Physics Force & Newton’s laws of motion Human tife would be dull without social interactions. Similarly, the physical universe would be dull without physical interactions. Social interactions with friends and family change our behaviour; physical interactions change the “behaviour” (e.g. mation) of matter. ‘An interaction between two objects can be described and measured in terms of two forces, one exerted on each ‘of the two interacting abjects. A force is a push or pull that one object exerts on another. ER ids of forces Force can be classified In many ways. For example, force ean be dluded in two kinds (1) Continuous forces (2) Momentary forces Continuous forces ‘A force constantly applied to an abject is called a continuous force. There are many examples of continuous forces. The downward force of Earth’ gravy on all objec is a continuous force called weight. All the automobile engines, aeroplane engines, rocket engines provide continuous forces to run them continuously. Momentary forces Not all forces are continuous. A moving object sometimes colldes with another object, causing a change in speed ‘or direction. This fs called a momentary force ‘A force applied to an object for a moment only is called @ momentary force. One example of a momentary force Is whon a cricket player his a cricket ball with a ba. Fist, the bower throws te bal toward the batsman. Then, the batsman sivings the bat. When the bat rakes contact with the ball, the momentary force applied by the bat causes the ball to move in another direction. The force applied by the bat is Imited and does not keep continuously ating on the ball. However, it Is Important to remember that other forces, such as gravitational force, do act on the ball continuously, Force can also be divided in two kinds (1) Contact forces (2) Non-contact force (or action-at-a-distance forces). Contact forces When you press the koys on @ computer keyboard, your fingers exert a force on the Keys. This force ean be exerted only when your fingers are touching the keys, A force thal Is exerted only when two objects are touching is a contact force. A contact force can be small, such asthe force you exer 10 push a penll across a sheet af paper, oF large, such 2s the force exerted by a traffic crane as It pulls a car along a stret. Contact forces include muscular forces, tension, frletion, normal forces Non-contact force (Action-at-a-distance force) ‘When you jump up in the ai, you are pulled back tothe ground, even though nothing seems to be touching you Forces can be exerted by one abject on another even though they arent touching each other. The force pulling you down to Earth is the gravitational force exered by Farlh. This force is a non-canlact force ‘A nonccontat force fsa force tha one object exerls on anather when they are nel touching. Non-contact forces include the gravitational force. the electric force, and the magnetic force Conservative force A force is conservative If work done by the force on a particle that moves through any round trip (complete cycle) is ze0, ©, graitatonal forces, electrostatic forces, elastic forces are conservative in nature Non-conservative force AA force Is non-conservative f work done by the force on a particle that moves through any round trp (complete cycle} Is not zero, 0g titional forces are non-conservative in nature. 33Class X | 22% Inertia Its “the natural tendency of an object to remain at rest or in molian at a constant speed along a straight line" is the tendency of an object to resist any atlempl to change In its velocty The mass of an object is a quantitative measure of inertia. More the mass, more will be the inertia of an object and vice-versa, Inotla of an object can be of tree typos (1) Ineria of rest, the tendency of an object to remain at rest. This means an object at rest remains at rest uni a sufficlenty large external force is applied on i {2) Inertia of motion, the tendency of an object to remain in the slate of uniform motion. This means an object in uniform motion remains continue to move uniformly unil an extemal force is applied on IL (3) Inertia of direction, the tendency of an object fo maintain its direction. This means an object moving in a paricular direction remains continue to move in that unl an external force is applied to change it Newton's first law of motion (Gallleo’s law of Inertia) ery object continues in its stale of rest, or of uniform molian in a straight line, unless itis compelled to change that state by forces impressed upon i Linear momentum (or momentum) 34 ‘When someone asks you whether you would hit more by a piece of chalk or by a cork ball af first, the answer ‘seems to be the cork ball. But if the chalk was moving at 250 m/s, then