EXERCISES OF MECHANICS

Transcription

1 ROVANIEMI UNIVERSITY OF APPLIED SCIENCE SCHOOL OF TECHNOLOGY Degree Programm of Informational Technology EXERCISES OF MECHANICS Course 504D3A Jouko Teeriaho 2007

2 Kinematics 1. A stone is dropped from the top of a 250 m high building. After how many seconds does the stone hit the ground? What is the speed of the stone then? 2. A Ferrari starts to brake from 200 km/h speed with a retardation (deceleration) of 5.5 m/s 2. Calculate: a) the braking distance; b) the velocity after 100 m of braking. 3. A bullet is shot with an initial speed of 450 m/s to 30 o angle. Calculate: a) the ight time; b) maximum height; c) horizontal distance. 4. A boat steers to compass direction 40 o. Its speed with respect to water is 12 knots. The direction of the stream is to the West and its speed is 3.0 knots. Calculate the velocity of the boat with respect to the land. (Hint: compass directions North=0, East=90, South=180 and West=270) 5. A car drives with a speed of 80 km/h a distance of 50 km, keeps a break and continues with a speed of 100 km/h a distance of 70 km. How long was the break if the average velocity during the whole trip was 70 km/h? 6. A hot air balloon is rising at the speed of 5.0 m/s. At the height of 21 m above the ground a sandbag is dropped from the balloon. How long does it take for the sandbag to hit the ground? 7. Car A drives at the speed of 60 km/h and car B at the speed of 90 km/h to the same direction. Car B is 30 m behind A. Both cars start to brake at the same time when the drivers see a reindeer on the road. Both cars have similar tires and therefore equal retardation of -3.0 m/s 2. Calculate, if the car B hits car A on back or not. Reason your answer. 8. Calculate from the picture 1 velocity curve: a) the maximum acceleration value; b) the distance traveled between 0-6 s. 9. The speed of a car is 100 km/h. It starts to brake with retardation (negative acceleration) of -4.0 m/s 2. Calculate: a) the time needed to stop the car and the braking distance; b) the velocity after 40 m of braking. (pic.1) 10. What would be the initial speed in the example above, if the velocity after 60 m of braking is still 90 km/h? 11. A sportsman throws a ball in shot put competition with an initial speed of 24.0 m/s to an angle of 43 o. What is the result of the throw, if the initial height from the ground is 190 cm? 2

3 12. What is the initial speed of a hammer, if the result of the throw is m? The initial height is 180 cm and the initial angle is 44 o. 13. A body moves according to s(t) = 22.6 t 1.85 t t 3 (t is the time). Calculate the velocity of the acceleration of the body at t=3.0s. (Hint: velocity v(t)=s'(t) and acceleration a(t)=v'(t)) 14. A motor boat is sailing in a stream. The captain of the boat steers to the compass direction of 190 o. The stream ows at the same time to the compass direction 285 o. Calculate the ground velocity (magnitude and direction) of the boat, assuming that the speed of the boat with respect to the water is 15 knots and the speed of the stream is 2.5 knots. 15. An aeroplane ies at the height of 1000 m with the horizontal speed of 200 m/s. A packet is dropped from the aeroplane. Calculate the time, the place and the speed when the packet hits the ground. 16. A train leaves the railway station. It accelerates for 60 s and reaches the speed of 25 m/s. Then it travels with a constant speed for 10 min. Finally, it slows down and stops at the next station. Braking takes 60 s. Draw the velocity curve. Calculate the distance between two stations. 17. On the picture you can see the velocity curve of a distance swam during one cycle. Estimate maximum acceleration of the swimmer and the distance swam during one cycle. 18. A stone is thrown upwards from the top of a 200 m high building with initial speed of 8 m/s. Calculate: a) time when the stone hits the ground; b) the speed of the stone when it hits the ground. 19. An open outdoor lift is rising with a speed of 2 m/s at the height of 10 m. A coin is dropped from the lift. After how many seconds does the coin hit the ground? 20. A stone is thrown up with an initial speed of 17 m/s. After exactly one second another stone is thrown upwards. The other stone hits the rst stone at 11 m height. Calculate the initial speed of the second stone. (Hint: start by calculating at what time the rst stone is at 11 m height) 21. A stone is thrown upwards from 1.8 m height with an initial speed of 15 m/s. It hits a wall at the height of 10 m. What is the speed of the stone then? 3

