W = mgh joule and mass (m) = volume density =

Transcription

1 1. A rain drop of radius 2 mm falls from a height of 500 m above the ground. It falls with decreasing acceleration due to viscous resistance of the air until at half its original height, it attains its maximum (terminal) speed and moves with uniform speed thereafter. What is the work done by the gravitational force on the drop in the first and second half of its journey? What is the work done by the resistive force in the entire journey if its speed on reaching the ground is 10 ms -1 Air resistance is not constant in the top half of the fall. Hence the formulae for uniform acceleration cannot be applied. Energy equation is therefore applied]. We know that work done = Force distance = F S = mgh joule. W = mgh joule and mass (m) = volume density = ρ Mass of the water drop m = π ( ) kg = kg... Potential energy at the top = mgh = J = J Work done by gravitational force = Kinetic energy gained by the drop + energy lost by air resistance. LHS = loss in potential energy during the respective falls. Work done by gravitational force during the first half is w 1 = Loss of potential energy during the first half of the fall = mg (h/2) = (32/3) π J = joules = joules. Since the distance covered in the second half journey is also 250 m, therefore work done is given by w 2 = = Joules Work done by the resistive force = Change in kinetic energy - work done by Gravitational force Change in kinetic energy = ½ mv 2-0 = ½ (10 10) joule = joules. Work done by gravitational force = mgh joule = J = joules Work done by resistive force = J = J. 2. Which of the following potential energy curves cannot possibly describe the elastic collision of two billiard balls? Here 'r' is the distance between centres of the balls and 'R' is the distance between the radiuses of each ball. 1

2 For elastic collision, (a) Momentum is conserved (b) Total energy is conserved (c) Mechanical energy is not converted into any other form such as sound, beat, light.(d) Forces involved during the interaction are of conservative nature.the potential energy of a system of two bodies varies inversely proportional to the distance between them i.e. as 1/r. So all the potential energy curve except the one shown in fig. (vi) cannot describe elastic collision. 3. The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative: (i) work done by a man in lifting a bucket out of a well by means of a rope tied to a bucket (ii) work done by gravitational force in the above case. (iii) work done by friction on a body sliding down an inclined plane, (iv) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity, (v) work done by the resistive force of air on a vibrating pendulum in bringing it to rest. Work done is considered positive when it is in the same direction as that of the applied force. (i) Movement of the bucket and the direction of application of the force are both upwards. Hence the work done is positive. (ii) Work done is -ve because the gravitational force acts in the downward direction while backed moves upward. (iii) Work done is -ve because the frictional force always acts in the opposite direction of the motion of the body. (iv) Work is done against the friction force which is in the negative direction. The applied force 2

3 moves the body in the same direction as its own. Hence it is positive. (v) Resistive force acts against the direction of vibration of pendulum and hence negative. 4. A body of mass 2 kg initially at rest moves under the action of an applied horizontal force of 7 N on a table with coefficient of kinetic friction = 0.1. Compute the (a) work done by the applied force in 10 s. (b) work done by friction in 10 s. (c) work done by the net force on the body in 10 s. (d) change in kinetic energy of the body in 10 s, and interpret the results. Reaction of the surface on the body = N = 19.6 N Forward force applied on the body F = 7N Dynamic friction force against the motion = reaction dynamic friction coefficient Frictional force = f r = mr = µ mg W 1 = 19.6 x (-0.1) = 1.96 N F' = F - f r Net force = 7N N = 5.04 N Acceleration produced in the body a = (5.04 N) / (2 Kg)= 2.52 m/s 2 (a) Distance traveled in 10 s = s = 126 m Work done by applied force = 7N x 126 m = 882 J (b) Work done by friction force W 2 = cos180 o =1.96 N x 126m x (-1) = -247 J acts in a direction opposite to that of motion of the body. (c) Work done by net force W 3 = F's = 5.04 N x 126 m = 635 J (d) Change in kinetic energy K e = 3

4 = (i.e v = at = 2.52 x 10 = 25.2 ms -1 ) = 635 J Change in kinetic energy = work done by net force. 5. Given in the figure below are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any in which the particle cannot be found for the given energy. Also indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant. [Magnitudes of potential energy and kinetic energy should be equal to the total energy. Situations which do not conform to this, are not possible]. a. Not possible in the region x>a. Minimum total energy = 0. b. Not possible in the region x<a, and x>b. Minimum total energy = -V 1. c. Not possible in the region -b/2<x>-a/2. Minimum total energy = -V The potential energy function of a particle executing simple harmonic motion is given by V(x) = ½ k x 2, where k is the force constant of the oscillator. For k = 0.5 N m -1 the graph of V(x) versus x is shown below. Show that a particle of total energy 1 J moving under this potential must "turn back" when it reaches x = ± 2 m. 4

