Chapter-4-DYNAMICS-01-THEORY-PART-I

4.1 Introduction. Dynamics is that branch of mechanics which deals with the study of laws governing motions of material system under the action of given forces. (1) Displacement : The displacement of a moving point is its change of position. To know the displacement of a moving point, we must know both the length and the direction of the line joining the two positions of the moving point. Hence the displacement of a point involves both magnitude and direction i.e., it is a vector quantity. (2) Speed : The speed of a moving point is the rate at which it describes its path. The speed of a moving particle or a body does not give us any idea of its direction of motion; so it is a scalar quantity. In M.K.S. or S.I. system, the unit of speed is metre per second (m/s). Average speed : The average speed of moving particle in a time-interval is defined as the distance travelled by the particle divided by the time interval. If a particle travels a distance s in time-interval t, then Average speed = Note :  The average speed of a particle gives the overall “rapidity” with which the particle moves in the given interval of time. Instantaneous speed : The instantaneous speed or simply speed of a moving particle is defined as the rate of change of distance along its path, straight or curved. If s is the distance travelled by a particle along its path, straight or curved, in time t, then Instantaneous speed = or speed at time Note :  The average speed is defined for a time interval and the instantaneous speed is defined at a particular instant. 4.2 Velocity and Acceleration. (1) Velocity : The velocity of a moving point is the rate of its displacement. It is a vector quantity. Let a particle starting from the fixed point O and moving along the straight line OX describes the distance OP = x in time t. If the particle describes a further distance in time , then is called the average velocity of the particle in time . And is called the instantaneous velocity (or simply velocity) of the particle at time t. Note :  Average velocity in time t is the mean of the initial and final velocity. (2) Acceleration : The rate of change of velocity of a moving particle is called its acceleration. It is a vector quantity. The acceleration of a moving point at time t at distance x is given by, In M.K.S. or S.I. system, the unit of acceleration is m/sec2. It should be noted that if v increases with time and if v decreases with time. A negative acceleration is called retardation. Clearly, retardation means decrease in the velocity. Example: 1 The average speed of a bicycle over a journey of 20 km; if it travels the first 10 km. at 15 km/hr and the second 10 km. at 10 km/hr, is (a) 12 km/hr (b) 10 km/hr (c) 15 km/hr (d) None of these Solution: (a) Time taken by the bicycle to travel the first 10 km = hour = hour Time taken by the bicycle to travel the second 10 km = hour = 1 hour Total distance travelled = 10 + 10 = 20 km Total time taken = hour. Average speed of bicycle = = km/hr. Example: 2 If a particle moves in a straight line according to the formula, , then the time interval during which the velocity is negative and acceleration is positive, is (a) [0, 5] (b) (c) (2, 5) (d) None of these Solution: (c) We have and Now and  and  and  . Example: 3 A particle moves in a fixed straight path so that . If v is the velocity at any time t, then its acceleration is (a) (b) (c) (d) Solution: (a) We have    . Example: 4 Two particles are moving with uniform velocities u and v respectively along X and Y axes, each directed towards the origin. If the particles are at distances a and b from the origin, the time at which they will be nearest to each other will be equal to (a) (b) (c) (d) Solution: (c) Let the two particles be at A and B respectively at and at time t their positions be P and Q respectively. Then, and and Now,   and For maximum or minimum value of , we must have   Clearly, for all t. Hence, is least at . 4.3 Resultant and Components of Velocities. (1) Resultant of velocities : The resultant of two velocities possessed by a particle is given by the law of parallelogram of velocities. Parallelogram law of velocities : If a moving particle has two simultaneous velocities represented in magnitude and direction by the two sides of a parallelogram drawn from an angular point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point. Let a particle at O has simultaneously two velocities u and v represented in magnitude and direction by OA and OB respectively of the parallelogram OACB. The diagonal OC represents the resultant velocity. Let R be the resultant velocity. If and , then the magnitude of their resultant velocity is and, the direction of this resultant velocity makes an angle  with the direction of u such that Case I : When , then Case II : When , then Case III : When , i.e., u and v are at right angle to each other, then and . Case IV : When , then and . Note :  The angle made by the direction of the resultant velocity with the direction of v is given by .  