4 Dynamics 22. Calculate the braking distance of a car from a speed of 100 km/h on the wet asphalt, where the coecient of friction is 0.40, if: a) the road is horizontal; b) the road is inclined at 6 o downwards; c) the road is inclined at 6 o upwards. 23. A truck is driving at the speed of 90 km/h. The front of the cabin is at.the trucks aerodynamical coecient is The height of the cabin is 3.0 m and the width 2.5 m. Calculate the value of the air resistance. 24. A spring can be used as a scale (for example, when measuring the weight of sh). Assume that the spring of a scale has a value of the spring coecient 600 N/m. How much is the length of the spring increased when a 5000 g sh is hanging at the end of the scale? 25. A child is sitting in a rocket of an amusement park. The rocket is going around a circle with the radius 5.0 m. The period (time spent on a one round) is 2.5 s. Calculate the centrifugal acceleration experienced by the child. Compare the result with g=9.81 m/s A 60 kg person jumps with a parachute. The area of the parachute is 40 m 2. Calculate the speed of the jumper during the landing. (Assume that the aerodynamic coecient of the parachute is 1.5) 27. A parachute jumper has a mass of 95 kg (parachute included). The aerodynamic coecient of the parachute is The nal speed of the jumper is 5.0 m/s. Calculate the area of the parachute. 28. A car drives over a top of a hill, which has a curvature (a radius) of 80 m. Calculate the maximum speed that the car can have without losing contact between the road and the tires. (pic.3) 29. A highway curve with a radius of the 200 m is not banked. Assume that the coecient of the friction between tires and dry asphalt is 0.65 and between tires and ice is Calculate the maximum safe speed traversing the curve on: a) dry days; b) icy days. 30. A highway curve (r=100 m) is planned for trac driving at the speed of 80 km/h. The curve is banked so that a car can travel at the curve without experiencing sidewise forces. Calculate the degree of banking (angle in degrees). 31. A body hanging at the end of a 120 cm rope is moving horizontally in a circle of radius 50 cm. Calculate the period and the speed of the motion. 32. Calculate the braking distance of a car driving at 120 km/h, if the coecient of friction is 0.40 (wet asphalt). 4

5 33. A space station is a 100 m long tube which rotates around its center. Calculate the period and the angular velocity of the space station, if an astronaut experiences a normal gravitation force at the end of the tube. 34. Calculate the braking time and braking distance of a car, which starts to brake at the speed of 100 km/h. The coecient of static friction is 0.60 between the tires and the road. The road is inclined down at 5 o. The car has ABS braking system, which uses static friction. 35. A force F acts on a system of two boxes with masses 2 kg and 5 kg. The acceleration of the system is 4.5 m/s 2. Calculate: a) the value of force F; b) the force between two boxes. (pic.4) 36. A car (2000 kg) pulls a caravan (1500 kg) on a horizontal road. When it speeds up the acceleration is 2.0 m/s 2. Calculate: a) the force between the road and the drive wheels; b) the pulling force between the car and the caravan. 37. A train has a locomotive (50 tn) and two wagons (10 tn each). Its maximum acceleration is 1.2 m/s 2. Calculate: a) the force between the drive wheels and the rails; b) the force between the two wagons; c) maximum acceleration of the train, if 3 extra (10 tn each) wagons are added to the train. (pic.5) 38. A car starts to brake from a velocity of 100 km/h. The coecient of friction is Calculate: a) the braking distance; b) the speed after 50 m of braking. Do the calculations based on energy principle. Gravitation 39. Calculate the period of a satellite, if the height of the orbit is 1200 km. 40. Calculate the mass of Moon from the following information: radius of Moon = 1737 km; acceleration due to gravitation of Moon = 1.62 m/s The period of Earth (the year of Earth) moving around Sun is 365 d. Calculate the length of the years in a) Venus; b) Mars; c) Mercury; 5