5 Potential energy = V(x) = ½ k x 2 = ½ ½ x 2 (given k= ½ N/m) Total energy = potential energy + kinetic energy = 1 J (given condition) When the particle turns back, the velocity and hence the kinetic energy should be zero. 1 J = ¼ x i.e. x ± 2 m [In the given sketch O is the turning point, and A and B both represent the centre point where the velocity is maximum]. 7. Answer the following: a. The casing of a rocket in flight burns up due to friction. At whose expense is the heat energy required for the burning obtained? The rocket or the atmosphere. b. Comets move around the sun in highly elliptical orbits. The gravitational force on the comet due to the sun is not normal to the comet's velocity in general. Yet the work done by the gravitational force over every complete orbit comet is zero. Why? c. An artificial satellite orbiting the earth in very thin atmosphere loses its energy gradually due to dissipation against atmospheric resistance, however small. Why then does its speed increase progressively as it comes closer and closer to earth? d. In the figure (i), the man walks 2 m carrying a mass of 15 kg on his hands. In figure (ii), he walks the same distance pulling the rope behind him. The rope goes over a pulley and a mass of 15 kg hangs at its other end. In which case is the work done greater? a. Rocket generates heat to burn itself by friction with the atmosphere. b. The gravitational forces that act on comets are conservative forces. For conservative forces the work done = - (total energy change). Over a complete cycle, the potential energy and the kinetic energy remain same. (Work done during a cycle involving conservative forces is calculated by considering the initial and final situations only and the path and other changes in between do not affect the calculation of work done) c. When the satellite comes closer and closer to earth, the potential energy decreases because of the decrease in height. Since the total energy is to remain constant, the kinetic energy increases and so 5

6 the speed. Of course, there is a small mount of frictional loss in energy, but the gain by velocity is more. Hence its speed continues to increase. d. Though the person carries the weight while walking, there is no relative movement with reference to gravitational force. Hence the only work done is the frictional force during walking. In the second case the a force of 15g N is applied and moved to lift the stone against gravity. Hence the work done is more in this case. 8. Underline the correct alternative: (a) When a conservative force does positive work on a body, the potential energy of the body increases / decreases / remains unaltered. (b) Work done by a body against friction always results in loss of its kinetic / potential energy. (c) The rate of change of momentum of a many-particle system is proportional to the external force/sum of the internal forces of the system. (d) In an inelastic collision of two bodies, the quantities, which do not change after the collision, are the total kinetic energy/total linear momentum total energy of the system of two bodies. (a) Decreases (e.g. When a mass falls freely under gravity, positive work is done by the gravity and the potential energy decreases. (b) Kinetic energy (e.g. When a mass slides down an inclined plane against friction, the velocity of the mass decreases. The potential energy, which is dependent on the height and mass of the body, remains unchanged. (c) External forces. (e.g. People walking inside a moving railway carriage do not constitute to the momentum of the carriage whereas an engine towing the carriage does.) (d) Total linear momentum does not change in an inelastic collision. The total energy also conserved if we consider all the affected components of the system (e.g. If the heat generated is also considered as within the system, the total energy will be conserved). 9. State if each of the following statements is true or false. Give reasons for your answer. (a) In an elastic collision of two bodies, the momentum and energy of each body is conserved. (b) Total energy of a system is always conserved, no matter what internal and external forces on the body are present. (c) Work done in the motion of a body over a closed loop is zero for every force in nature. (d) In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system. (a) False. Only the total energy and momentum of both the bodies together are conserved. (b) False. The external forces affect the energy of the system. (c) False. It is true only for conservative forces. (d) True if no other external or internally generated force influence the system. 10. Answer carefully with reasons: (a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact)? (b) Is the total linear momentum conserved during the short time of elastic collision of the two balls? (c) What are the answers to (a) and (b) for an inelastic collision? (d) If the potential energy of the two billiard balls depends only on separation distance between the centres, is the collision elastic or inelastic? (Note, we are talking here of potential energy corresponding to the force during collision, not gravitational potential 6

