DYNAMICS-02-THEORY-PART-II

Example: 24 A man falls vertically under gravity with a box of mass ‘m’ on his head. Then the reaction force between his head and the box is (a) mg (b) 2 mg (c) 0 (d) 1.5 mg Solution: (c) Let reaction be R. Since the man falls vertically under gravity From , we have . . Example: 25 A man of mass 80 kg. is travelling in a lift. The reaction between the floor of the lift and the man when the lift is ascending upwards at 4 m/sec2 is (a) 1464.8 N (b) 1784.8 N (c) 1959.8 N (d) 1104.8 N Solution: (d) When the lift is ascending, we have = = 1104.8 N. Example: 26 A particle of mass m falls from rest at a height of 16 m and is brought to rest by penetrating into the ground. If the average vertical thrust exerted by the ground be 388 kg. wt, then the mass of the particle is (a) 2 kg (b) (c) (d) Solution: (c) Let be the retardation produced by the thrust exerted by the ground and let . be the velocity with which the mass m penetrates into the ground. Then,  Since the mass m penetrates through in the ground. Therefore, its final velocity is zero. So,    [ ]   . Hence, the mass of the particle is 4 kg. 4.10 Motion of two particles connected by a String. (1) Two particles hanging vertically : If two particles of masses and are suspended freely by a light inextensible string which passes over a smooth fixed light pulley, then the particle of mass will move downwards, with Acceleration Tension in the string Pressure on the pulley (2) One particle on smooth horizontal table : A particle of mass attached at one end of light inextensible string is hanging vertically, the string passes over a pulley at the end of a smooth horizontal table, and a particle of mass placed on the table is attached at the other end of the string. Then for the system, Acceleration Tension in the string Pressure on the pulley . (3) One particle on an inclined plane : A particle of mass attached at one end of a light inextensible string is hanging vertically, the string passes over a pulley at the end of a smooth plane inclined at an angle to the horizontal, and a particle of mass placed on this inclined plane is attached at the other end of the string. If the particle of mass moves vertically downwards, then for the system, Acceleration Tension in the string Pressure on the pulley Example: 27 The sum of two weights of an Atwood's machine is 16 lbs. The heavier weight descends 64 ft in 4 seconds. The value of weights in lbs are (a) 10, 6 (b) 9, 7 (c) 8, 8 (d) 12, 4 Solution: (a) Let the two weights be m1 and (lbs.) ......(i) Now, the distance moved by the heavier weight in 4 seconds from rest is 64 ft. If acceleration = f, then  But   ......(ii) From (i) and (ii), lbs, . Example: 28 Two weights W and are connected by a light string passing over a light pulley. If the pulley moves with an upward acceleration equal to that of gravity, the tension of the string is (a) (b) (c) (d) Solution: (d) Suppose W descends and ascends. If f is the acceleration of the weights relative to the pulley, then since the pulley ascends with an acceleration f, the actual acceleration of and are (downwards) and (upwards) respectively. Let T be tension in the string. and or and Subtracting, We get or kg. wt. Example: 29 To one end of a light string passing over a smooth fixed pulley is attached a particle of mass M, and the other end carries a light pulley over which passes a light string to whose ends are attached particles of masses and . The mass M will remain at rest or will move with a uniform velocity, if (a) (b) (c) (d) Solution: (c) If the mass M has no acceleration, then But or 4.11 Impact of Elastic Bodies. (1) Elasticity : It is that property of bodies by virtue of which they can be compressed and after compression they recover or tend to recover their original shape. Bodies possessing this property are called elastic bodies. (2) Law of conservation of momentum : It states “the total momentum of two bodies remains constant after their collision of any other mutual action.” Mathematically where mass of the first body, initial velocity of the first body, final velocity of the first body. Corresponding values for the second body. (3) Coefficient of restitution : This constant ratio is denoted by e and is called the coefficient of restitution or coefficient of elasticity. The values of e varies between the limits 0 and 1. The value of e depends upon the substances of the bodies and is independent of the masses of the bodies. If the bodies are perfectly elastic, then and for inelastic bodies . (4) Impact : When two bodies strike against each other, they are said to have an impact. It is of two kinds : Direct and Oblique. (5) Newton's experimental law of impact : It states that when two elastic bodies collide, their relative velocity along the common normal after impact bears a constant ratio of their relative velocity before impact and is in opposite direction. If and be the velocities of the two bodies before impact along the common normal at their point of contact and and be their velocities after impact in the same direction, Case I : If the two bodies move in direction