6 d) Calculate also the orbit velocity of Earth around Sun. P lanet M ass(kg) Radius(km) M ean distance f rom the Sun(km) Mars Earth V enus Mercury Calculate the value of the gravitation eld g (gravitational acceleration) on the surface of Moon. The mass of Moon is kg and its radius is 1739 km. 43. The international space station ISS travels now with a speed of 7352 m/s. Calculate a) the radius of its orbit; b) the height of the orbit; c) the period of its motion around Earth. 44. Calculate the velocity needed to leave our solar system starting from Earth's surface (the third cosmic velocity). The mass of Sun is kg. The average distance of Earth and Sun is 150 Gm. The mass and radius of Earth are kg and 6380 km. (Hint: the potential energy of a rocket is the sum of the two potential energies due to Earth and Sun. The kinetic energy of a rocket must exceed the absolute value of this sum.) 45. The mass of the planet Mars is kg and its radius is 6787 km. Calculate the velocity needed to leave the gravitational eld of Mars (so called the second escape velocity). 46. The mass of Earth is kg and its radius is 6380 km. A bullet is shot upwards with an initial velocity of 3000 m/s. How high does it rise? Work, Energy and Power 47. A crane in the port lifts up a load of 5000 kg with a speed of 50 cm/s. Calculate the power input of the motor of the crane, if its eciency is 65%. 48. Electric stairs bring people up from Helsinki metro station. Calculate the power consumption of the stairs during the rush hour if the stairs elevate 120 persons per minute up to the street level 30 meters above. Assume that the average weight of passengers is 70 kg and the eciency of the stairs is 65%. 49. Calculate the electric power produced by a wind mill, when the wind speed is 10 m/s and the radius of the rotor circle (wing length) is 15 m. The density of air is 1.25 kg/m 3. The eciency of the wind mill is 45%. 50. A Norwegian hydroelectric power plant is built on a small river. The water falls 400 m from the mountains to the turbines. The ow is 80 m 3 /s. Calculate: a) the produced electric power; b) the value of production of one day (unit price = 10 cnt/kw h). The eciency of turbines is 93%. 6

7 51. A hydroelectric power plant has an average water ow of 250 m 3 /s through turbines which transform mechanical energy to electric power with eciency of 90%. The dierence of water levels between the upper and the lower pools is 28 m. Calculate: a) the amount of the electric power the plant produces; b) the energy produced in one month (month = 30 days). 52. A wind power plant can transform 40% of the kinetic energy of the wind into the electric energy. The length of its rotor wings is 9.0 m. Calculate the power output of the windmill if the wind speed is 8.0 m/s. The density of air is 1.25 kg/m The air resistance on a car can be calculated with the formula F = 1 Cw ρ A v2 2 where Cw = the aerodynamic coecient of the car ρ = 1.25 kg/m 3 the density of air A = the front area of the car v = speed of the car A car drives at the speed of 25 m/s. Its front area is 2.5 m 2 and Cw = Calculate: a) the power output of the car at this speed; b) the maximum speed of the car if the power value given by the manufacturer is 50 kw. 54. Water is pumped up 6.0 m distance from a well. The ow is 100 liter/min and the speed of the water when it comes from the upper end of the water pipe is 5.0 m/s. Calculate the power input needed for the pump when its eciency is 70 percents. 55. A truck (mass 15 tn) is driving at the speed of 90 km/h. The aerodynamic constant of the truck is 0.40 and its front area is 6.0 m 2. Calculate: a) the air resistance of the truck; b) the power the truck uses to win the air resistance; c) the additional force the truck needs when it is driving up on a slope of 5.0 degrees. The density of air is 1.25 kg/m 3. Mechanics of Rotation 56. A solid ball rolls on a horizontal plane with a speed of 8.0 m/s. In front of the ball is a 3.0 m high hill. a) Does the ball have enough speed to roll over the hill? b) If it does not, how far (vertically) from the top of the hill does it goes? If it does, calculate the speed of the ball on the top of the hill. 57. Explain the principle of gyrocompass (its structure and why does it turn to the North). 58. Give 3 examples of applications of the conservation of angular momentum. 7

