Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

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1 LAWS OF MOTION PLOM 1 53 The Law of Inertia : Q What is inertia? Solution : If the net external force is zero, a body at rest continues to remain at rest and a body in body continues to move with a uniform velocity This property of the body is called inertia 54 Newton s First Law of Motion : Q What is Newton s first law of motion? Solution : Every body continues to be in its state of rest or of uniform motion in a straight line unless compelled by some external force to act otherwise This law can be expressed in another ways as : If the net external force on a body is zero, its acceleration is zero Acceleration can be non zero only if there is a net external force on the body Q An athelete runs a certain distance before taking a jump Why? Solution : An athelete runs a certain distance before taking a long jump This is because velocity acquired by running is added to the velocity of the athelete at the time of jump Hence he can jump over a longer distance Q An astronaut accidentlly gets separated out of his small spaceship accelerating in inter steller space at a constant rate of 100 m s What is the acceleration of the astronaut the instant after he is outside the spaceship? (Assume that there are no nearby stars to exert gravitational force on him) [NCERT Solved Example 51] Solution : Since there are no nearby stars to exert gravitational force on him and the small spaceship exerts negligible gravitational attraction on him, the net force acting on the astronaut, once he is out of the spaceship, is zero By the first law of motion the acceleration of the astronaut is zero 55 Newton s Second Law of Motion : Q What is momentum or linear momentum? Is it a vector or scalar quantity? Solution : Momentum of a body is defined to be the product of its mass m and velocity v, and is denoted by p : p mv Momentum is clearly a vector quantity Q Write down Newton s second law of motion? Solution : 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 acts Q Express force in terms of rate of change in linear momentum Hence prove that force equals to product of mass and acceleration Solution : If under the action of a force F for time interval t, the velocity of a body of mass m changes from v to v v ie its initial momentum p mv changes by p m v According to the Second law, p p p F or F k, where k is a constant of proportionality Taking the limit t 0, the term t t t dp dp becomes Thus F k dt dt dp d dv For a body of fixed mass m, (mv) m ma ie, the second law can also be written as dt dt dt F kma which shows that force is propoetional to the product of mass m and acceleration a For dp simplicity, we choose k = 1 The second law then is F ma dt

2 PLOM Q A bullet of mass 004 kg moving with a speed of 90 m s 1 enters a heavy wooden block and is stopped after a distance of 60 cm What is the average resistive force exerted by the block on the bullet? [NCERT Solved Example 5] Solution : 70 N Q The motion of a particle of mass m is described by particle [NCERT Solved Example 53] Solution : mg y ut 1 gt Find the force acting on the Q Define impulse Is it a vector or scalar quantity? Write down its units and dimensions Solution : The product of force and time, which is the change in the momentum of body is called impulse It is a vector quantity It has the same unit and dimensions as the momentum Its unit is kg m/s or Ns Its dimensional formula is [MLT 1 ] Q What is impulse momentum theorem? Solution : Impulse momentum theorem is given by : Impulse = Force time duration = Change in momentum Hence a given change in linear momentum can be produced by applying a larger force for a smaller time or by applying a smaller force for a larger time Q Give reason for the following : A cricket player lowers his hand while catching the cricket ball Solution : The player draws in his hands backward in the act of catching the ball and he allows a longer time for his hands to stop the ball As impulse = force time = change in linear momentum Therefore by increasing the time of a catch, the player has to apply a smaller force against the ball in order to stop it The ball in turn, exerts a smaller force on his hands and his hands are not injured Q A batsman deflects a ball by an angle of 45 0 without changing its initial speed which is equal to 54 km/h What is the impulse imparted to the ball? (Mass of the ball is 015 kg) [NCERT Solved Example 54] Solution : 36 N s Q Is the force necessary to change the direction of velocity (motion) if the speed is constant? Give example Solution : If the direction will change keeping the speed constant then its velocity will change and hence the motion is accelerated That s why from Newton s second law, the force is necessary to change the direction of motion if the speed is constant Any particle in uniform circular motion will experience a centripetal force Q Is the force necessary to change the direction of momentum, even if its magnitude is constant? Solution : If the direction will change keeping the magnitude is constant then its momentum will change and hence the force is necessary 56 Newton s Third Law of Motion : Q Write down Newton s third law? Solution : To every action, there is always an equal and opposite reaction Q Prove that the second law is the real law of motion Solution : To establish it, we shall show that (a) the first law is contained in the second law, and (b) the third law is contained in the second law (a) First law is contained in the second law : According to Newton s second law of motion, F ma If no external force is applied on a body, F = 0 m a = 0 As m 0, therefore, a = 0 Thus there will be no acceleration in the body if no external force is applied This is Neuton s first law of motion Hence the first law is contained in the second law (b) Third law is contained in the second law : Consider an isolated system of two bodies A and B Let them collide During collision, let the body A exert a force F 1 (action) on body B for a time t Let the body B exert a force F (reaction) on body A for the same time t

