Worksheet 1 Horizontal Circular Motion 1. Will the acceleration of a car be the same if it travels Around a sharp curve at 60 km/h as when it travels around a gentle curve at the same speed? Explain. 2. Sometimes it is said that water is removed from clothes in a spin dryer by centrifugal force throwing the water outward. Is this correct? 3. When a figure skater goes into a tight axel spin, her skirt seems to fly outward. How can this be if the force acting is directed towards the center? 4. For each of the following cases of bodies moving in a circle, state what force or forces act on the body towards the center of the circle: a) a car going round a bend b) a train on a curved track c) the moon orbiting the earth 5. The orbit of the Earth round the Sun is an ellipse, but a good approximation is to take it as a circle of radius1.496 x 10 11 m. At what speed is the Earth moving in its orbit round the Sun? (2.979 x 10 4 m/s) 6. A roller coaster has a loop-de-loop with a radius of 9.00 m. If the coaster cars go around the loop in 5.00 s, what is the average velocity of the cars around the loop? (11.3 m/s) 7. A jet plane travelling at 1800 km/h pulls out of a dive by moving in a circular arc of radius 3.00 km. What is the plane's acceleration? How many time the gravitational field strength at earth is this? (83.3 m/s 2-8.49 g's) 8. A mass of 2.0 kg at the end of a 1.5 m string experiences a centripetal acceleration of 2.00 m/s 2. What is the velocity of the mass at the end of the string? (1.7 m/s) 9. A 150 g ball at the end of a string is swinging in a Horizontal circle of radius 1.15 m. The ball makes exactly 2.00 revolutions in 1.00 s. What is the centripetal acceleration?(182 m/s 2 ) 10. A mass of 50.0 g is swung horizontally at a velocity of 150 cm/s at the end of a string which is 40.0 cm long. Calculate the centripetal force in Newtons. (0.281 N) 11. A 70.0 kg boy on an amusement park ride with radius of 9.50 m experiences a centripetal acceleration of 8.00 m/s 2. What is the velocity of the boy as he travels a circular motion? (8.72 m/s) 12. A rotating circular space station produces an artificial centripetal gravity of 9.9 m/s 2. If the rotational velocity of the space station is 25 m/s, what is the diameter of the space station? (1.3 x 10 2 m) 13. Describe what would happen if the astronaut dropped a 1.00 kg mass. Would it "fall", or hover in mid-air? To support your answer, prepare diagrams showing the velocity vectors of the space station and the mass just prior to and after its release. 14. Determine the size of the centripetal force acting on a 1000 kg sports car rounding a curve of radius 100 m at a speed of 30 m/s without skidding? (9.0 x 10 3 N) 15. A stunt pilot is flying his 2500 kg aircraft in a tight circular path of radius 250 m. If he is flying with a maximum speed of 500 km/h, determine the maximum centripetal force acting on his aircraft. (1.93 x 10 5 N) 16. A horizontal force of 26 N is applied to a 0.60 kg stone to keep it rotating in a horizontal circle of radius 0.40 m. Calculate its speed. (4.2 m/s) 17. A 2.00 kg mass is swung on a horizontal frictionless surface at the end of a wire which is 0.700 m long. If it makes 2.00 revolutions per second, calculate: a) the linear speed of the mass (8.80 m/s) b) its acceleration (111 m/s 2 ) c) the centripetal force (221 N) S Molesky @ Notre Dame 1
d) if the force needed to beak the wire is 880 N, determine the maximum speed the mass can be whirled. (17.5 m/s) 18. A car of mass 750 kg is taking a corner with a flat horizontal surface and a radius of 10 m. If the sideways friction forces cannot exceed 1/10 of the weight of the vehicle, what is the maximum speed at which it can take the bend. (3.13 m/s) 19. How fast can a 1.6 x 10 3 kg car round an unbanked curve (radius 55 m) if the frictional force between the car and the road is 9.42 x 10 3 N? (18 m/s) 20. How large must the force of friction be between the tires and the road if a 1500 kg car is to round a level curve of radius 62.0 m at a speed of 55.0 km/h? (5.65 x 10 3 N) 21. A 10.0 g coin is placed 18.0 cm from the axis of rotating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 58 rpm (revolutions per minute) is reached, at which point the coin slides off. What is the force of friction between the coin and the turntable? (0.066 N)) 22. A rock is attached to a cord and then swung in a circular arc. If the length of cord is halved and the rock swung at three times the previous speed as before, how will the tension in the rope change? (18 x the old force) 23. An object which has a mass of 100 g is swung in a Horizontal circle at 4 rev/s at the end of a chord which is 28.0 cm long. Calculate: a) its circular speed (7 m/s) b) its centripetal acceleration (2.0 x 10 2 m/s 2 ) c) its centripetal force (2 x 10 1 N) Worksheet 2: Curve Banking 1. For a vehicle taking a bend at speed v, the banking of the curve to eliminate side slip (any tendency to skid) is given by Tan = v2 gr where ø is the angle between the road surface and the horizontal. British Rail's new high-speed trains are designed to travel at 200 km/h. a) Calculate