8 59. A person (m = 60 kg) stands on a horizontal bar at a point situated 2 m from its left end (pic.6). The length of the bar is 6 meters and its weight is 20 kg. The bar lays on two supports which are situated at both ends of the bar. Calculate the supporting forces acting on the bar. 60. Calculate the moment of inertia of a 5.0 m long, 20 kg bar with respect to an axis point P, which is inside the bar 1.5 m from the end. (Hint: use Steiner's rule that is explained in the lecture slides.) 61. A solid ball is thrown along the oor horizontally with an initial speed of 4.0 m/s. It starts to rotate gradually due to the gliding friction. The coecient of friction is How far from the start it rolls purely? The mass of the ball is 3.0 kg and its radius is 20 cm. 62. A solid ball (m = 0.2 kg,r = 0.2 m) starts to roll down from the rest along a surface inclined at 30 degrees to the horizontal. Rolling is not pure because of the small value of the coecient of static friction (µ = 0.05). Calculate the linear speed V and angular velocity ω of the ball when it rolled 10 m along the surface. (Hint: do not use energy principle, but rather Newton's laws : F=ma and T=Iα) 63. Calculate the rotational energy of Earth in rotation around its axis. Assume that Earth is a solid sphere with uniform density. Radius of Earth Mass of Earth P eriod of rotation m kg 24 hours = seconds 64. A solid ball starts rolling down from the top of a hill. The height of the hill is 8.0 meters. Calculate the nal speed of the ball when it reaches the bottom. 65. A gure skater is spinning around her axis with rotational frequency 2.0 RP S. She pulls her arms and legs closer to the body and manages to decrease the value of her moment of inertia to 35% of the original. What is her new frequency? (pic.7) 66. A solid wheel (mass 8.0 kg, radius 25 cm) rotates at 600 RP M. Is it stopped applying a 15 Newtons tangential force on the sphere of the wheel? a) Calculate the value of angular acceleration (deceleration). b) How long does it take for the wheel to stop? c) How many rounds does the wheel rotate while stopping? 67. The diameter of a bicycle wheel is 78 cm. Calculate the rotational frequency of the wheel in the speed of 12 m/s. 68. A ball with a radius of 15 cm rolls with 3.5 RP S rotational frequency and linear speed of 4.0 m/s. Is the rolling in this case pure rolling or does the wheel glide also? 8

9 69. Think of the following situation: you are standing with a rotating wheel in your hand. The spin vector of the wheel (=axis direction) points straight forward (pic.8). To what direction does the axis tends to if you try to turn the axis of the wheel: a) to the left; b) to the right; c) up; d) down. (pic.8) 70. A solid wheel of an engine has a mass of 40 kg and a radius of 30 cm. Its rotating frequency is 1200 RP M. How many Joules are stored to its rotation? 71. Calculate the nal velocities of the following bodies when they roll purely (starting from the rest) down at 4.0 m high hill. A - the hollow cylinder, B - the solid ball, C - the solid cylinder. 72. A car engine has maximum power output of 150 kw. The maximum rotational frequency is 5000 RP M. Calculate the torque the engine can create. 73. A solid ball rolls, and a block of ice glides down from the roof of a house (pic.9). Calculate their nal speeds when hitting the ground. The coecient of friction of between the roof and the ice is 0.2. (pic.9) 74. A wooden bar (20 kg, 3.0 m) falls down from the vertical position. Calculate the speed of its upper end when it hits the ground. (The lower end does not glide on the ground) 75. A man sits on a freely rotating chair and holds a rotating wheel in his hands (pic.10). Rotation axis in the beginning is horizontal and the spin vector points to the end B of the axis (B = right hand side). In the beginning the chair does not rotate. What happens when: a) the man forces the end B down; (nor completely, a few degrees is enough) b) the man forces the end B up. (pic.10) 76. A wheel is rotating so that the spin points to positive y - axis (pic.11). You try to pull the axis end B down. Where does the axis turn (alternatives: positive x, negative x, positive z, negative z - direction)? (pic.11) 9