3 As, change in linear momentum = Force time change in linear momentum of B F 1 t and change in linear momentum of A F t Total change in linear momentum of A and B = F t F 1 t PLOM 3 As no external force is acting on the system, therefore according to Newton s second law of motion, the total change in linear momentum of the system is zero Thus F1t Ft 0 or F1 t Ft or F 1 F It means action is equal and opposite to reaction Thus third law is contained in the second law Q Two identical billiard balls strike a rigid wall with the same speed but at difference angles, and get reflected without any change in speed as shown in figure What is (i) the direction of the force on the wall due to each ball? (ii) the ratio of the magnitudes of impulse imparted to the balls by the wall? [NCERT Solved Example 55] Solution : (i) towards positive x direction (ii) 1 57 Conservation of Momentum : Q Which laws of motion lead to the conservation of momentum? Explain it Solution : The second and third laws of motion lead to the law of conservation of momentum Q Prove that the total momentum of (bullet + gun) system is conserved Solution : A bullet is fied from a gun If the force on the bullet by the gun is F, the force on the gun by the bullet is F, according to the third law The two forces act for a common interval of time t According to the second law, F t is the change in momentum of the bullet and F t is the change in momentum of the gun Since initially, both are at rest, the change in momentum equals the final momentum for each Thus if p b is the momentum of the bullet after firing and p g is the recoil momentum of the gun, p g pb ie, p p 0 That is, the total momentum of the (bullet + gun) system is conserved b g Q What is law of conservation of momentum? Solution : The total momentum of an isolated system of interacting particles is conserved, ie, if net external force acting on the system equals to zero, then linear momentum of the system remains conserve dp If F 0 0 p = constant (conserve) dt Q Explain conservation of Linear momentum by an example OR Prove the following statement : Momentum will be conserved in the collision of two bodies Solution : Consider two bodies A and B, with initial momenta p A and pb The bodies collide, get apart, with final momenta pa and pb respectively By the Second Law FAB t pa pa and FBAt pb pb

4 PLOM 4 (where we have taken a common interval of time t for both forces ie, the time for which the two bodies are in contact) Since F AB F by the third law, BA p A p A (pb pb ) ie, pa pb pa p B which shows that the total final momentum of the isolated system equals its initial momentum 58 Equilibrium of a Particle : Q What is equilibrium of a particle? Solution : Equilibrium of a particle refers to the situation when the net external force on the particle is zero Q A mass of 6 kg is suspended by a rope of length m from a ceiling A force of 50 N in the horizontal direction is applied at the mid-point of the rope What is the angle the rope makes with the vertical in equilibrium? (Take g = 10 m s ) Neglect the mass of the rope [NCERT Solved Example 56] Solution : Common Forces in Mechanics : Q What is spring force? Solution : When a spring is compressed or extended by an external force, restoring force is generated This force is usually proportional to the compression or elongation (for small displacements) The spring force F is written as F = k x where x is the displacement (compression or extension) and k is the force constant The negative sign denotes that the force is opposite to the displacement from the unstretched state Q Draw the graph for the variation of spring force with compression or extension What is the physical meaning of slope of this graph? Solution : The slope of this graph tan equals to k, where k is the spring constant Q What is the unit and dimensional formula of force constant? Solution : The unit of force constant is N/m and dimensional formula is [MT ] Q A spring of force constant or spring constant k cuts into two equal parts Find the spring constant of each part Solution : The force constant of spring is inversly proportional to length of spring If it cuts into two equal parts then spring constant of each part is k Q What is static friction or static frictional force? Also define the limiting static friction? Write down the law of limiting static friction Also define the coefficient of limiting static friction Solution : Static friction opposes impending motion The term impending motion means relative motion that would take place under the applied force if the surface is frictionless, but does not actually take place ie, there is a tendency of relative motion of point of contact between two surfaces As the applied force exceeds a certain limit, the body begins to move The value of static friction just before the begining of relative motion of point of contact is known as limiting static friction Law of limiting static friction : (i) It is found experimentally that the limiting value of static friction (f s ) max is independent of the area of contact (ii) It varies with the normal force(n) approximately as : (f s ) max = µ s N, where µ s is a constant of proportionality depending only on the nature of the surface in contact The constant µ s is called the coefficient of static friction The law of static friction may thus be written as f s µ s N

