Work, Power & Energy Exercise - I

WORK, POWER & ENERGY Page # 10 Exercise - I UNSOLVED PROBLEMS Q.1 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 : (a) work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket. (b) work done by gravitational force in the above case. (c) work done by friction on a body sliding down an inclined plane. (d) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity, (e) work done by the resistive force of air on a vibrating pendulum in bringing into rest. Q.2 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 your results. Q.3 Given in figure 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. Q.4 The potential energy function for a particle executing linear simple harmonic motion is given by V(x) = kx2/2, where k is the force constant of the oscillator. For k = 0.5 N m–1, the graphj of V(x) versus x is shown in figure. Show that a particle of total energy 1 J moving under this potential must 'turn back' when it reaches x = ± 2 m. V(x) x Q.5 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 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 of the 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 as it comes closer and closer to the earth ? (d) In figure 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 ? 15kg 15kg (i) (ii) Q.6 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 a losses of its kinetic/potential energy. (c) The rate of change of total momentum of a manyparticle system is proportional to the external force/ sum of the internal forces on 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. Q.7 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. WORK, POWER & ENERGY (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. Q.8 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 an elastic collision of two balls ? (c) What are the answers to (a) and (b) for an inelastic collision ? (d) If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collision elastic or inelastic ? (Note, we are talking here of potential energy corresponding to the force during collision, not gravitational potential energy). Q.9 Which of the following potential energy curves in Figure cannot possibly describe the elastic collision of two billiard balls ? Here r is the distance between centres of the balls. No.of -particles per unit energy interval Q.10 Consider the decay of a free neutron at rest : n  p + e–1 Show that the two-body decay of this type must necessarily give an electron of fixed energy and, therefore, cannot account for the observed continuous energy distribution in the -decay of a neutron or a nucleus (figure). [Note : The simple result of this exercise was one among the several arguments advanced by W. Pauli to predict the existence of at third particle in the decay products of -decay. This particle is known as neutrino. We now know that it is a particle of intrinsic spin ½ (like e–1, p or n), but is neutral, and either massless or having an extremely small mass (compared to electron's mass) and which interacts very weakly with matter. The correct decay process of neutron is : n  p + e– + v] Kinetic energy of -particle emitted Page # 11 Q.11 A body is initially at rest. It undergoes on dimensional motion with constant acceleration. The power delivered to it at time t is proportional to (i) t1/2 (ii) t (iii) t3/2 (iv) t2 Q.12 A body is moving uni-directionally under the influence of a source of constant power. I ts displacement in time t is proportional to : (i) t1?2 (ii) t (iii) t3/2 (iv) t2 Q.13 A body constrained to move along the z-axis of a coordinate system is subject to a constant force F given by F=  î  2 ĵ  3 k̂ N , where î, ĵ, k̂ 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 ? Q.14 An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy 10 keV, and the second with 100 keV. Which is faster, the electron or the proton ? Obtain the ratio of their speeds. (Electron mass = 9.11 × 10–31 kg, proton mass = 1.67 × 10–27 kg, 1 eV = 1.60 × 10–19 J). Q.15 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 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 m s–1 ? Q.16 A molecule in a gas container hits a horizontal wall with speed 200 m s–1 and angle 30° with the normal, and rebounds with the same speed. Is momentum conserved in the collision ? Is the collision elastic or inelastic ? Q.17 A pump on the ground floor of a building can pump up water to fill a tank of volume 30 m3 in 15 min. 