bar PQ pivoted at a point O in the middle. At the two ends P and Q of the bar, two equal and opposite forces of magnitude F are exerted. Ihe perpendicular distance between two forces is called the couple arm, denoted by d. Each force has the turning effect on the bar in the same direction (anticlockwise in the figure). These two forces form a couple and rotate the bar around the point O. Fig. 1.14 Moment of couple Now, moment of force at the end P= F x OP (anticlockwise) Moment of force at the end Q = F x OQ (anticlockwise) Total moment of the couple = F x d (anticlockwise) = Either force x Couple arm (perpendicular distance between two forces) the resultant of a number of forces (two or more) acting on the body is zero such that the state of the body, whether rest or motion, remains unchanged. So we can say, a body in equilibrium has balanced forces acting on it. 1.4.1 Types of Equilibrium There are two types of equilibrium. 1. Static equilibrium: A body is said to be in static equilibrium if it remains in a state of rest under the influence of applied forces. Examples: A box lying on the table. The force exerted by the weight of the box in the downward direction is balanced by the reaction force of the table in the opposite direction, i.e., vertically upwards. The box remains in its state of rest as the resultant force is zero. It is thus in static equilibrium. (b) If a wooden block placed on a desk is pushed on the left side and the right side along the same line with an equal force, the block does not move because the two forces applied on the block are equal and opposite in direction, and thus, the net force is zero. (c) A beam balance is in static equilibrium when the anticlockwise moment of force on its left pan balances the clockwise moment of force on its right pan. The beam has zero rotational motion in this state. (a) 2. Dynamic equilibrium: A body is said to be in dynamic equilibrium if it remains in a state of motion under the influence of applied forces. Examples: (a) The movement of electrons around the nucleus of an atom. Here, the force of attraction provides the force required for the motion of electrons. (b) The revolution of earth around the sun. The gravitational force of attraction provides the force required for the motion of the earth. (c) The circular motion of a stone. A stone tied at the end of a string when whirled in a circular path with a uniform speed, the tension in the string provides the centripetal force required for the circular motion. Conditions for equilibrium The following two conditions must be satisfied for a body to be in the state of equilibrium. 1. The resultant of all the forces acting on the body must be zero. 2. The resultant of moments of all the forces acting on the body about the point of rotation must be zero, i.e., the sum of the clockwise moments about the axis of rotation must be equal to the sum of the anticlockwise moments about the same point. 1.5 PRINCIPLE OF MOMENTS If a body is in equilibrium under the action of a number of forces, then the algebraic sum of the moments of the forces about any point on the body is equal to zero. The principle of moments states that, the algebraic sum of anticlockwise moments is equal to the sum of clockwise moments when a number of forces act on a rigid body in equilibrium. While calculating the algebraic sum, the anticlockwise moment is considered positive, whereas the clockwise moment is considered negative. A beam balance is a device that works on the principle of moments. i.e., clockwise moment = anticlockwise moment. This verifies the principle of moments. 1. A force of 15 N is applied at a perpendicular distance of 0.2 m from a pivoted point. Calculate the moment of force. Ans. 3 N m 2. 2. The moment of force of 10 N about a fixed point is 6 N m. Calculate the distance of the point from the line of action of force. Ans. 0.6 m 3. A nut can be opened by a wrench of length 50 cm with a force of 120 N. If a smaller force of 75 N is applied, what will the required length of the handle of the wrench? Ans. 80 cm . 