5 PLOM 5 Q What is kinetic or sliding friction? Write down the law of kinetic friction Also define the coefficient of kinetic friction Solution : If the applied force F exceeds limiting static friction (f s ) max the body begins to slide on the surface It is found experimentally that when relative motion has started, the frictional force decreases from the static maximum value (f s ) max Frictional force that opposes relative motion between surfaces in contact is called kinetic or sliding friction and is denoted by f k Law of kinetic friction : (i) Kinetic friction, like static friction, is found to be independent of the area of contact (ii) Further, it is nearly independent of the velocity (iii) It is proportional to normal reaction ( N ) f µ N, where µk is the coefficient of kinetic friction Q Does the static and kinetic friction depend on area of contact? Solution : The static and kinetic friction does not depend on area of contact Q Does the kinetic friction depend on the velocity of body? Solution : The kinetic friction does not depend on the velocity of body Q Is the law of friction are fundamental laws? Solution : The law of friction are not fundamental laws like conservation of linear momentum, conservation of energy etc Q Which one is greater : µ s or µ k? Solution : µ s is greater than µ k Q Does coefficient of friction depend on nature of surfaces in contact? Solution : The coefficient of friction µ s and µ k depends on nature of surfaces in contact Q Explain angle of friction Relate angle of friction and coefficient of friction Solution : The angle of friction between any two surfaces in contact is defined as the angle which the net contact force (P) makes with normal reaction R The net contact force is the resultant of static friction and normal reaction R Consider a body on which limiting static friction and normal force is shown in the figure : k k Here is the angle of friction between the two bodies in contact The value of angle of friction depends on the nature of materials of the surfaces in contact and the nature of the surfaces fs Relation between µ and : tan µ s ( f R s = µ s R) Hence µ s = tan, ie, coefficient of limiting friction between any two surfaces in contact is equal to tangent of the angle of friction between them Q Draw the graph for the variation of force of friction with applied force? Solution : Q What is angle of repose? Prove that angle of repose equals to angle of friction? Solution : Angle of repose or angle of sliding is defined as the minimum angle of inclination of a plane with the horizontal, such that a body placed on the plane just begins to slide down

6 PLOM 6 The value depends on material and nature of the surfaces in contact In figure, AB is an inclined plane such that a body placed on it just begins to slide down BAC = = angle of repose The various forces acting on the body is as shown in the figure (also known as force body diagram or free body diagram of the body) In equilibrium, f s = mg sin, R = mg cos Dividing both, we get f s R mg sin, ie, µ mgcos s = tan ( f s = µ s R) Also µ s = tan, where is the angle of friction between two surfaces Hence we obtain, tan = tan = Q What is rolling friction? Solution : The frictional force that comes into play when one body is actually rolling over the surface of the other body is called rolling friction The frictional force acting on a disc, sphere or any type of rounded body during rolling motion over a frictional surface is the rolling frictional force Q Explain : Friction is a necessary evil Solution : Friction is called a necessary evil It is a necessity because we cannot do work without it At the same time, it is an evil because it involves unnecessary waste of energy The following facts make the point clear : (a) Friction is a Necessity (Advantages of friction) : (i) Walking will not be possible without friction (ii) No two bodies will stick to each other if there is no friction, brakes of the vehicles will not work without friction and many other mechanical phenomena cannot work without friction (b) Friction is an evil (Disadvantages of friction) (i) Friction always opposes the relative motion between any two bodies in contact Therefore, extra energy has to be spent in overcoming friction Thus friction involves unnecessary expense of energy (ii) Friction causes wear and tear of the parts of machinery in contact Thus their life time reduces Hence we conclude that friction is a necessary evil Q Write down some ways of reducing friction Solution : Some ways of reducing friction are (a) Ball bearing placed between moving parts of a machine (b) Compressed cushion of air between surfaces in relative motion Q A person starts his motion on a frictional ground towards east What is the direction of frictional force acting on the person? Solution : The direction of friction force is directed towards east Q Why is it easier to drive a bicycle when its tyres are fully inflated? Solution : When the tyres of a vehicle is fully inflated then rolling friction reduces Therefore it is easier to drive a bicycle when its tyres are fully inflated Q Comment on the following statement : Static friction is a self adjusting force Solution : Static friction adjusts itself so that its magnitude is equal to the magnitude of the applied force and its direction is opposite to that of the applied force Hence static friction is a self adjusting force Q Determine the maximum acceleration of the train in which a box lying on its floor will remain stationary, given that the co-effecient of static friction between the box and the train s floor is 015 [NCERT Solved Example 57] Solution : 15 m s

7 PLOM 7 Q A mass of 4 kg rests on a horizontal plane The plane is gradually inclined until at an angle = 15 0 with the horizontal, the mass just begins to slide What is the coefficient of static friction between the block and the surface? [NCERT Solved Example 58] Solution : 07 Q What is the acceleration of the block and trolley system shown in the figure, if the coefficient of kinetic friction between the trolley and the surface is 004? What is the tension in the string? (Take g = 10 m s ) Neglect the mass of the string [NCERT Solved Example 59] Solution : 096 m/s, 71 N 510 Circular Motion : Q A stone of mass m is rotated with speed v in a circle by a string of length l Which force will provide the centripetal force? What is the value of this force? mv Solution : Tension in the string will provide centripetal force which equals to l Q Which force will provide the centripetal force for the motion of planet around the sun? Solution : Gravitational force between planet and the sun will provide centripetal force Q Find the maximum speed of a car on a level circular road of radius R? The coefficient of static friction between the tyre of car and road is µ s Solution : Consider a car which is moving with speed v on a circular level road, as shown in figure : Three forces act on the car are : (i) The weight of the car, mg (ii) Normal reaction, N (iii) Frictional force, f These force are shown in the figure : As there is no acceleration in the vertical direction, N mg = 0 N = mg The centripetal force rquired for circular motion is along the surface of the road is provided by static frictional force Static friction opposes the impending motion of the car moving away from the circle mv Hence, f R mv f µ sn µ R s mg v µ sgr v µ sgr