you'te basicaly dealing with a bullet ‘and the choice becomes obvious. Momentum is an important concept when considering impacts, colisions, and how objects in general intoract. It is not just an object's mass or an object's velocity that is important; itis the product of its mass and velocity. ‘The product of the mass (mn) & velocity (W) is called linear momentum ' pm] F quantity. Its direction Is “the direction along the Linear momentum is a velocity’ The linear mamentum of a particle is drectly proportional to () its mass (i) its velocity. - Unit of linear momentum : SI unit : kg m/s or kg m/s” or Newion-second (Ns) Fig.1 Sign convention CGS. unit: g cm/s or g cm sor Dyne-second for tinear momentum Linear momentum can be positive or negative depending on its alrection. For a given velocity, the momentum Is directly proportional to the mass of the object (p se m). This means more ‘the mass, more willbe the momentum and vice-versa, If a car and a truck has same velocity, then, the momentum ‘of truck Is more than the momentum of car as the mass of a truck is greater than the mass of a car For a given mass, the momentum is directly proportional to the velocity of the object (p< v). This means more the velocity, more will be the momentum and vice-versa. If two bodies with same masses move with different velocities then, the body having more velocity will have more momentum, For a given momentum, the velocity is inversely proportional to the mass of the object (v cc 1 / m). This means smaller the mass, mare will be the velocity of an object and vice-versa. If a car and a truck has same momentum, the velocity of car will be more than the velocity of trck as the mass of a car is smaller than the mass of a truck Fig. Diferant graphs related to momentum When a object Is moving along a circular path, its velocity is tangential to the circular path hence, Its momentum is also tangential to the circular pathTol Physics Momentum (p) of a photon [A photon Is considered as massless, chargeless particle of an electromagnetic wave like visible light, X rays ultraviolet rays, radio waves, ete. but It carries energy, Planck's constant 6.63 « 108" Is vy = frequency of electromagnetic wave > 2 = wavelength of electromagnetic wave. wton's second law of motion Where, E = energy carried by a photon “The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acs’. Mathematically, it can be represented as, Pa=Pi_miv=u t t If force Is constant Le., F = ma = constant, then, the acceleration produced in the body Is inversely proportional to its mass, Le, ac 1/m. This means, if same force F Is applied to masses m, and m, and the resulting accelerations in them are a, and a, respectively, then, ma, = m, IF =ma fa, _m am, newton Is the amount of force that produces an acceleration of 1m s* in an abject of 1 kg mass. Similarly, 1 dyne fs the amount of force that produces an acceleration of 1 em s? in an object of 1 g mass 1 N = 10% dynes 1 1 t F F a f = constant A= constant fy constant Fig. Graphs related to force, mass and acceleratlon Impulse (J) The product of force and time is ealled ‘mpube’. 1is also the change in momentum of the body. It isa vector quanti 1 P= Pip, =Mv— A large force acting for a short lime that produces a significant change in momentum is called an Impulsive force. Y) =F] — a Iv force F acting is variable then, impulse, [T= Fa, ‘Area under the forcectime graph gives impulse (see fig 4). Impulse = A, + A, Newton's third law of motion Fig.4 Area under Ft graph gives impulse Whenever one body exerts a force on a second body, the second body cexeris an oppositely directed force of equal magnitude on the first body’ ‘To every action, there is abvays an equal and opposite reaction’ Forces always exist in pairs : When two objects interact, two forces will always be involved. One force is the action force and the other Is the reaction force 38Class X 36 Consider a pair of bodies A and B. According to the Newton's third law, Fy, == Faq Where, F,, = force on A due to B and F,,: ‘Though action-reaction pair are equal in magnitude and opposite in direction but the reaction force aluays acls on a diferent object than the action force. Thus, these forces do not cancel out each other. Hence there can be an acceleration in an object. Newton's third law Is applicable to non-contact forces also. For example, the Earth pulls an object downwards due to gravity. The object also exert the same force an the Earth but in upward direction. But, we hardly see the effect of the stone on the Earth because the Earth Is very massive and the effect of a small force ‘nts motion is negligible, That is, the acceleration of Earth Is negligible due to its huge mass Even though the aclion and reaction