shown in diagram given below, then (a) and (b) Case II : If the direction of motion of two bodies before and after the impact are as shown below, then by the laws of direct impact, we have (a) or and (b) or Case III : If the two bodies move in directions as shown below, then by the laws of direct impact, we have (a) or and (b) or (6) Direct impact of two smooth spheres : Two smooth spheres of masses and moving along their line of centres with velocities and (measured in the same sense) impinge directly. To find their velocities immediately after impact, e being the coefficient of restitution between them. Let and be the velocities of the two spheres immediately after impact, measured along their line of centres in the same direction in which and are measured. As the spheres are smooth, the impulsive action and reaction between them will be along the common normal at the point of contact. From the principle of conservation of momentum, …..(i) Also from Newton’s experimental law of impact of two bodies, …..(ii) Multiplying (ii) by and subtracting from (i), we get …..(iii) (7) Oblique impact of two spheres : A smooth spheres of mass , impinges with a velocity obliquely on a smooth sphere of mass moving with a velocity . If the direction of motion before impact make angles and respectively with the line joining the centres of the spheres, and if the coefficient of restitution be e, to find the velocity and directions of motion after impact, Let the velocities of the sphere after impact be and in directions inclined at angles and respectively to the line of centres. Since the spheres are smooth, there is no force perpendicular to the line joining the centres of the two balls and therefore, velocities in that direction are unaltered. …..(i) …..(ii) And by Newton’s law, along the line of centres, …..(iii) Again, the only force acting on the spheres during the impact is along the line of centres. Hence the total momentum in that direction is unaltered. …..(iv) The equations (i), (ii), (iii) and (iv) determine the unknown quantities. (8) Impact of a smooth sphere on a fixed smooth plane : Let a smooth sphere of mass m moving with velocity u in a direction making an angle with the vertical strike a fixed smooth horizontal plane and let v be the velocity of the sphere at an angle to the vertical after impact. Since, both the sphere and the plane are smooth, so there is no change in velocity parallel to the horizontal plane. …..(i) And by Newton’s law, along the normal CN, velocity of separation = e.(velocity of approach) ….(ii) Dividing (i) by (ii), we get Particular case : If then from (i), ; and from (ii) Thus if a smooth sphere strikes a smooth horizontal plane normally, then it will rebound along the normal with velocity, e times the velocity of impact i.e. velocity of rebound = e.(velocity before impact). Rebounds of a particle on a smooth plane : If a smooth ball falls from a height h upon a fixed smooth horizontal plane, and if e is the coefficient of restitution, then whole time before the rebounding ends And the total distance described before finishing rebounding . Example: 30 A ball is dropped from a height h on a horizontal plane and the coefficient of restitution for the impact is e, the velocity with which the ball rebounds from the floor is (a) eh (b) egh (c) (d) Solution: (d) Let u be the velocity with which the ball strikes the ground. Then,  .....(i) Suppose the ball rebounds with velocity v, then   , [from (i)]. Example: 31 A sphere impinges directly on a similar sphere at rest. If the coefficient of restitution is , the velocities after impact are in the ratio (a) 1 : 2 (b) 2 : 3 (c) 1 : 3 (d) 3 : 4 Solution: (c) From principle of conservation of momentum , ......(i) From Newton's rule for relative velocities before and after impact. ......(ii) Here , and let Then from (i) or ......(iii) From (ii), .......(iv) Adding (iii) and (iv) , or ......(v) From (iii) and (v), . Hence . Example: 32 A ball impinges directly upon another ball at rest and is itself reduced to rest by the impact. If half of the K.E. is destroyed in the collision, the coefficient of restitution, is (a) (b) (c) (d) Solution: (d) Let masses of the two balls be and . By the given question, we have or .....(i) and i.e., .....(ii) Again for the given condition, K.E. (before impact) = 2 × K.E. after impact .....(iii) or But from (i), , i.e., . Putting in (ii), we get , . 