4. A wheel of diameter 2 m is shown in Fig. 1.28 with axle at O. A force F = 2 N is applied at B in the direction as shown in figure. Calculate the moment of force about (a) the centre O, and (b) the point A. Fig. 1.28 Ans. (a) 2 N m (clockwise), (b) 4 N m (clockwise) 5. Fig. 1.29 shows two forces Fl = 5 N and F2=3 N acting at points A and B of a rod pivoted at a point O, such that OA = 2 m and 0B = 4 m Fig. 1.29 Calculate: (a) The moment of force Fl about O. (b) The moment of force F2 about O. (c) Total moment of the two forces about O. Ans. (a) 10 N m (anticlockwise), (b) 12 N m (clockwise), (c) 2 N m (clockwise) 6. Two forces of equal magnitude are acting on a uniform bar AB of length 6 m, which is pivoted on the centre as shown in Fig. 1.30. Determine the magnitude of moment of force at the end A, (b) at the end B (c) total moment of couple (a) Fig. 1.30 Ans. (a) 15 N m anticlockwise, (b) 15 N m anticlockwise, (c) 30 N m anticlockwise 7. Fig. 1.31 shows two forces each of magnitude 10 N acting at points A and B at a separation of 50 cm, in opposite directions. Calculate the resultant moment of the two forces about the point (a) A, (b) B, and (c) O situated exactly at the middle of the two forces. ION Ans. (a) 5 N m clockwise, (b) 5 N m clockwise; (c) 5 N m clockwise. 8. A steering wheel of diameter 0.5 m is rotated anticlockwise by applying two forces each of magnitude 5 N. Draw a diagram to show the application of forces and calculate the moment of the forces applied. Ans. 2.5 N m 9. A uniform metre scale of weight 10 gf is pivoted at its 0 mark. What moment of force depresses the scale? (b) How can it be made horizontal by applying a least force? (a) Ans. (a) 500 gfcm (b) By applying a force 5 gfupwards at the 100 cm mark 10. A uniform metre scale is pivoted at its mid-point. A weight of 50 gf is suspended at one end of it. Where should a weight of 100 gfbe suspended to keep the scale horizontal? Ans. At distance 25 cm from the other end. 11. A nut is opened by a wrench of length 20 cm. If the least force required is 2 N, find the moment of force needed to loosen the nut. Ans. 0.4 N m 12. A uniform metre scale of mass 100 g is balanced on a fulcrum at mark 40 cm by suspending an unknown ass m at the mark 20 cm. a. Find the value of m. b. To which side the scale will tilt if the mass m is moved to the mark 10 cm? c. What is the resultant moment now? d. How can it be balanced by another mass of 50 g? 13. A physical balance has its arms of length 60 cm and 40 cm. What weight kept on the pan of the longer arm will balance an object of weight 100 gf kept on the other pan? Ans. 66.67 gf 13. When a boy weighing 20 kgf sits at one end of a 4 m long see-saw, it gets depressed at this end. How can it be brought to the horizontal position by a man weighing 40 kgf? Ans. If the man sits at a distance 1 m from the centre on the side opposite to the boy 14. A uniform half metre scale can be balanced at the 29.0 cm mark when a mass 20 g is hung from its one end. (a) Draw a diagram of the arrangement. (b) Find the mass of the half metre scale. (c) In which direction would the balancing point shift if 20 g mass is shifted inside from its one end? Ans. (b) 105 g (c) towards 25 cm mark 1 .6 CENTRE OF GRAVITY We know that the Earth attracts every body or particle towards its centre through the force of gravity due to the weight of a body. A rigid body of weight W can be considered to be made up of a number of minute particles, each particle of weight w (Fig. 1.33). w wet Fig. 1.33 Centre of gravity Centre of gravity (C.G.) of a rigid body is defined as The point at which the entire weight of the body acts and the algebraic sum of moments of weights of particles constituting the body is zero about this point. The position of C.G. of a body of a given mass depends on the shape of the body and distribution of its mass. Depending on these two factors, the C.G. may lie within the body or outside where there is no material. Also, if a body is deformed, the position of its C.G. changes. Examples: 1. The C.G. of a uniform bar or rod lies at the midpoint of its axis. 2. The C.G. of a ring lies at its centre where there is no material. 3. A wire has its C.G. at its midpoint. But if the wire is bent in the form of a circle, its C.G. will be at the centre of the circle; i.e. position of C.G. changes. Centre of Gravity of Regular Bodies Regular bodies have a definite geometrical shape and uniform distribution of mass. If a body has a regular geometrical shape and its mass distribution or density is uniform, then, in most cases, its C.G. coincides with its geometrical centre. The shape and position of C.G. for a few regular bodies S.No. Body Position of centre of gravity 1. Uniform straight wire Midpoint of the wire 2. Uniform beam or rod Midpoint of the axis of rod 3. Circular disc or ring Geometric centre 4. Sphere (solid or hollow) Geometric centre 5. Cylinder Midpoint on the axis of cylinder 6. Solid cone At a height, 1/4th from its base on its axis 7. Hollow cone At a height, 1/3rd