forces are always equal in magnilude, these forces may nat produce accelerations of equal magnitudes. This is because each force acts on a diferent object that may have a different masses, force on B due to A Co When the net external force on a system of objects Is 2er0, the total linear momentum of the system remalns conslant’. In other words ‘the total momentum of an isolated system of abjects remains constant The term ‘colfsion’ is used to represent the event of two paticles coming together for a short time and thereby producing “impulsive forces’ on each other. These forces are assumed to be much greater than any external forces present because they act for a very short time interval, Momentum is conserved for al types of callisons that take place in Teal world In the absence of any external force. Rocket propulsion or the recoll of gun are based on law of conservation ‘of momentum as well as Nowton’s third law. This is because the law ‘of conservation of momentum is derived using Newton's third law. Solving problems on conservation of momentum Recoll of a gun : Initial momentum = Final momentum rvation of linear momentum Intlal momentum = 0 m ord = MV- mw or aa] Gee ia) v ‘© A bullet Is fired on a wooden block and It gets embedded in t, after i that they move together with a common velocity (see fig.) Initial momentum = Final momentum M Tm Final momentum = my - MV = 0 Mem Fig Rel of a gun or pista A bomb of mass M explodes in wo paris having masses m, and m, Gee fi 7 Final momentum = Ital momentum amy, Or my, = mY, or mu = (Mam) Voor o> [a In momertim +o mxu+Mx0=mu Initial veel, u = 0 M+ ‘Mass of bomb ~ M inal momentum = v Initial momentum = M x0 = 0 my, — my > r (Gerore explosion) ‘ater explosion) Final momentum = Fig.7 (temy Fig.Physics x=x= A775 00 N Force applied on the wall = - Force applied on the body = (- 2500 N) = + 2500 N ‘A bulet of mass 0.04 kg moving with a speed of 90 m s" enlers a heavy wooden block and i stopped afler @ dislance of 60 cm. Whal is the average resistive force exerted by the block on the bullet ? Solution Mass of bullet, m = 0.04 kg ; intial speed, u = 90 ms" ; final speed, v = 0 distance, s = 60 cm ~ (60/100) m = 0.6 m From third equation of motion, v? = u + 2a or (OF = (90 + 22(0.6) (0 2x06 Force, F = ma = (0.04) (6750) = = 270 N = = 6750 ms? ‘A shel of mass 0.020 kg Is frea by a gun of mass 100 kg. Irthe muzzle speed of the shells 80 m/s, what is the recoil speed of the gun ? Solution Given, mass of but, m. = 0.020 kg : muzzle speed (speed of bul, v, = 80 m/s ‘mass of gun, m, = 100 kg ; reco spec Initial momentum, p, = 0 Final momentum, p, = m.v, — my, By conservation of momentum Final momentum = Initial momentum oF By = P, or my - m or (0.020)80) - (100},) = 0 or 1.6 - 100¥,, or v, = (1.6/100) m/s = 1.6 « 102 m/s 37Class X Ea Tension In strings 38 ‘Strings are assumed to be inextensible Le., they cannot be stretched. Due to this assumption ‘acceleration of masses connected through a string Is alvays same. They are assumed to be massless unless It Is mentioned, Due to this assumption ‘tension in the string is same every where If the string has mass, tension at different points will be different. I is maximum al the end at which force is applied and minimum at the other end connected to a mass. Minimum T Maximum T ‘A massless string ‘A string having mass Fig Tensions In the sting The direction of tension at body (ora point) is abways outward along the string ie., away from the body along. the sting, A tension always have pling action Free body diagrams A system diagram isa sketch of all he objects involved In a situation. ffe-body diagram (FD) fa drawing in which only the cbject Being analyed is drawn, with arrows showing all the frces acting on the object (1) Free body dlagrams represent all forces acting on one object (2) Forces that the object exerts on other objects do not appear in free body diagrams because they have no effec on the motion of the objet salt (3) In drawing a free body dlagram, you can represent the object as a single dot or a simplified shape the object, (4) In FBD each force acting on the object is represented with an arrow. The arrow’s direction shows the dltection of the fore and the arraw’s relate length provides information about the magnitude ofthe force (5) Forces that have the same magnitude should be sketched with approximately the same length, forces that ate larger should be longer, and smaller forces should be shorter. {6) In case of objects in motion, the direction of acceleration should be made on the FBD In the direction of greater force (or net force) Ball » () ad rin ad ) Hand w w x system System slagram Free body diagrams Free body oe agra diag Fig.10 Making free body diagramsARLEN