10 77. A lorry has a mass of 15 tn. The distance between its front axle and back axle is 6.0 m. The center of mass of the lorry is 2.0 m behind the front axle. Calculate the axle weights of the lorry. Collisions and recoil problems 78. A canon bullet (6 kg) is shot with an initial speed of 500 m/s. During the ring the pipe of the canon (600 kg) gets a recoil speed backwards. The recoil motion is stooped after 0.80 m by the damping system. Calculate: a) the recoil speed of the canon pipe; b) the average damping force acting on the pipe. 79. A van (2500 kg) and a smaller car (1300 kg) collide on a road. At the moment of collision the speed of the van is 15 m/s and the speed of the car is 10 m/s to the opposite direction. Calculate the speeds of both cars just after the accident, if: a) collision is completely inelastic (both cars continue moving together after the collision); b) collision is partially elastic with coecient of restitution e= In a completely elastic collisions a ball with speed of 4 m/s and mass of 100 g hits another ball with mass of 200 g, which is originally at rest. Collision is central. Calculate the velocity of both balls just after the collision. 81. A rocket has a mass of 2000 kg and it moves with a constant speed in the space. To get more speed it starts its engine for 30 s. During that time 5 kg of exhaust gases escape from the rocket each second with relative speed of 800 m/s. Calculate, how much does the speed of the rocket increase. 82. A ball (mass 110 g, velocity 2.5 m/s) hits completely elastically another ball (mass 100 g, velocity 1.0 m/s to the opposite direction). Calculate the velocity of the balls after the collisions. The collision is central. 83. A stone is thrown upwards with initial speed of 15 m/s. Another stone is thrown up 1.0 s later with initial speed of 20 m/s. At which height the two stones collide with each other? 84. A bullet (15 g) is shot horizontally into a block of wood (600 g), which is hanging in the end of a 300 cm weightless rope. The collision is fully inelastic. Calculate the velocity of the bullet, if the rope makes an angle of 45 o to the vertical when it is at the highest position. 85. Two railway wagons with masses 15 tn and 20 tn come from the opposite directions with velocities 2.0 m/s and 1.0 m/s respectively. In the contact they connect and continue together. a) What is the speed of the two wagon system after the collisions? b) How many percents of the kinetic energy is lost in the collision? 86. A car of mass 1500 kg hits with a velocity of 60 km/h a reindeer (mass 200 g). The collision is linear and the coecient of restitution is Calculate the speed of the car and the reindeer just after the collision. 87. A ball with mass 200 g and velocity 3.0 m/s hits completely elastically another ball (mass 250 g), which is at rest. Calculate the velocity of the ball after the collision. The collision is central. 10