8 PLOM 8 which is independent of the mass of the car This shows that for a given value of µ s and R, there is a maximum speed of circular motion of the car possible, namely Q What is banking of road? Why the road is banked? vmax µ srg Solution : By raising outer edge of the curved road above the inner edge, we can reduce the contribution of friction to the circular motion of the car This phenomena is known as banking of road By doing so, a component of normal reaction of the road will provide the centripetal force Q Find the expression of the maximum speed of a car on a banked circular road of radius R? The angle of banking is and coefficient of static friction is µ s Solution : The force body diagram of a car on a bank circular road is shown in figure : Since there is no acceleration along the vertical direction, the net force along this direction muse be zero Hence, N cos = mg + f sin (i) mv N sin + f cos = R But f µ s N Thus to obtain v max we put f = µ s N Then eqs (i) and (ii) become N cos = mg + µ s N sin N sin + µ s N cos = mv /R (ii) (iii) (iv) Solving equation (iii), we obtain mg N cos µ s sin Substituting value of N in eq (iv), we get mg(sin µ s cos ) mv cos µ sin R s max or v max 1 µ s tan Rg 1 µ s tan Q Write true or false : The maximum possible speed of a car on a banked road is greater than that on a flat road Solution : True Q Find the expression of the speed of a car on a frictionless banked circular road of radius R? The angle of banking is Prove that OR v tan, where symbols have their usual meaning gr Solution : The FBD of a car moving on a frictionless banked circular road is shown in the figure : As there is no vertical acceleration, hence N cos = mg and N sin will provide centripetal force ie, N sin = mv /R

9 PLOM 9 Dividing Nsin Ncos mv R mg v tan v gr tan gr Q Why does a cyclist bend inwards from vertical while moving on a curved road or taking a turn? Also find this angle with the vertical if the cyclist has the speed v and radius of curved road is R Solution : When a cyclist takes a turn, a centripetal force will be required which can be provided by frictional force between the tyre and road and normal reaction by the road on the tyre By bending inwards the cyclist reduces the contribution of frictional force The free body diagram of cyclist is as shown in figure : As there is no vertical acceleration, hence N cos = mg and N sin will provide centripetal force ie, N sin = mv /R Dividing Nsin Ncos mv R mg v tan v gr tan gr Q A cyclist speeding at 18 km/h on a level road takes a sharp circular turn of radius 3 m without reducing the speed The co-efficient of static friction between the tyres and the road is 01 Will the cyclist slip while taking the turn? [NCERT Solved Example 510] Solution : The cyclist will slip while taking the circular turn Q A circular racetrack of radius 300 m is banked at an angle of 15 0 If the coefficient of friction between the wheels of a racecar and the road is 0, what is the (a) optimum speed of the racecar to avoid wear and tear on its tyres, and the (b) maximum permissible speed to avoid slipping? [NCERT Solved Example 511] Solution : (a) 81 m/s (b) 381 m/s 511 Solving Problems in Mechanics : Q A wooden block of mass kg rests on a soft horizontal floor When an iron cylinder of mass 5 kg is placed on top of the block, the floor yields steadily and the block and the cylinder together go down with an acceleration of 01 m s What is the action of the block on the floor (a) before and (b) after the floor yields? Take g = 10 m s Identify the action-reaction pairs in the problem [NCERT Solved Example 51] Solution : (a) 0 N (b) 673