Physics Motion of bodies connected by strings Let us conser to toes m, and mpl! on orzo ones plane connected by @ masses stn. Le the mas m, pul by a fe FAS a res he Whol stm motes in the deton of apples ec with an acceleration a. Let the tension in the string be T (see fig.17). im — a a F Teer FBD form, — FBD form, ‘Amasses sing Fig.11 Motion of Bodies connected by srngs For mass m,, F = T=m,a —~ (1) [F is greater force as its inthe dlecton of acceleration a] For mass m,, T =m, 2 (2) (Here, Tis the ony force acting on m,} i) > F-N+T=ma + ma ¥ im. + mja or From (2), we have, T = ma Motion of bodies connected by string passing over a light pulley {Atwood’s Machine) Let us consider two masses m, and m, passing over a Hght poley connected through a sting (Ge fig.12), The term tight pulley’ means the mass of pulley is neglected, i is assumed tobe massless, Since the two botles ae connected with each other, both move with same acceleration a, Let m, > m, then, m, will go downwards while m, will go upwards, f 1 Ten | ™ form, T-mg-ma —() tee Tg aT sine deci ot xcaeatn) = GY |S em For my mg T= ma — (2) FBD orm, BD form [Here, m, g > T, as m,g Is In the diestion of acceleration a ns im, ()+@) > (T-mg)+img-TN=ma + ma or (m,— mg = im, + ma mo hme Fig. 12 Motion of massos connocted by a pulley or (m,+m,) (Since a + g, two bodles are not free fang bodles}) —_ am mag| mem, Putting the value of a in eq(), we get, Motion of bodies In contact Let two bodles of masses m, and m, respectively are placed side by side touching each other. A push force "Is ‘applied on m, such that both the bodies start moving together with an acceleration ‘a’ Since both the bodies are touching each other there Is a palr of action reaction force between them at place of thelr contact, These forces ‘are called normal contact forces (see fig.13) and obviously they are equal in magnitude but opposite in direction (Newton's third fan) 39Class X Applied fewer or force F 7 nm FBD form, — FBD form, Fig.18 Motion of bodies In contact For mass m,, — (1) [Fis greater forco as itis in the direction of acceleration a} For mass m,, f= m, a —~ (2) [Here {1s the only force acting on m (1 Qs F-Oet oma + ma o From (2), we have, Welght of an object In a lift ‘A weighing machine measures the normal force not the ‘irue weight. Thus, if the normal force changes, the weighing machine does not give reading of true weigh, it gives a reading of normal force which we can be called ‘apparent welght’ of the object Lot us consider a git standing in a ft ee R R dt AF PLU ‘or maving itn Upwards with FBO downwards with FBO Uniform velacty 10° il uniform acceleration fOr Gi ynifarm acceleration fora @ w o Fig.14 Weight of an object in a lit (1) When the lit is at rest or in uniform motion, nel acceleration of the system is zero [see fig.14all. Thus, net force on it is ze, Allis at ros Allis moving Net force, FL,=mg-R=0 of R=mg The R represents the apparent weight, Le, W = R = mg [Apparent weight rue weight) (2) When the lit is moving up with uniform acceleration a (sce fig.14(b]. Thus, net force on it fs not zor. Net force, F,, = R'=mg = ma IR’ Is greater force as it Is in the direction of acceleration a] or R= masmg=miesg) he R represents the apparent weight, Le., W" m(a + 9) [Apparent weight > true weight) (3) When the lit fs moving down with uniform acceleration a [see fig. 14]. Thus, net force on it Is not 2210. mg ~ R= ma_ mg is greater force as itis in the diection of acceleration 3] or RY = mg =m ma-a) he R" represents the apparent weight, Le., W" Suppose the rope of the lift breaks, then it will fall feely under gravity Le., a = g. In this situation, apparent Im(g ~ 9) = 0. That fs, the weighing machine will read zero weight. R" = mig = 2) (Apparent weight < true weight)i“ Physics E> _ Friction It is a force that opposes the movernent between two surfaces in contact. Some Important points related to friction are (1) The magnitude of the friction force depends on the types of surfaces in contact, The fictional force is usualy larger on the rough surfaces and smaller on the smooth surfaces. Friction depends on both the surfaces that are in contac, therefore, the value of friction is diferent for diferent pais of surfaces (2) Friction is always parallel to the surface in contac. (3) I an objec is allowed to move on a surface then, more the distance travelled by the abject on the surface, fess willbe the friction between them and vice-versa (4) Friction is caused by the iregularies on the two surfaces in contact (6) There are many kinds of friction that exist in diferent media () Static friction : 1k exists when two surfaces try to move across each other but not enough force is applied to cause motion (i) Siding friction : It