If the direction of resultant velocity R makes an angle with the direction of , then and (2) Components of a velocity in two given directions : If components of a velocity V in two given directions making angles and with it, then by the law of parallelogram of velocities the required components of V are represented by AB and AC = BD along AX and AY respectively. i.e., and Hence, the components of velocity V in the directions making angle and are and respectively. Important Tips  If components of velocity V are mutually perpendicular, then . Thus components of V are along AX and along AY. Example: 5 Drops of water falling from the roof of the tunnel seems from the window of the train to be falling from an angle making from the horizontal. It is known that their velocities are 24 decimeter/sec. Assuming the resistance of air negligible, what will be the velocity of the train (a) 42 decimeter/sec. (b) 48 decimeter/sec. (c) 45 decimeter/sec. (d) 44 decimeter/sec. Solution: (b) Let the velocity of train is v decimeter/sec. Let the real velocity of drops in magnitude and direction may be represented by OB and the velocity of train in opposite direction is represented by OD, then , Completing parallelogram OBCD, OC will represent the relative velocity of drops. If the drops are seen in a direction making an angle with the horizontal, i.e., , then or or In , from sine formula, or or decimeter/second. Example: 6 If u and v be the components of the resultant velocity w of a particle such that , then the angle between the velocities is (a) (b) (c) (d) Solution: (a) Let be the angle between the components of the resultant velocity w. Then     . Example: 7 A man swims at a speed of 5 km/hr. He wants to cross a canal 120 metres wide, in a direction perpendicular to the direction of flow. If the canal flows at 4 km/hr, the direction and the time taken by the man to cross the canal are (a) (b) (c) (d) None of these Solution: (b) Suppose the man swims in a direction making an angle with the direction of current. Since the man wants to cross the canal in a direction perpendicular to the direction of flow. Therefore,      Let V be the resultant velocity, then  Time taken by the man to cross the canal = 4.4 Relative Velocity. Let A and B are two bodies in motion. The velocity of A relative to B is the velocity with which A appears to move as viewed by B i.e., velocity of A relative to B = Resultant of velocity of A and reversed velocity of B. Apparent velocity of A as seen from B means velocity of A relative to B. The relative velocity of A with respect to B is also called the velocity of A relative to B and is denoted by and . If two bodies A and B are moving with velocities of magnitudes and respectively inclined at an angle with each other, then If the direction of the relative velocity of A with respect to B makes an angle with the direction of velocity of B, then Case I : When the particles A and B move parallel to each other in the same direction with velocities u and v respectively. Here , then in the direction of A in the direction of B Case II : When the particles A and B move parallel to each other in opposite directions with velocities u and v respectively. Here , then in the direction of A. in the direction of B. Note :  True velocity of A = Resultant of the relative velocity of A with respect to B and the true velocity of B i.e., [Since ] Example: 8 A train of length 200 m travelling at 30 m/sec. overtakes another of length 300 m travelling at 20 m/sec. The time taken by the first train to pass the second is (a) 30 sec. (b) 50 sec. (c) 10 sec. (d) 40 sec. Solution: (b) Distance covered by the first train to pass the second = (200 + 300)m = 500 metres. Since both the trains are travelling in the same direction. Velocity of first train relative to second = (30 – 20) = 10 m/sec. Time taken by the first train to pass the second = second. Example: 9 To a man running at a speed of 20 km/hr, the rain drops appear to be falling at an angle of from the vertical. If the rain drops are actually falling vertically downwards, their velocity in km/hr. is (a) (b) 10 (c) (d) 40 Solution: (c) Velocity of rain relative to man = Actual velocity of rain – Velocity of man Resolving horizontally and vertically, and = 20 i.e., ,  . Example: 10 A man is walking at the rate of 3 km/hr and the rain appears to him falling vertically with a velocity of . If the actual direction of the rain makes an angle with