10 NCERT EXERCISE 51 Give the magnitude and direction of the net force acting on (a) (b) (c) (d) a drop of rain falling down with a constant speed, a cork of mass 10 g floating on water, a kite skillfully held stationary in the sky, a car moving with a constant velocity of 30 km/h on a rough road, PLOM 10 (e) a high-speed electron in space far from all gravitating objects, and free of electric and magnetic fields 5 A pebble of mass 005 kg is thrown vertically upwards Give the direction and magnitude of the net force on the pebble (a) (b) (c) during its upward motion, during its downward motion, at the highest point where it is momentarily at rest Do your answers change if the pebble was thrown at an angle of say 45 0 with the horizontal direction? Ignore air resistance 53 Give the magnitude and direction of the net force acting on a stone of mass 01 kg, (a) (b) just after it is dropped from the window of a stationary train, just after it is dropped from the window of a train running at a constant velocity of 36 km/h (c) just after it is dropped from the window of a train accelerating with 1 m s, (d) lying on the floor of a train which is accelerating with 1 m s, the stone being at rest relative to the train Neglect air resistance throughout 54 One end of a string of length l is connected to a particle of mass m and the other to a small peg on a smooth horizontal table If the particle moves in a circle with speed v the net force on the particle mv mv (directed towards the centre) is : (i) T, (ii) T (iii) T (iv) 0 l l 55 A constant retarding force of 50 N is applied to a body of mass 0 kg moving initially with a speed of 15 m s 1 How long does the body take to stop? 56 A constant force acting on a body of mass 30 kg changes its speed from 0 m s 1 to 35 m s 1 in 5 s The direction of the motion of the body remains unchanged What is the magnitude and direction of the force? 57 A body of mass 5 kg is acted upon by two perpendicular forces 8 N and 6 N Give the magnitude and direction of the acceleration of the body 58 The driver of a three-wheeler moving with a speed of 36 km/h sees a child standing in the middle of the road and brings his vehicle to rest in 40 s just in time to save the child What is the average retarding force on the vehicle? The mass of the three-wheeler is 400 kg and the mass of the driver is 65 kg 59 A rocket with a lift-off mass 0,000 kg is blasted upwards with an initial acceleration of 50 m s Calculate the initial thrust (force) of the blast 510 A particle of mass 040 kg moving initially with a constant speed of 10 m s 1 to the north is subjected to a constant force of 80 N directed towards the south for 30 s Take the instant the force is applied to be t = 0, the position of the particle at that time to be x = 0, and predict its position at t = 5 s, 5 s, 100 s 511 A truck starts from rest and accelerating uniformly with 0 m s At t = 10 s, a stone is dropped by a person standing on the top of the truck (6 m high from the ground) What are the (a) velocity, and (b) acceleration of the stone at t = 11 s? (Neglect air resistance)

11 PLOM A bob of mass 01 kg hung from the ceiling of a room by a string m long is set into oscillation The speed of the bob at its mean position is 1 m s 1 What is the trajectory of the bob if the string is cut when the bob is (a) at one of its extreme positions, (b) at its mean position 513 A man of mass 70 kg stands on a weighing scale in a lift which is moving (a) upwards with a uniform speed of 10 m s 1 (b) downwards with a uniform acceleration of 5 m s, (c) upwards with a uniform acceleration of 5 m s, (d) What would be the readings on the scale in each case? What would be the reading if the lift mechanism failed and it hurtled down freely under gravity? 514 Figure shows the position-time graph of a particle of mass 4 kg What is the (a) force on the particle for t < 0, t > 4 s, 0 < t < 4 s? (b) impulse at t = 0 and t = 4 s? (Consider one-dimensional motion only) 515 Two bodies of masses 10 kg and 0 kg respectively kept on a smooth, horizontal surface are tied to the ends of a light string a horizonatal force F = 600 N is applied to (i) A, (ii) B along the direction of string What is the tension in the string in each case? 516 Two masses 8 kg and 1 kg are connected at the two ends of a light inextensible string that goes over a frictionless pulley Find the acceleration of the masses, and the tension in the string when the masses are released 517 A nucleus is at rest in the laboratory frame of reference Show that if it disintegrates into two smaller nuclei the products must be emitted in opposite directions 518 Two billiard balls each of mass 005 kg moving in opposite directions with speed 6 m s 1 collide and rebound with the same speed What is the impulse imparted to each ball due to the other 519 A shell of mass 000 kg is fired by a gun of mass 100 kg If the muzzle speed of the shell is 80 m s 1, what is the recoil speed of the gun? 50 A batsman deflects a ball by an angle of 45 0 without changing its initial speed which is equal to 54 km/h What is the impulse imparted to the ball? (Mass of the ball is 015 kg) 51 A stone of mass 05 kg tied to the end of a string in a horizontal plane is whirled around in a circle of radius 1 m with a speed of 40 rev/min What is the tension in the string? What is the maximum speed with which the stone can be whirled around if the string can withstand the maximum tension of 00 N? 5 If, in Exercise 51, the speed of the stone is increased beyond the maximum permissible value, and the string breaks suddenly, which of the following correctly described the trajectory of the stone after the string breaks : (a) the stone moves radially outwards, (b) the stone flies off tangentially from the instant the string breaks, (c) the stone flies off at an angle with the tangent whose magnitude depends on the speed of the particle? 53 Explain why (a) a horse cannot pull a cart and run in empty space (b) passengers are thrown forward from their seats when a speeding bus stops suddenly (c) it is easier to pull a lawn mower than to push it (d) a cricketer moves his hands backwards while holding a catch