7 energy). (a) No. In the short time of collision the kinetic energy is stored as elastic strain energy and reconverted to kinetic energy. (b) Yes. By elastic deformation and also by joint movement together in some cases. (c) Linear momentum is conserved during the collision. Kinetic energy is not conserved even after impact. (d) Elastic. Whatever deformation caused during the first half of collision, should recover during the second half. If the potential energy depends only on the centre distance, whatever energy stored should be recovered. There will be no loss. Hence the collision will be elastic. 11. A body is initially at rest. It undergoes one-dimensional motion with constant acceleration. The power delivered to it at time t is proportional to (i) t 1/2 (ii) t (iii) t 3/2 (iv) t 2 (ii) t 12. A body is moving uni-directionally under the influence of a source of constant power. Its displacement in time t is proportional to (i) t 1/2 (ii) t (iii) t 3/2 (iv) t 2 (iii) t 3/2 13. A body constrained to move along the z-axis of a co-ordinate system is subject to a constant force given by: Λ Λ Λ Λ where i, j, k are unit vectors along the x, y and z - axis of the system respectively. What is the work done by this force in moving the body a distance of 4 m along the z-axis. Work done is defined as the component of force along the direction of motion and displacement. The body moves 4 m along z - axis only... Work done, W = Nm 14. An electron and a proton are detected in a cosmic ray experiment, the first with a kinetic energy of 10 kev, and the second with 100keV. Which is faster, the electron or the proton? Obtain the ratio of their speeds. (Electron mass = kg, proton mass = kg, 1eV = J). 7

8 Kinetic energy = E = ½ m v 2 V = (2 E / m) ½ Kinetic energy of electron E e = 10 Kev = 10 4 x 1 ev = 10 4 x 1.6 x J = 1.6 x J. Kinetic energy of proton, E p = 100 kev = 10 5 x 1 ev = 10 5 x 1.6 x J = 1.6 x J. 2 2 E e = (1/2 ) m e V e E p = (1/2) m p V p V e = (2 E e / m e ) = = 5.93 X 10 7 ms -1 V p = (2 E p / m p ) = = 4.38 x 10 6 ms -1 The required ratio V e /V p = 5.93 X 10 7 = x A molecule in a gas container hits a horizontal wall with a speed of 200 m/s, and an angle of 30 o with normal, and rebounds with the same speed. Is momentum conserved in the collision? Is the collision elastic or inelastic? Momentum is conserved for all systems during collision. Kinetic energy is conserved in elastic collisions only. In the given case, the momentum of the molecule and the wall system is conserved. Due to the momentum of the incoming molecule, the wall deflects and rebounds with the same momentum. This momentum is then transferred to the ball, which rebounds with an equal momentum. At any moment, the total momentum of the ball and the wall system is constant. However, it is to be noticed that the recoil momentum of the wall produces very little velocity, because of the very large mass of the wall compared to that of the ball. Coefficient of resolution = e = - (Relative velocity of the ball along the common normal with respect to the wall after collision.) / (Relative velocity of the ball along the common normal respect to the wall before collision) Therefore e = - e = 1 If e = 1, then it is perfectly elastic and if e < 1, then it is inelastic. [The above events are concerned with the component of the momentum normal to the wall. The component resolved along the surface of the wall remains same throughout the event] with 16. A pump on the ground floor of a building can pump up water to fill a tank of volume 30 m 3 in 15 minutes. If the tank is 40 m above the ground, and the efficiency of the pump is 30%, how much electric power is consumed by the pump? 8

9 Power required to lift water for 1 s = 30 / (15 60) J Since this output will be only 30% of input, The input power required for 1 s = {30 / (15 60) } / 0.3 J = 43,600 W The pump consumes 43.6 kw of power. 17. Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed V. If the collision is elastic, which of the following is a possible result after collision? The momentum and the kinetic energy of the moving balls after collision in the above three cases are given below: Case Momentum Kinetic energy Case 1 2 ( m v/2 ) = mv 2 {½ m (v/2) 2 } = ¼ (m v 2 ) Case 2 1 (m v ) = mv 1 ½ m (v) 2 = ½ (m v 2 ) Case 3 3 ( m v/3) = mv 3 {1/2 m (v/3) 2 = 1/6 (mv 2 ) 18. The bob A of a pendulum released from 30 o to the vertical hits another bob B of the same mass at rest on a table as shown in the figure. How high does the bob A rise after the collision? Neglect the size of the bobs and assume the collision to be elastic. Let the velocity of A while hitting B be v, and let us assume that A moves with a velocity v 1 and B moves with a velocity v 2 after impact. Equating momentum m v 1 + m v 2 = m v v 1 + v 2 = v Equating kinetic energies ½ m v ½ m v 2 2 = ½ m v 2 2 v 1 + v 2 2 = v 2 = (v 1 + v 2 ) 2 For the above situation v 1 should be zero and v 2 should be = v. For an elastic collision, A will stop fully and B will move with a velocity v. 9