11 88. A canon's pipe has a mass of 600 kg and length of 200 cm. A 6 kg bullet is shot at the speed of 700 m/s. Calculate: a) the recoil speed of the canon; b) the force needed to stop the recoil motion, if the pipe moves 70 cm backwards during the braking. 89. A bullet (15 g) is shot with a speed of 400 m/s horizontally into a piece of wood (200 g), which lies on the ice. When the bullet hits the wood, it (bullet inside) starts to glide along the ice and stops after 200 m of gliding. Calculate the coecient of friction between wood and ice. Fluid Mechanics 90. A body weights 50 g in the air, 42 g in the water. What is the density of the body? The density of water is 1000 kg/m A body oats on the water. 30% of its volume is under the water. Determine the density of the body. 92. Determination of the density of unknown liquid: weight of a body in the air is m 1 = 80 g, weight of the body in the water is m 2 = 20 g, weight of the body in liquid is m 3 = 15 g. What is the density of the liquid ρ? 93. What is the absolute (total) pressure in 800 m depth in the ocean? The density of sea water is 1030 kg/m 3 and assume that it decreases linearly to zero. 94. Estimate the height of the atmosphere h using the fact that the air pressure on sea level is 1.25 kg/m 3 and assuming that it decreases linearly to zero. 95. The width of a river is 230 m and the average depth is 280 cm. The speed of the stream is 1.8 m/s. Calculate the ow in the river (unit m 3 /s). 96. A level of water in a water tank is at the height of 300 cm. Water is taken form a tap, which is situated at 40 cm height. Calculate the speed of the water coming out from the tap. 97. During a storm the wind speed is 25 m/s. Calculate the suction force of the storm on a horizontal roof with area of 30 m A Formula 1 car has a small back wing, where the speed of air under the wing is 10% greater than above it, where it is same as the speed of the car. The area of the wing (one side) is 0.40 m 2. Calculate the downforce due to the wing at the speed of 270 km/h. 99. A Pitot tube in a submarine shows pressure 60 mmhg. What is the speed of the submarine? 100. A Pitot tube in an old aircraft shows pressure 60 mmhg. what is the speed of the aircraft? 11

12 Kinematics s, 62.6 m/s; 2. a)281 m b)160 km/h; 3. a)45.9 s b)2580 m c)17900 m; knots; min; s; 8. a)2 m/s 2 b)8 m; 9. a)6.9 s b)21.2 m/s; knots, compass direction o ; s, 2860 m ahead, velocity components=(200 m/s, -140 m/s), the speed=length of the velocity vector=244 m/s; km; 18. a)7.3 s b)63.2 m/s; s; m/s; m/s. Dynamics 22. a)98 m b)134 m c)78 m; kn; cm; m/s 2 (a little over 3 g); m/s; m 2 ; km/h; 29. a)35.7 m/s b)22.1 m/s; o ; s, 1.4 m/s; m; s; s, 57 m; 35. a)31.5 N b)22.5 m; 36. a)7 kn b)3 kn; 37. a)84 kn b)12 kn c)0.84 m/s 2 ; 38. a)98 m b)19.5 m/s. Gravitation min; kg; 41. a)225 d b)686 d c)87 d d) km/h; m/s 2 ; 43. a)7404 km c)1.76 h; km/s; km/s; km. Work, Energy and Power kw ; kw ; kw ; MW ; 51. a)62 MW b)160 T J=44 GW h; kw ; 53. a)8.5 kw b)162 km/h; W ; 55. a)938 N b)23.4 kw c)12.8 kn. Mechanics of Rotation 56. a)yes b)4.7 m/s; 59. N2=294.3 N, N1=490.5 N; Nm; m; J; m/s; RP S; 66. a)8.0 rad/s 2 b)4.2 s c)21 rounds; RP S; 68. rolling is not pure, because ω r<linear velocity; kj; m/s, 7.5 m/s, 7.2 m/s; Nm; 73. ice 8.4 m/s, ball 8.2 m/s; m/s; 75. a)the chair starts to rotate left b)the chair starts to rotate right; 76. negative x; tn (front axle) and 5 tn (back axle). Collisions and recoil problems 78. a)5 m/s b)9.5 kn; 79. a)6.4 m/s b)3.0 m/s and 13 m/s; m/s, 2.7 m/s; m/s; m/s, 2.67 m/s; m; m/s; 85. a)0.29 m/s b)96.5%; km/h; m/s, 2.67 m/s; 88. a)-7 m/s b)21 kn; Fluid Mechanics kg/m 3 ; kg/m 3 ; kg/m 3 ; MP a=81.8 bar; m; m 3 /s; m/s; kn; N; m/s; m/s. 12