12 PLOM 1 54 Figure shows the position-time graph of a particle of mass 004 kg Suggest a suitable physical context for this motion What is the time between two consecutive impulses received by the particle? What is the magnitude of each impulse? 55 Figure shows a man standing stationary with respect to a horizontal conveyor belt that is accelerating with 1 m s What is the net force on the man? If the coefficient of static friction between the man s shoes and the belt is 0, up to what acceleration of the belt can the man continue to be stationary relative to the belt? (Mass of the man = 65 kg) 56 A stone of mass m tied to the end of a string revolves in a vertical circle of radius R The net forces at the lowest and highest points of the circle directed vertically downwards are : [Choose the correct alternative] Lowest Point Highest point (a) mg T 1 mg + T (b) mg + T 1 mg T (c) mg + T 1 (mv 1 )/R mg T + (mv 1 )/R (d) mg T 1 (mv 1 )/R mg + T + (mv 1 )/R 57 A helicopter of mass 1000 kg rises with a vertical acceleration of 15 m s The crew and the passengers weigh 300 kg Give the magnitude and direction of the (a) (b) (c) force on the floor by the crew and passengers, action of the rotor of the helicopter on the surrounding air, force on the helicopter due to the surrounding air 58 A steam of water flowing horizontally with a speed of 15 m s 1 gushes out of a tube of cross-sectional area 10 m, and hits a vertical wall nearby What is the force exerted on the wall by the impact of water, assuming it does not rebound? 59 Ten one-rupee coins are put on top of each other on a table Each coin has a mass m kg Give the magnitude and direction of (a) (b) (c) the force on the 7th coin (counted from the bottom) due to all the coins on its top, the force on the 7th coin by the eight coin, the reaction of the 6th coin on the 7th coin 530 An aircraft executes a horizontal loop at a speed of 70 km/h with its wings banked at 15 0 What is the radius of the loop? 531 A train rounds an unbanked circular bend of radius 30 m at a speed of 54 km/h The mass of the train is 10 6 kg What provides the centripetal force required for this purpose? The engine or the rails? The outer or the inner rails? Which rail will wear out faster, the outer or the inner rail? What is the angle of banking required to prevent wearing out the rails?

13 53 A block of mass 5 kg is raised by a 50 kg man in two different ways as shown in figure PLOM 13 What is the action on the floor by the man in the two cases? If the floor yields to a normal force of 700 N, which mode should the man adopt to lift the block without the floor yielding? 533 A monkey of mass 40 kg climbs on a rope in figure, which can stand a maximum tension of 600 N In which of the following cases will the rope break if the monkey (a) climbs up with an acceleration of 6 m s (b) climbs down with an acceleration of 4 m s (c) climbs up with a uniform speed of 5 m s 1 (d) falls down the rope nearly freely under gravity? (Ignore the mass of the rope) 534 Two bodies A and B of masses 5 kg and 10 kg in contact with each other rest on a table against a rigid partition The co-efficient of friction between the bodies and the table is 015 A force of 00 N is applied horizontally at A What are (a) the reaction of the partition (b) the action-reaction forces between A and B? What happens when the partition is removed? Does the answer to (b) change, when the bodies are in motion? Ignore the difference between µ s and µ k 535 A block of mass 15 kg is placed on a long trolley The co-efficient of static friction between the block and the trolley is 018 The trolley accelerates from rest with 05 m s for 0 s and then moves with uniform velocity Discuss the motion of the block as viewed by (a) a stationary observer on the ground (b) an observer moving with the trolley 536 The rear side of a truck is open and a box of 40 kg mass is placed 5 m away from the open end The co-efficient of friction between the box and the surface below it is 015 On a straight road, the truck starts from rest and accelerates with m s At what distance from the starting point does the box fall off the truck? (Ignore the size of the box)

14 PLOM A long playing record revolves with a speed of 33 rev/min and has a radius of 15 cm Two coins 3 are placed at 4 cm and 14 cm away from the centre of the record If the co-efficient of friction between the coins and the record is 015, which of the two coins will revolve with the record? 538 You may have seen in a circus a motorcyclist driving in vertical loops inside a death-well (a hollow spherical chamber with holes, so the spectators can watch from outside) Explain clearly why the motorcyclist does not drop down when he is at the uppermost point, with no support from below What is the minimum speed required at the uppermost position to perform a vertical loop if the radius of the chamber is 5 m? 539 A 70 kg man stands in contact against the inner wall of a hollow cylindrical drum of radius 3 m rotating about its vertical axis with 00 rev/min The co-efficient of friction between the wall and his clothing is 015 What is the minimum rotational speed of the cylinder to enable the man to remain stuck to the wall (without falling) when the floor is suddenly removed? 540 A thin circular wire of radius R rotates about its vertical diameter with an angular frequency Show that a small bead on the wire remains at its lowermost point for g / R What is the angle made by the radius vector joining the centre to the bead with the vertical downwards direction for g / R? Neglect friction ANSWERS 51 (a) to (d) No net force according to the First Law (e) No force, since it is far awar from all material agencies producing electromagnetic and gravitational forces 5 The only force in each case is the force of gravity, (neglecting effects of air) equal to 05 N vertically downwards The answers do not change, even if the motion of the pebble is not along its vertical The pebble is not at rest at the highest point It has a constant horizontal component of velocity throughout its motion 53 (a) 1N vertically downwards (b) same as in (a) (c) same as in (a); force at an instant depends on the situation at that instant, not on history (d) 01 N in the direction of motion of the train 54 (i) T s N in the direction of motion 57 m/s at an angle of 37 0 with 8 N 58 1kN 59 F = N km 511 (a) 4 m s 1 at an angle of tan 1 (½) with the horizontal (b) 10 m s vertically downwards 51 (a) At the extreme position, the speed of the bob is zero If the string is cut, it will fall vertically downwards (b) At the mean position, the bob has a horizontal velocity If the string is cut, it will fall along a parabolic path 513 (a) = 70 kg (b) 35 kg (c) 105 kg (d) zero 514 (a) In all the three intervals, acceleration and, therefore, force are zero (b) 3 kg m s 1 at t = 0; (c) 3 kg m s 1 at t = 4 s 515 a = 0 m s, T = 00 N, a = 0 m s, T = 400 N 516 a = m s, T = 96 N 517 By momentum conservation principle, total final momentum is zero Two momentum vectors cannot sum to a null momentum unless they are equal and opposite 518 Impulse on each ball = = 06 kg m s 1 in magnitude The two impulses are opposite in direction cm s kg m s m s 1 5 Alternative (b) is correct, according to the First Law