4.12 Projectile Motion. When a body is thrown in the air, not vertically upwards but making an acute angle with the horizontal, then it describes a curved path and this path is a Parabola. The body so projected is called a projectile. The curved path described by the body is called its trajectory. The path of a projectile is a parabola whose equation is Its vertex is . Focus is (1) Some important deductions about projectile motion: Let a particle is projected with velocity in a direction making angle with . Let the particle be at a point after time t and v be its velocity making an angle with . (i) Time of flight (ii) Range of flight i.e., Horizontal Range Maximum horizontal Range and this happens, when (iii) Greatest height Time taken to reach the greatest height (iv) Let be the position of the particle at time t. Then , (v) Horizontal velocity at time t Vertical velocity at time t (vi) Resultant velocity at time t . And its direction is (vii) Velocity of the projectile at the height h and its direction is (2) Range and time of flight on an inclined plane Let OX and OY be the coordinate axes and OA be the inclined plane at an angle to the horizon OX. Let a particle be projected from O with initial velocity u inclined at an angle with the horizontal OX. The equation of the trajectory is . (i) Range and time of flight on an inclined plane with angle of inclination are given by and maximum range up the plane , where (ii) Time of flight (iii) Range down the plane (iv) Maximum range down the plane , where (v) Time of flight , down the plane (vi) Condition that the particle may strike the plane at right angles is . Example: 33 Two stones are projected from the top of a cliff h metres high, with the same speed u so as to hit the ground at the same spot. If one of the stones is projected horizontally and the other is projected at an angle to the horizontal then equals (a) (b) (c) (d) Solution: (d) ; .....(i) Now , ‘t’ from (i) , , ; . Example: 34 A gun projects a ball at an angle of with the horizontal. If the horizontal range is 39.2 m, then the ball will rise to (a) 9.8 m (b) 4.9 m (c) 2.45 m (d) 19.6 m Solution: (a) Horizontal range =   Greatest height attained = = = . Example: 35 If the range of any projectile is the distance equal to the height from which a particle attains the velocity equal to the velocity of projection, then the angle of projection will be (a) (b) (c) (d) Solution: (b) Range = , where is the angle of projection. Also,  By the given condition,  , or or . Hence, required angle of projection = . Example: 36 A particle is thrown with the velocity v such that its range on horizontal plane is twice the maximum height obtained. Its range will be (a) (b) (c) (d) Solution: (c) Range = 2  (maximum height)     or But , , . Now  . Example: 37 Let u be the velocity of projection and be the velocity of striking the plane when projected so that the range up the plane is maximum, and be the velocity of striking the plane when projected so that the range down the plane is maximum, then u is the (a) A.M. of (b) G.M. of (c) H.M. of (d) None of these Solution: (b) Let u be the velocity of projection and be the inclination of the plane with the horizon. Let R be the maximum range up the plane. Then, The coordinates of the point where the projectile strikes the plane are ( ) . It is given that is the velocity of striking the plane when projected so that the range up the plane is maximum.   .....(i) Replacing by in (i) , we get .....(ii) From (i) and (ii), we get  i.e., u is the G.M. of . 4.13 Work, Power and Energy. (1) Work : Work is said to be done by a force when its point of application undergoes a displacement. In other words, when a body is displaced due to the action of a force, then the force is said to do work. Work is a scalar quantity. Work done = force  displacement of body in the direction of force , where  is the angle between F and d. (2) Power : The rate of doing work is called power. It is the amount of work that an agent is capable of doing in a unit time. 1 watt = 107 ergs per sec = 1 joule per sec., 1 H.P. = 746 watt. (3) Energy : Energy of a body is its capacity for doing work and is of two kinds : (i) Kinetic energy : Kinetic energy is the capacity for doing work which a moving body possesses by virtue of its motion and is measured by the work which the body can do against any force applied to stop it, before the velocity is destroyed. (ii) Potential energy : The potential energy of a body is the capacity for doing work which it possesses by virtue of its position of configuration. Example: 38 Two bodies of masses m and 4m are moving with equal momentum. The ratio of their K.E. is (a) 1 : 4 (b) 4 : 1 (c) 1 : 1 (d) 1 : 2 Solution: (b) Momentum of first body = mu, where u is the velocity of the first body. Momentum of second body = 4mv, where v is the velocity of the second body. By the given condition, Ratio of K.E. = . Hence, the ratio of K.E. is 4 : 1. Example: 39 A bowler throws a bumper with a speed of The moment the ball touches the ground, it losses its energy by 1.5 kgm. If the weight of the ball is 225 gm, then speed of the ball at which it hits the bat is (a) 2.22 m/sec. (b) 22.2 m/sec. (c) 4.44 m/sec. (d) 44.4 m/sec. Solution: (b) Weight of the ball = gm = 0.225 kg Velocity of the ball = v = 25 m/sec. Loss of energy = 1.5 kg.m K.E. of the ball before it touches the ground = . K.E. of the ball after bumping = K.E. before touching the ground – loss of K.E. = 7.17 – 1.5 = 5.67 kg-m. Let be the velocity of the ball at which it hits the bat. Now K.E. of the ball at the time of hitting the bat   Example: 40 A particular starts from rest and move with constant acceleration. Then the ratio of the increase in the K.E. in mth and second is (a) (b) (c) (d) None of these Solution: (c) Required ratio = = = = .