vertical, then = (a) (b) (c) (d) Solution: (d) Let denotes the velocity of man. Then, It appears to the man that rain is falling vertically at the rate of . This means that the relative velocity of rain with respect to man is , in a direction perpendicular to the velocity of the man. Let denote the relative velocity of rain with respect to man and be the velocity of rain. Let , . Complete the parallelogram OACB. Then, the diagonal represents true velocity of the rain. [Using  Let be the angle between the direction of velocity of man and velocity of rain. Then,  Thus, the actual velocity of the rain is 6 km/hr in a direction making an angle of with the direction of motion of the man. Example: 11 The particles start simultaneously from the same point and move along two straight lines, one with uniform velocity and the other from rest with uniform acceleration . Let be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time (a) (b) (c) (d) Solution: (d) After t, velocity = For max. and min., 4.5 Rectilinear motion with uniform Acceleration. A point or a particle moves in a straight line, starting with initial velocity u, and moving with constant acceleration f in its direction of motion. If v be its final velocity at the end of time t and s be its distance at the instant, from its starting point, then the equations describing the motion of the particle are, (i) (ii) (iii) (iv) If is the distance travelled by the particle in nth second, then Important Tips  If a particle moves in a straight line with initial velocity u m/sec. and constant acceleration f m/sec2, then distance travelled in t seconds is given by, = where = (Average velocity) × t {Average velocity = } Example: 12 A body is moving in a straight line with uniform acceleration. It covers distances of 10 m and 12 m in third and fourth seconds respectively. Then the initial velocity in m/sec. is (a) 2 (b) 3 (c) 4 (d) 5 Solution: (d) Let initial velocity is u m/sec. and acceleration is . So, or .....(i) and or ......(ii) Subtracting (i) from (ii), we get or Substituting value of f in equation (i) , or or Example: 13 Two trains A and B, 100 kms apart, are travelling to each other with starting speed of 50 km/hr for both. The train A is accelerating at 18 km/hr2 and B is decelerating at 18 m/h2. The distance where the engines cross each other from the initial position of A is (a) 50 kms (b) 68 kms (c) 32 kms (d) 59 kms Solution: (d) Let engine of train A travel x km and cross engine of train B in t hours. , Acceleration , so , .....(i) So distance travelled by engine of train B will be in t hour. Acceleration = –18 km/hr  …..(ii) Adding (i) and (ii), we get hour. From (i), . Example: 14 A particle is moving with a uniform acceleration. If during its motion, it moves x, y and z distance in pth, qth and rth seconds respectively, then (a) (b) (c) (d) Solution: (c) Let u be the initial velocity and f be the acceleration of the particle. Then , , Now, multiplying (i) by (ii) by , (iii) by and adding, we get . Example: 15 A body travels a distance s in t seconds. It starts from rest and ends at rest. In the first part of the journey, it moves with constant acceleration f and in the second part with constant retardation r. The value of t is given by (a) (b) (c) (d) Solution: (d) Portion OA, OB corresponds to motion with acceleration ‘f’ and retardation ‘r’ respectively. Area of and . Let and = and Also, and ;  . 4.6 Motion Under Gravity. When a body is let fall in vacuum towards the earth, it will move vertically downward with an acceleration which is always the same at the same place on the earth but which varies slightly from place to place. This acceleration is called acceleration due to gravity. Its value in M.K.S. system is in C.G.S. system and in F.P.S. system It is always denoted by g. (1) Downward motion: If a body is projected vertically downward from a point at a height h above the earth’s surface with velocity u, and after t second its velocity becomes v, the equation of its motion are , , , In particular, if the body starts from rest or is simply let fall or dropped, then, , , . (2) Upward motion : When a body be projected vertically upward from a point on the earth’s surface with an initial velocity u and if the direction of the upward motion is regarded as +ve, the direction of acceleration is –ve and it is, therefore, denoted by – g. The body thus moves with a retardation and its velocity gradually becomes lesser and lesser till it is zero. Thus, for upward motion, the equations of motion are, , , , (3) Important deductions (i) Greatest height attained : Let H be the greatest height. From the result, . We have, . Hence greatest height (ii) Time to reach the greatest height : Let T be the time taken by the particle