from its base on its axis 8. Triangular lamina Centroid-the point of intersection of medians 9. Rectangular lamina or square Point of intersection of its diagonals 10. Rhombus (or parallelogram) Point of intersection of its diagonals Cube or cuboid Midpoint of the line joining the centres of opposite sides 11. Fig. 1.34 Centre of gravity of some regular bodies Centre of gravity and the balance point Each solid body can be balanced by supporting it at its centre of gravity. When a body is freely suspended from a point, it comes to rest in such a position that its centre of gravity lies vertically below the point of suspension. For example, we often try to balance a notebook on the tip of our finger at its centre. Similarly, an object can be balanced on a knife by keeping it exactly below the centre of gravity of the object. 1.7 UNIFORM CIRCULAR MOTION Circular motion is the simplest type of rotational motion. It is commonly seen in both microscopic and macroscopic systems. When a particle moves with a constant speed in a circular path, its motion is then termed as uniform circular motion. In a uniform circular motion, a particle moves equal distances along the circular path in equal intervals of time, so the speed of particle is uniform, but the direction of motion of the particle changes at each point of the circular path. The continuous change in the direction of motion implies that the velocity of the particle is non-uniform (or variable), i.e., the motion is accelerated. Difference between uniform linear motion and uniform circular motion The uniform linear motion is an unaccelerated motion, i.e., the speed and velocity are constant and acceleration is zero. The uniform circular motion is an accelerated motion, as the velocity is variable (although the speed is uniform). 1.8 CENTRIPETAL FORCE We know that a force is required to change the direction of motion of a particle (or to change the velocity of a particle), i.e., to produce acceleration. A particle moving in a circular path continuously changes its direction of motion at each point of its path. This change in direction of motion can not be brought without a force. Thus, the motion in a circular path is possible only under the influence of a force. This force is termed as the centripetal force. At each point ofthe circular path, this force is directed towards the centre of the circle as shown in Fig. 1.37. Thus, the direction of force and also of acceleration changes at each point of the circular path, but its magnitude remains the same, i.e., acceleration is variable (or nonuniform). Hence, for a body moving in a circular path, a force is required, which acts as the centripetal force. Thus, centripetal force is the force that acts on a body moving in a circular path, and it is directed towards the centre of the circular path. c Fig. 1.37 Direction of force in uniform circular motion The word centripetal means centre seeking. Examples: When a stone tied at the end of a thread is whirled in a circular path, the thread needs to be continuously pulled inwards. The tension developed in the thread held by the hand provides the centripetal force (Fig. 1.38). If the thread breaks or is released, the stone immediately leaves the circular path and moves along the tangent to the circular path from that particular point. This happens due to the fact that as the tension in the thread vanishes, the centripetal force also vanishes. 1. Path travelled by stone Fig. 1.38 Centripetal force 2. The centripetal force required for the movement of an electron around the nucleus in an atom is exerted by the electrostatic force of attraction between negatively charged electrons and positively charged nucleus. 3. The planets revolving around the Sun or the satellites revolving around the planets receive centripetal force from the force of gravitational attraction between the two bodies. A car travelling on the road has three forces acting on it. The gravitational force is due to the weight of the car that gets balanced by the normal reaction force exerted by the road on the car. The third force is the force of friction that the road exerts on the wheels of the car. When a car makes a turn around a curved road, it is this force of friction acting upon the wheels of the car that provides the centripetal force required for taking a smooth turn. 5. In all the above examples, the body moves in a circular path with a uniform speed under the influence of a centripetal force and it is in a state of dynamic equilibrium. 4. 