10 19. The bob of a pendulum is released from a horizontal position A as shown in the figure. If the length of the pendulum is 1.5, what is the speed with which the bob arrives at the lowermost point B, given that it dissipates 5% of its initial energy against air resistance? There is a loss of 5% of the initial energy when the bob moves from A to B Hence 0.95 Potential energy of the bob at A = Kinetic energy of the bob at B 95% of this energy is converted into kinetic energy Þ 0.95 (m ) = ½ m v 2 v 2 = v = 5.3 m/s. 20. A trolley of mass 300 kg carrying a sandbag of 25 kg is moving uniformly with a speed of 27 km/h on a frictionless track. After a while, sand starts leaking out of a hole on the trolley's floor at the rate of 0.05 kg/s. What is the speed of the trolley after the entire bag is empty? Since the trolley is moving on a frictionless track with uniform speed, there is no external force acting on it. The sand falling from the bag falls in the trolley itself and there is no change in total mass of the moving bodies. Hence the speed of 27 km / h will continue to remain constant. 21. A particle of mass 0.5 kg travels in a straight line with velocity v = ax 3/2 where a = 5 m -1/2 s -1. What is the work done by the net force during its displacement from x = 0 to x = 2m? W = v = ax 3/2, dv = 3/2ax 1/2 dx = max 3/2 3/2 ax 1/2 dx = 3/2 ma 2 x 2 dx = 1/2 x 0.5 x 25 x 8 = 50J. 22. The blades of a windmill sweep out of a circle of area A. (a) If the wind flows at a velocity v perpendicular to the circle, what is the mass of the air passing through it in time t? (b) What is the kinetic energy of the air? (c) Assume that the windmill converts 25% of the wind's energy into electrical energy and that A = 30 m 2, v = 36 km/h and the density of air is 1.2 kg m -3. What is the electrical power produced? 10

11 (a) Density of air = ρ Time taken = t Velocity of air = v Mass of the air = density x volume = ρ Avt (b) Kinetic energy = mv 2 = ρavt v 2 = ρav 3 t (c) Velocity, v = 36 km/h = = 10 m/s ρ = 1.2 kg m -3 A = 30 m 3 Work done = = J Efficiency = 25% Therefore work done = J 23. A person trying to loose weight (dieter) lifts a 10 kg mass 0.5 m, 1000 times. Assume that the potential energy lost each time she lowers the mass is dissipated. (a) How much does she do against the gravitational force? (b) Fat supplies J of energy per kilogram which converted to mechanical energy with a 20% efficiency rate. How much fat will the dieter use up? (a) Mass m = 10 kg Distance moved d = 0.5 m Number of times lifted n = 1000 Therefore the work done against the gravitational force = mgh = = J (b) J of energy is given by 1 kg fat J of energy is given by = = = kg Efficiency = 20% Fat used up by the dieter = = kg. 24. A large family uses 8 kw of power. (a) Direct solar energy is incident on the horizontal surface at an average rate of 200 W per square meter? If 20% of this energy can be converted to useful electrical energy, how large an area is needed to supply 8 kw? (b) Compare this area to that of the roof of a typical house. Power used P = 8 kw (a) 200 W solar energy is incident on 1 square metre, 20% of this energy can be converted to useful 11

12 energy Hence = 40 W 40 W solar energy is incident on 1 square metre Therefore W solar energy is incident on = 200 m 2 (b) It is comparable to the roof of a large house of dimension 14m 14 m. 25. A bullet of mass kg and horizontal speed of 70 m/s strikes a block of wood of mass 0.4 kg and instantly comes to rest with respect to the block. The block is suspended from the ceiling by means of thin wires. Calculate the height to which the block rises. Also, estimate the amount of heat produced in the block. [Momentum is conserved in all collisions. The bullet hits the block and imparts momentum conserving the momentum, by which the kinetic energy of the block with the embedded bullet can be found out. This kinetic energy will raise the block, but will not account for the initial kinetic energy of the bullet. The difference is given out as heat during impact.] Velocity of the block with the embedded bullet after impact = (mass of the bullet Velocity of the bullet) / (mass of block + bullet) = ( ) / (0.412) m/s = 2.04 m/s Kinetic energy of the block + bullet =½ mv 2 = ½ ( ) ( 2.04) 2 = J This gets converted into potential energy of the combination. If h is the height of rise, then Height of the (block + bullet) will rise = / {( ) 9.8} = m Initial kinetic energy of bullet = ½ (0.012) (70) 2 = 29.4 J Heat generated = Initial kinetic energy of bullet - Kinetic energy of the block + bullet = = 28.5 J. 26. Two inclined frictionless tracks, one gradual and the other steep meet at A from where two stones are allowed to slide from rest, one on each track. Will the stones reach the bottom at the same time? Will they reach there with the same speed? Explain. Given θ = 30, and h = 10 m, what are the speeds and times taken by the two stones? 12