to reach the greatest height. From the result, We have, i.e., Therefore time to reach the greatest height (iii) Time for a given height : Let t be the time taken by the body to reach at a given height h. Then, . Hence, and Clearly and are real; if . If , then , which is the time taken to reach the highest point. So, let . The lesser value of gives the time when the body is going up i.e., it is time from O to P and the larger value gives the total time taken by the body to reach the highest point and then coming back to the given point, i.e., it is the time from O to A and then A to P. (iv) Time of flight : It is the total time taken by the particle to reach the greatest height and then return to the starting point again. When the particle returns to the starting point, . Therefore, from the result, . We have, or t = 0 or t = 0 corresponds to the instant with the body starts, and t = 2 u/g corresponds to the time when the particle after attaining the greatest height reaches the starting point. Time of flight . Example: 16 A stone is dropped from a certain height which can reach the ground in 5 sec. If the stone is stopped after 3 seconds of its fall and then time taken by the stone to reach the ground for the remaining distance is (a) 2 seconds (b) 3 seconds (c) 4 seconds (d) None of these Solution: (c) Let distance travelled in 5 sec. is metre. or Let distance travelled in 3 seconds is metre. So, or So remaining distance or or . Let time taken by the stone to reach the ground for the remaining distance 8g is t second. or sec. Example: 17 A man in a balloon, rising vertically with an acceleration of 4.9 m/sec2 releases a ball 2 seconds after the balloon is let go from the ground. The greatest height above the ground reached by the ball is (a) 14.7 m (b) 19.6 m (c) 9.8 m (d) 24.5 m Solution: (a) Velocity after 2 seconds = (4.9)2 = 9.8 m/sec. Distance covered in 2 seconds = Again after the release of ball, velocity of the ball = 9.8 m/sec. ; Using , we get  metre. Hence greatest height attained = 9.8 + 4.9 = 14.7 m. Example: 18 A particle is dropped under gravity from rest from a height and then it travels a distance in the last second. The height h is (a) 100 metre (b) 122.5 metre (c) 145 metre (d) 167.5 metre Solution: (b) and     ; Since , rejecting this value. We have ; . 4.7 Laws of Motion. (1) Newton’s laws of motion (i) First law : Every body continues in its state of rest or of uniform motion in a straight line except when it is compelled by external impressed forces to change that state. (ii) Second law : The rate of change of momentum is proportional to the impressed force, and takes place in the direction of the straight line in which the force acts.  Force = mass  acceleration. The S.I. unit of force is Newton and C.G.S. unit of force is dyne. 1 Newton = 1 kg-m/sec2, 1 Dyne = 1 gm-cm/sec2 (iii) Third law : To every action there is an equal and opposite reaction, or the actions and reaction are always equal and opposite. Note :  1 Newton = 105 Dynes.  The action and reaction do not act together on the same body or the same part of the body. (2) Weight : The weight of a body is the force with which it is attracted by the earth towards its centre. For a body of mass m, the weight W is given by (By Newton's second law of motion) Note :  Let and be the weights of two bodies of masses and respectively at a place on the earth then, and  1 gm.wt. = g dynes = 981 dynes 1 kg. wt. = g Newtons = 9.81 N. (3) Momentum of a body : It is the quantity of motion in a body and is equal to the product of its mass (m) and velocity (v) with which it moves. Thus, momentum of the body is mv. The units of momentum are gm-cm/sec or kg-m/sec. Momentum = Mass × Velocity = m.v. (4) Impulse of a body : The impulse of a force in a given time is equal to the product of the force and the time during which it acts. The impulse of a force F acting for a time t is therefore F.t. Example: 19 A train weighing W tons is moving with an acceleration f ft/sec2. When a carriage of weight w tons is suddenly detached from it, then the change in the acceleration of the train is (a) (b) (c) (d) Solution: (c) Mass of train = W × 2240 lbs. Pull of the engine = W × 2240 × f Poundals When a carriage of mass W tons is detached, mass of the train = (W – w) tons = (W – w) . 2240 lbs Pull is the same as before i.e., W × 2240 × f Poundals. New acceleration = f1 = Change in acceleration = = = . Example: 20 A hockey stick pushes a ball at rest for 0.01 second with an average force of 50 N. If the ball weighs 0.2 kg, then the velocity of the ball just after being pushed is (a) 3.5 m/sec. (b) 2.5 m/sec. (c) 1.5 m/sec. (d) 4.5 m/sec. Solution: (b) If v m/sec. is the velocity of the ball just after being pushed, then 50 × 0.01 = 0.2 × v [ Impulse = change of momentum] v = 2.5 m/sec. Example: 21 A mass m is acted upon by a constant force P lb.wt., under which in t seconds it moves a distance of x feet and acquires a velocity v ft/sec. Then x is equal to (a) (b) (c) (d) Solution: (d) Force = P lb.wt.