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 ? Q.18 Two identical ball bearings in contact with each of the 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 figure) is a possible result after collision ? WORK, POWER & ENERGY 1 2 3 Page # 12 1 2 3 (i) V V=0 V 2 Q.23 The blades of a windmill sweep out 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 m2, v = 36 km/h and the density of air is 1.2 kgm–2. What is the electrical power produced ? Q.24 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 work does she do against the gravitational force ? (b) Fat supplies 3.8 × 107 J of energy per kilogram which is converted to mechanical energy with a 20% efficiency rate. How much fat will the dieter use up ? Q.19 The bob A of a pendulum released from 30° to the vertical hits another bob B of the same mass at rest on a stable as shown in figure. How high does the bob A rise after the collision ? Neglect the size of the bobs and assume the collision to be elastic. Q.25 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. Q.26 The heart rate (number of heart beats per minute) scales as 1/L where L is the characteristic length of the mammal. (a) Can you explain this ? (b) The scale factor of a human relative to a rhesus monkey is about 2.5. What is the monkey's heart rate ? Q.20 The bob of a pendulum is released from a horizontal position A as shown in figure. If the length of the pendulum is 1.5 m, what is the speed with which the bob arrives at the lowermost point B, given that it dissipated 5% of its initial energy against air resistance ? Q.21 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–1. What is the speed of the trolley after the entire sand bag is empty ? Q.22 A particle of mass 0.5 kg travels in a straight line with velocity v = ax3/2 where a = 5 m–1/2 s–1. What is the work done by the net force during its displacement from x = 0 x = 2m ? Q.27 In a nerve impulse about 10 5 neurons are fired. Estimate the energy used. Q.28 A bullet of mass 0.012 kg and horizontal speed 70 ms–1 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. Q.29 Two inclined frictionless tracks, one gradual and he other steep meet at A from where two stones are allowed to slide down from rest, one on each track (Figure). Will the stones reach the bottom at the same time ? Will they reach there with the same speed ? Explain. Given 1 = 30°, 2 = 60°, and h = 10 m, what are the speeds and times taken by the two stones ? WORK, POWER & ENERGY Page # 13 Q.34 Estimate the amount of energy released in the nuclear fusion reaction 12 H  2 1H  13He  n ( 12 H : deuterium, 32He : isotope of helium, given that M( 32He ) = 3.0160 , Mn = 1.0087 u and M( 12 H ) = 2.0141 Q.30 A 1 kg block situated on a rough incline is connected to a spring of spring constant 100 N m–1 as a shown in figure. The block is released from rest with the spring 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. Q.31 The potential energy of a 2 kg particle free to 4 2 x x move along the x-axis is given by U(x) =   – 5   J, b b where b = 1 m. Plot this potential, identifying the extremum points. Identify the regions where particle may be found and its maximum speed given that the total mechanical energy is (1) 36 J ; (ii) 4 J. Q.32 A blot of mass 0.3 kg falls from the ceiling of an elevator moving down with an uniform speed of 7 ms–1. It hits the floor of the elevator (length of the elevator = 3m) and does not rebound. What is the heat produced by the impact ? Would your answer be different if the elevator were stationary ? Q.33 A trolley of mass 200 kg moves with a uniform speed of 35 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 ms–1 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 has the trolley moved from the time the child begins to run ? u where 1 u = 1.661 × 10-27 kg. Express your answer in units of MeV. Q.35 When slow neutrons are incident on a target containing 235 92 U  n  235 92 U , 141 36 Kr a possible (fission) reaction is  3n Estimate the amount of energy released using the following data : M(235U) = 235.04 u, M(141Ba) = 140.91 u, M(92Kr) = 91. 926 u, Mn = 1.0087 u. The energy released in a fusion reaction (problem 6.34) is much less than the energy release in a fission reaction of the heavy nucleus. Why then is a hydrogen bomb (based on nuclear fusion) far more powerful than the atom bomb (based on nuclear fission). Q.36 The nucleus 57Fe emits a -ray of energy 14.4 keV. If the mass of the nucleus is 56.935 u, calculate the recoil energy of the nucleus. You will notice that the recoil energy of a nucleus is inversely proportional to its mass. Now, if a nucleus such as 57Fe is embedded in a lattice, the recoil momentum due to -emission is shared, roughly speaking. By the entire crystal. Consequently, the recoil energy is very greatly reduced nad the entire energy of the decay is converted (without any loss due to recoil) into the energy of the -ray. This the principle of recoilless emission employed in Mossbauer spectroscopy invented in 1958 and now an important tool for high resolution nuclear spectroscopic studies. Q.37 Discuss how the work-energy theorem would stand modified if we consider motion in an accelerated frame of reference. The following two problems are calculator / computer based Q.38 A 2 kg particle starts at the origin and moves along the positive x-axis. The net force acting on it measured at intervals of 1 m is 27.9, 28.3, 30.9, 34.0 34.5,46.9, 48.2, on the particle in this interval ? Q.39 A 0.5 kg particle moves along the x-axis from x = 5 m to x = 17.2 m under the influence of a force 200 F(x) F = N Estimate (with at least 5% 2x  x 3 accuracy) the total work done by this force during this displacement.