15 PLOM (a) The horse-cart system has no external force is empty space The mutual forces between the horse and the cart cancel (Third Law) On the ground, the contact force between the system and the ground (friction) causes their motion from rest (b) Due to inertia of the body not directly in contact with the seat (c) A lawn mower is pulled or pushed by applying force at an angle When you push, the normal force (N) must be more than its weight, for equilibrium in the vertical direction This results in greater friction f(f N) and, therefore, a greater applied force to move Just the opposite happens while pulling (d) To reduce the rate of change of momentum and hence to reduce the force necessary to stop the ball kg m s 1 after every s from the walls at x = 0 and x = cm 55 m s 56 (a) is correct 57 (a) N downwards (b) N downwards (c) N upwards N 59 (a) 3 m g (down) (b) 3 m g (down) (c) 4 m g (up) km N, 50 N; mode (b) should be adopted 533 (a) 640 N (b) 40 N (c) 400 N (d) 0, the rope will break in case (a) 534 (a) 00 N (b) 00 N, N; opposite to motion, N; in the direction of motion 535 (a) Maximum frictional force possible for opposing impending relative motion between the block and the trolley = = 7 N, which is more than the frictional force of = 75 N needed to accelerate the box with the trolley When the trolley moves with uniform velocity, there is no force of friction acting on the block (b) For the accelerated (non-inertial) observer, frictional force is opposed by the pseudo-force of the same magnitude, keeping the box at rest relative to the observer When the trolley moves with uniform velocity there is no pseudo-force for the moving (inertial) observer and no friction m cm from the centre m s s

16 ADDITIONAL QUESTIONS AND PROBLEMS PLOM 16 Q Derive an expression for (i) acceleration of a body down a rough inclined plane (ii) acceleration of a body up a rough inclined plane The coefficient of kinetic friction is µ k and angle of inclination is A (i) g sin µ k g cos (down the incline) (ii) g sin + µ k g cos (down the incline) Q A lift is accelerated upward Will the apparent weight of a person inside the lift increase, decrease or remain the same relative to its real weight? If the lift is going with uniform speed, then what is the appearent weight A The apparent weight will increase If the lift is going with uniform speed then the aparent weight equals to real weight Q The distance travelled by a moving body is directly proportional to time Is any external force acting on it? A The external force is zero Q A person sitting in the compartment of a train moving with uniform speed throws a ball in the upward direction What path of the ball will appear to him? What to a person standing outside? A The path of the ball will be straight line wrt the person in the comparement The path of the ball will be parabolic wrt the person standing outside Q If force is acting on a moving body perpendicular to the direction of motion, then what will be its effect on the speed and direction of the body? A The speed will be constant but the direction of the body will change Q A thief jumps from the roof of a house with a box of weight W on his head What will be the weight of the box as experienced by the thief during jump? A Zero Q A bullet fired from a gun is more dangerous than an air molecule hitting a person, though both of them have almost the same speed Why? A Mass of a bullet is much greater than the mass of an air molecule Therefore, even when both have almost the same speed, linear momentum of bullet is much larger than the linear momentum of an air molecule Hence the force required to stop a bullet will be much bigger than the force required stop an air molecule That is why a bullet fired from a gun is more dangerous than an air molecule Q Rocket can move in air free space, but jet planes cannot Why? A A rocket is self contained, as both the fuel and the oxygen needed to burn the fuel are available inside the rocket Therefore, a rocket can move in air free space A jet plane has fuel only It needs to take oxygen from the atmosphere to burn the fuel That is why a jet plane cannot operate in air free space Q The distance travelled by a moving body is directly proportional to time Is any external force acting on it? A When s t, acceleration = 0 Therefore, no external force is acting on the body Q The length of an ideal spring increases by 01 cm when a body of 1 kg is suspended from it If this spring is laid on a frictionless horizontal table and bodies of 1 kg each are suspended from its ends, then what will be the increase in its length? A Increase in length will be 01 cm only Q The two ends of a spring-balance are pulled each by a force of 10 kg wt What will be the reading of the balance? A The reading of the balance will be 10 kg wt Q A retarding force is applied to stop a motor car If the speed of the motor car is doubled, how much more distance will it cover before stopping under the same retarding force? A As s v, therefore, motor car will cover a distance four times longer than before Q A ball of 05 kg mass moving with a speed of 10 m/s rebounds after striking normally a perfectly elastic wall Find the change in momentum of the ball A Change in linear momentum = mv mv = mv = = 10 kg ms 1