exists when two surfaces slide across each other. (i) Rolling friction : It exists when one object roll over another object. (iv) Ale frletion (air resistance) : It exists when air moves around an object () Viscous friction : it exists when objects move through water or other liquids. (6) Force of friction increases if the two surfaces are pressed harder. The greater the force pressing the two surfaces together, the greater will be the force of friction between them. (1) Friction increases with weight. For a heavy object, the weight is quite large, therefore, the push force (pressing force) between the object and the floor is also large, Thus, the fiction force between them is large. (8) For hard contact surfaces, the force of friction does not depend on the ‘area of contact’ between the two surfaces. Bul, it is not true if the surfaces are wet, or if they are soft. Rubber is soft as compared to the surface of a toad. The friction between rubber and surface of road also depends on how much rubber is contacting with the surface of road. Thus, wide tres (made of rubber) have mor friction than narrow tires. Static friction It is the force exerted on an object at rest that prevents the object from siding The direction of static frcton Is opposite to the applied force, Also, it acts in a direction opposite to the direction in which an object tends to move ‘The maximum value of static friction is called the starting friction or limiting friction. It is the amount of force that must be overcome to start 2 stationary abject moving The law of stale felon may be witen as, [aN Where, 4, = coefficient of static friction, depends only on the nature of surfaces in contact N= normal force (or normal reaction. Limiting (maximum) value of state friction is gen by. f, =u, N. Ifthe applied force F exceeds 4, , the body begins to slide on the surface. If applied force F is lss than f, thon, F = fie. applied force is equa to the value of stati fiction and body wil Sllding friction (or kinetic friction) Its the force exerted on an ject in motion that opposes the rotion ofthe object asi sides on another object Sliding or kinetic ction is smaller than the limiting value of static fiction, This is because I takes more force to break the interlocking between two surfaces than it does to keep them sliding once they ae already moving. Kinetic frtion, tke sla frtion, is also found to be independent of sig-18 variation of tcton the area of contact. Further, itis nearly independent of the velocity Oth applied force, of the body, Kinetic friction ‘Applied force > aformal force fiction may be written a, N Where pis the coefficient of kinetic friction, depends only on the nature of surfaces in contact, Hs > Hi] . 1, oF 14 has not units as they are ratio of two forces, Force of gravity (Weight = ma) Normal force Normal force on a horizontal plane is, N = mg while normal force on an inclined plane is mgcos8, where “0” is the angle made by the plane with the horizontal SS Note that it is not motion, but relative motion that the frictional force opposes. 3 Angle of repose (0) If a body is placed on an inclined plane and if its angle of inclination is gradually increased, then at some angle of inclination @ the body will start Novmat eres = macose © just sliding down. This angle of the inclined plane at which the body just stars ig. Nocmal forces on sliding is called angle of repose (0) horizontal and inclined plane From fig., we have, f, = mgsiné io} R = mgcosd -—~ (2) _,£_ masino QR” mgcoso iané or ffand (1) If angle of inctination (a) of the plane is less than angle of repose (0), the body will remain at rest. 19.17 Angle of repose (2) Ifa = 6, then body will ust slide Le. wil move uniformly TT Ange rece (3) If a > 6, the body will accelerate downwards, The acceleration can be found by the 9.18 Fg = msn = f= mgsina = 4 = mgsina -y,9c05a ortna = misina =, 96050) ora = glsina -1,¢0sa) IC there Is no con, acceleration on the incned plane, a = gsina, where ais the angle made by the Inclined plane with the horizon Rolling friction The rolling motion of the wheel is a combination of both spin (rotational) motion and near (translational) motion, When one body rolls aver the surface of another body, the resistance (opposition) to Its mation is called the rolling friction ‘© Roling reduces the friction significantly. Since the rolling tition is smaller than the sliding ftition, sliding Is replaced in most machines by rolling by the use of ball bearings. '® Roling friction increases with the deformation of tyre or wheel. Thus, roling tition of a tyre or wheel made of rubber is more than a tyre ar wheel made of iron, This is because the iron wheel deform negligibly while rubber tyre deform significantly.