13 [The gravitational force on the mass m = mg and acts vertically down. This can be resolved into two components mg sin θ down the plane, and mg cos θ normal to the plane. The normal component is balanced by the reaction of the plane on the body. The component down the plane accelerates the body downwards] For a mass 'm' at a height 'h' on the plane inclined at θ to the horizontal, the acceleration down the plane = mg sin θ Acceleration produced in the 1 st stone a 1 = g sinθ 1 and a 2 = g sinθ 2 Velocity at bottom = v = u + at = 0 + (m g sin θ )x t, where t = time to reach the bottom. Considering the energy equation, m g h = ½ m v 2 v = (2 g h) 1/2 Potential energy at energy = kinetic energy at B or C Since v is independent of the slope and mass, both the stones will reach the bottom at the same speed. Considering the time to reach the bottom, if t 1 is the time for the stone in the gradual slope and t 2 for the stone in steep slope, and θ 1 and and and and and θ 2 their inclination angles, t 1 = v / (g sin θ 1 ) and t 2 = v / (g sinθ 2 ) θ 2 > θ 1, sin θ 2 > sin θ 1, t 2 < t 1 Stone on the steeper slope will take less time to reach the bottom. mgh = mv 2 = v Therefore v = v1 = v2 = 14 m/s Therefore t1 = 2.86 s t2 = 1.65 s Hence t2 < t1. = 14 m/s 27. A 1 kg block situated on a rough incline is connected to a spring of spring constant 100 Nm -1 as shown in the figure. The block is released from rest with the spring constant in the unstretched position. The block moves 10 cm down the incline before coming to rest. Find the coefficient of friction between the block and the incline. Assume that the spring has negligible mass and the pulley is frictionless. Mass of the block m = 1 kg Distance moved x = 10 cm = 0.1 m Spring constant k = 100 Nm -1 Force of friction F = kx = = 1000 N 13

14 Coefficient of friction µ = = = A bolt of mass 0.3 kg falls from the ceiling of an elevator moving down with a uniform speed of 7 m/s. It hits the floor of the elevator (length of the elevator = 3 m) and does not rebound. What is the heat produced by the impact? Would your answer be different if the elevator were stationary? [Since the bolt is falling in the elevator itself, we have to consider the relative velocity only. The relative acceleration is g only, since the elevator does not have any acceleration] Relative to the elevator, let the velocity of the bolt hitting the floor = v m/s v 2 = m/s 2 3m = 58.8 (m/s) 2 Kinetic energy of the bolt with reference to the elevator = ½ m v 2 = ½ = 8.82 J Since the relative velocity w. r. to the elevator after impact, the total kinetic energy of the system is zero. The bolt does not have a potential energy w. r. to while on floor, the only way the kinetic energy is to be dissipated is through heat generated. Heat generated is 8.82 J. 29. A trolley of a mass 200 kg moves with a uniform speed of 36 km/h on a frictionless track. A child of mass 20 kg runs on the trolley from one end to the other (10 m away) with a speed of 4 m/s relative to the trolley in a direction opposite to the trolley's motion, and jumps out of the trolley. What is the final speed of the trolley? How much the trolley moved from the time the child begins to run? [When the child starts moving on the platform, an impact is given to the trolley by the reaction of the starting force. Afterwards the child does not impart any force on the trolley since the movement is at constant speed of 4 m/s. The impact force will accelerate the trolley with the child.] Initial speed of the trolley = 36 km/h = 10 m/s = u m/s say. Speed of the child w. r. to the trolley = 4 m/s final speed of the trolley = v say Mass of the trolley m 1 = 200 kg Mass of the child m 2 = 20 kg Consider the speeds and momentum with reference to the ground. Speed of the child w. r. to the ground = (v - 4) m/s Equating the momentum before and after impact, ( ) 10 = {20 (v - 4)} v v = 10.36m/s = The final speed of the trolley Time taken by the child to run the 10 m length of the trolley = 10/4 s = 2.5 s Distance covered by the trolley during this 2.5 s = m = 25.9 m. 14