= Pg poundals and mass = m lbs Now initial velocity = 0, final velocity = v ft/sec. Distance = x and time = t Hence x = = . Example: 22 An engine and train weight 420 tons and the engine exerts a force of 7 tons. If the resistance to motion be 14 lbs. wt. per ton, then the time, the train will take to acquire a velocity of 30 m/hr. from rest is (a) 2.2 m (b) 2.6 m (c) 2.8 m (d) 3 m Solution: (a) Effective force = Pull of the engine – Resistance = 7 tons – 420 × 14 lbs = 7 × 2240 lbs – 5880 lbs = 9800 lbs = 9800 × 32 Poundals Also acceleration = Again initial velocity = 0 and final velocity = 30 m.p.h. = 44 ft/sec. Use we get second = minute; minute = 2.2 minute. 4.8 Motion of a Body released from a Balloon or a Lift. (1) When a lift is ascending with uniform acceleration of f m/sec2 and after t second a body is dropped from it, then at the time when the body is dropped: (i) Initial velocity of the body is same as that of the lift and is in the same direction. So, the velocity of the body is ft m/sec. (ii) Initial velocity of the body relative to the lift = Velocity of the body – Velocity of the lift = ft – ft = 0 (iii) Acceleration of the body = g m/sec2 in downward direction. (iv) Acceleration of the lift = f m/sec2 in upward direction. (v) Acceleration of the body relative to the lift = Acceleration of the body – Acceleration of the lift = g – (– f) = f + g in downward direction. (2) When a lift is ascending with uniform acceleration of and after t second a body is thrown vertically upward with velocity , then at that time, we have the following : (i) Initial velocity of the body = v + velocity of lift = v + ft, in upward direction (ii) Initial velocity of the body relative to the lift = Velocity of the body – Velocity of lift = (v + ft) – ft = v m/sec. (iii) Acceleration of the body relative to the lift in vertically downward direction is (f + g) m/sec2. (3) When a lift is descending with uniform acceleration and after time t a body is dropped from it. Then at that time, we have the following (i) Velocity of the body = Velocity of the lift = ft m/sec in downward direction (ii) Acceleration of the body relative to the lift in downward direction = Acceleration of the body – Acceleration of lift = 4.9 Apparent weight of a Body resting on a moving Horizontal plane or a Lift. Let a body of mass m be placed in a lift moving with an acceleration f and R is the normal reaction, then (1) When the lift is rising vertically upwards : Effective force in upward direction = Sum of the external forces in the same direction. Clearly, the pressure R exerted by the body on the plane is greater than the actual weight mg of the body. This pressure is also known as apparent weight. If a man of mass m is standing on a lift which is moving vertically upwards with an acceleration, then  R = (weight of the man)  R = Weight of the man + (weight of the man) Thus, the apparent weight of the man is times more than the actual weight. (2) When the lift is descending vertically downwards : Effective force in downward direction = Sum of the forces in the same direction.   ......(ii) Clearly, the pressure exerted by the body on the plane is less than its actual weight when the plane moves vertically downwards. If a man of mass m in is standing on a lift, which is moving vertically downward with an acceleration f, then the pressure =  R = weight of the man – (weight of the man) Thus, the apparent weight of the man is times less than his actual weight. Note :  Effective force stopping a falling body =  If R be the resistance of sand on a body of mass m falling in sand, then effective force = . Important Tips  If the plane moves vertically upward with retardation equal to g i.e., , then from , we get . Thus there is no pressure of the body on the plane when the plane rises vertically with retardation equal to g.  If the plane moves down freely under gravity i.e., with acceleration equal to g, then from , we get . Thus there is no pressure of the body on the plane, when it moves vertically downwards with an acceleration equal to g. Example: 23 A paracute weighing 112 lbs. wt. falling with a uniform acceleration from rest, describes 16 ft. in the first 4 sec. Then the resultant pressure of air on the parachute is (a) 85 lbs. wt. (b) 95 lbs. wt. (c) 105 lbs. wt. (d) 115 lbs. wt. Solution: (c) Let f be the uniform acceleration. Here , , Since , . Equation of motion is (Downward) = 112 (32 – 2) = 112 × 30 Poundals lbs. wt. = 105 lbs. wt.