1.9 CENTRIFUGAL FORCE The force which appears to act on a body moving in a circular path and is directed away from the centre around which the body is moving, is known as centrifugal force. While the centripetal force acts towards the centre of a circular path, centrifugal force acts in its opposite direction. Even though the magnitude of the centrifugal force is same as that of the centripetal force and its direction is opposite to that of the centripetal force, it is not a reaction force to the centripetal force. In fact, centrifugal force is not a real force; it is an apparent force or fictitious force or virtual force. It is an apparent force because it is not a part of an interaction but is a result of rotation. It is not a reaction force and is considered only to understand a particular type of motion (i.e., circular motion). This can be illustrated by the following experiment. Experiments 15 Consider one end of a string tied to a ball and the other end of the string tied at the centre of a merrygo-round. baX at Merry-go- Fig. 1.39 A ball tied at the end of string moving in a circular path on a merry-go-round Initially, when the platform of the merry-go-round is stationary, the ball also appears stationary and the string is loose. As the platform of merry-go-round starts to rotate, the ball rolls towards the edge of the merry-go-round and the string becomes tight due to tension T developed in it. Suppose that the motion of the ball is observed by two persons (i) standing outside the merry-go-round on the ground at X, and (ii) standing on the platform of the merry-go-round at A. The person who is standing on the ground at X outside the merry-go-round observes that the ball is moving in a circular path (denoted by dotted line), while the person who is standing on the merry-goround at A, observes that the ball is stationary placed just in front of him at B. The different observations of the same motion by the two persons at X and A can be explained easily as follows. Explanation; For the person at X, the ball moves in a circular path because the tension T in the string provides the centripetal force which is needed for the circular motion. The person at A observes the ball as stationary. He considers the following two forces to be acting on the ball in order to explain his observation. 1. The tension T of the string acting towards the centre of merry-go-round, and 2. The centrifugal force which is directed away from the centre. These two forces are equal and opposite; hence, the net force on the ball is zero and the ball appears stationary. However, for the person standing on the ground, only one force is visible; which is, the centripetal force due to the tension in the string, and thus to him, the ball appears to be rotating in a circular path. Merry-go- Fig. 1.40 Centrifugal force Conclusion: Ibe centrifugal force is not a real force. It is a virtual force. The only real force involved here is the force of tension in the string acting towards the centre (i.e., the centripetal force). A force that really does not exist, but is considered to describe (or understand) a certain motion, is called a fictitious force (or virtual force). Now, if the string is cut when the ball is at a particular position, there is no longer a force of tension in the string. To the person standing on the ground, it appears that the ball is moving in a straight-line path; it is a tangent to the point where the ball lies on the circular path. However, to the person standing on the platform, the ball appears to be moving radially away from him but always remains in front of him as his own position changes with the rotation of the platform. For this person, the centripetal force has ceased, but the centrifugal force still acts on the ball along the radius of the platform in a direction away from its centre. Difference between Centripetal and Centrifugal Force Centripetal force Centrifugal force 1. Centripetal force is a force which acts on a 1. Centrifugal force is a force which appears to act body moving in a circular path and is on a body moving in a circular path and is directed towards the centre around which directed away from the centre around which body is moving. body is moving. 2. It is a real force. 2. It is an imaginary or virtual force. 3. It is an inward force. 3. It is an outward force. Example: satellite orbiting a planet. Example: passenger in a turning car feels as if they are pushed outward. _________________________________________________________________________________

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