17 PLOM 17 Q A body of kg is suspended on a spring balance hung vertically in a lift If the lift is falling downward under acceleration due to gravity g, then what will be the reading of the balance? If going upward with the same acceleration, then? A Reading of the balance = apparent weight W = m (g a) = m (g g) = zero If lift is going upwards with same acc ; W = m (g + a) = m (g + g) = m g = 4 kg Q A 5-kg body is suspended from a spring-balance, and an identical body is balanced on a pan of a physical balance If both the balances are kept in an elevator, then what would happen in each case when the elevator is moving with an upward acceleration? A The reading of the spring balance will increase, but there will be no effect on the equilibrium of common physical balance Q A body is dropped from the ceiling of a transparent cabin falling freely towards the earth Describe the motion of the body as observed by an observer (a) sitting in the cabin, (b) standing on earth A (a) The body will appear stationary in air (b) The body will appear to be falling freely under acceleration due to gravity Q Action and reaction forces do not balance each other Why? A This is because force of action and reaction act always on two different bodies Q What is the principle of working of a rocket? A A rocket works on the principle of conservation of linear momentum Q Why does a gun recoil when a bullet is fired? A The recoiling of a gun is accounted for by the principle of conservation of linear momentum The total linear momentum of the gun and the bullet on firing is equal to their total linear momentum before firing, which is zero (as both are at rest initially) Q A soda water bottle is falling freely Will the bubbles of the gas rise in the water of the bottle? A Bubbles will not rise in water This is because water in freely falling bottle is in the state of weightlessness No upthrust acts on the bubbles Q A bird is sitting on the floor of a closed glass cage and the cage is in the hand of a girl Will the girl experience any change in the weight of the cage when the bird (i) starts flying in the cage with a constant velocity (ii) flies upwards with acceleration (iii) flies downwards with acceleration? A In a closed glass cage, air inside is bound with the cage Therefore, (i) there would be no change in weight of the cage if the bird flies with a constant velocity (ii) the cage becomes heavier, when bird flies upwards with an acceleration (iii) the cage appears lighter, when bird flies downwards with an acceleration Q When a ball is thrown upwards, its momentum first decreases and then increases Is conservation of linear momentum violated in this process? A No, the momentum conservation principle applies only when no external forces act on the system Q When a wheel is rolling on a level road, what is the direction of frictional force between the wheel and the road? A As the wheel is moving forward, the portion of the wheel in contact with the road moves backwards Hence the force of friction must be acting in the forward direction, along the tangent to the surface of the road and the wheel in contact Q A stone tied to one end of a string is whirled in a circle If the string breaks, the stone flies off tangentially Why? A The velocity of the stone at any instant is along the tangent to the circle at that instant When the string breaks, centripetal force whirling the stone vanishes Therefore, on account of inertia, the stone flies off tangentially Q Why does a child in a merry-go-round press the side of his seat radially outward? A This is because of centrifugal force acting radially outwards on the child rotating actually in the mettry go round

18 Q A bucket containing water is rotated in a vertical circle Explain why water does not fall PLOM 18 A This is because weight of water in the bucket is spent in supplying the necessary centripetal force Q Why does a pilot not fall down, when his aeroplane loops a vertical loop? A This is because at the highest point of the (vertical) loop, weight of the pilot is spent in providing the necessary centripetal force Q A block is gently placed at the top of an inclined plane 64 m long Find the time taken by the block to slide down to the bottom of the plane The plane makes an angle 30 0 with the horizontal Coefficient of friction between the block and the plane is 0 Take g = 10 m/s A 198 s Q How does a lubricant help in reducing friction? A When a lubricant is added to a machine, it spreads between the two surfaces rubbing each other, fills the irregularities present on the surface and forms a thin layer between the surfaces in contact As a result of it, the contact between the two hard surfaces is replaced by the contact between the hard surface and lubricant layer Due to it, the force of friction is reduced considerably Q What are the factors on which coefficient of friction depends? A Coefficient of friction between any two surfaces in contact depends on (i) material of the surfaces (ii) nature of the surfaces Q Rubber tyres are preferred to steel tyres Why? A This is because coefficient of friction between rubber tyres and road is smaller than the coefficient of friction between steel tyres and road Q Is friction a self adjusting force? A No, all types of friction are not self adjusting Only static friction is a self adjusting force Q Why are wheels of an automobile made circular? A Circular wheels roll on the road Therefore, motion of the vehicle is opposed by rolling friction which is much smaller than the sliding friction Q Automobile tyres are generally provided with irregular projections over their surfaces, why? A Irregular projections over the surface of automobile tyres increase the force of friction between the tyres and the road