the banking needed on a track of radius 0.25 km if the speed is to be sustained without the train trying to push the rails out of place. (52 ) b) What will passengers feel if the train takes this banked curve at speeds below 200 km/h? c) What will passengers feel if the train takes the curve at speeds above 200 km/h? 2. An airplane travelling at a speed of 115 m/s makes a complete 180 horizontal turn in 1.20 x 10 2 s. What is the banking angle? (17.1 ) 3. Calculate the angle at which a frictionless curve must be banked if a car is to round it safely at a speed of 22 m/s The radius of the curve is 475 m. (5.9 ) 4. A car travels around a curve (radius = 60.0 m) at a speed of 22.0 m/s. At what angle must the curve be banked so that the car does not have to rely on friction to remain on the road? (39.4 ) 5. What is the maximum speed of a car rounding a curve or radius 125 m in the highway under very icy conditions if the banking angle is 20.0? (21.1 m/s) 6. A car is rounding a curve, in the highway, of radius 515 m. If he curve is banked at an angle of 12.0, what is the maximum speed of the car? (32.8 m/s) 7. An engineer is to design a curved exit ramp from a freeway for traffic with a maximum speed of 20.0 m/s. If she decides to bank the curve at an angle of 20.0, at what radius of curve must it be built if the vehicles are not to rely on friction? (1.12 x 10 2 m) 8. An airplane flying at a speed of 205 m/s makes a complete horizontal turn while banking at 29.0, What is the radius of the turn? (7.73 x 10 3 m) 9. A 60.0 kg child is riding the swing ride at the fair in a circle of radius 6.00 m. If the angle the swing makes with the vertical is 25 o, how fast is the child moving? (5.24 m/s) 10. A 40.0 kg child is on the swing ride and moving in a circle of 5.00 m. If her speed is 10 m/s, what angle does she make to the vertical, and what is the tension in the swing cable? (63.9 o ) Worksheet 3: Vertical Circular Motion 1. You are riding a bike on a track that forms a vertical circular loop. If the diameter of the loop is 10.0 m, how fast would you have to be moving when you reach the top of the loop so that you will stay on the track? (7.00 m/s) 2. You are rotating a bucket of water in a vertical circle. Assuming that the radius of the rotation of the water is 0.95 m, what is the minimum velocity of the bucket at the top of its swing if the water is not to spill? (3.1 m/s) 3. An amusement park ride spins in a vertical circle. If the diameter of this ride (circular) is 5.80 m, what minimum speed must the ride travel so that a 75.0 kg student will remain against S Molesky @ Notre Dame 2
the wall when he is in the highest position of the ride? (5.33 m/s) 4. You are riding a bike (total mass = 95.0 kg) over a rise (radius of curve = 10.0 m) on a bike path. How fast must you be travelling so that your bike loses contact with the path? (9.9 m/s) 5. A 915 kg car goes over a hill. If the effective radius of the top of the curve of the hill is 43 m, how fast must the car travel so that it exerts no force on the road at the crest? (21 m/s) 6. A student has a weight of 655 N. While riding on a roller coaster this same student has an apparent weight of 1.96 x 10 3 N at the bottom of a dip that has an effective radius of 18.0 m. What is the speed of the roller coaster at this position? (18.8 m/s) 7. A string requires a 186 N force in order to break. A 1.50 kg mass is tied to this string and whirled in a vertical circle with a radius of 1.90 m. What is the maximum speed that this mass can be whirled without breaking the string? (14.7 m/s) 8. An object of mass = 50.0 g is swung in a vertical circle of radius = 1.10 m using a cord that will break if it is subjected to a force greater than 262 N. What is the maximum speed that this mass can travel as it passes through the bottom of its circle? (75.8 m/s) 9. A 2.2 kg object is whirled in a vertical circle whose radius is 1.0 m. If the time of one revolution is 0.97 s what is the tension in the string (assume uniform speed) a) When it is at the top? (71 N) b) When it is at the bottom? (1.1 x 10 2 N) 10. A 2.5 kg ball is tied to a 0.75 m string and whirled in a vertical circle (assume a constant speed of 12 m/s) a) Why is the tension in the string greater at its low point than at its high point? b) Calculate the tension in the string at its low point. (5.0 x 10 2 N) c) Calculate the tension in the string at its high point. (4.6 x 10 2 N) 2. When a man tries to bend horizontally at his waist, he falls forward. He can make the same bend however when he has a pair of skis on. Explain 3. Brian stands on the plank as shown and does not fall off the edge. Explain how this is possible. 4. If the boy at right is 80 kg and is located 1.5 m from the fulcrum, where should the hanging boy who is 55 kg be located? What will happen if he chooses a distance greater than this from the fulcrum? (consider the board to be mass-less) (2.2 m) 5. A 10.0 kg mass is placed 0.45 m from the fulcrum along a 4.00 kg board, of uniform composition, which is 2.00 m long. Where should a 2.00 kg mass be placed so that the torques balance? (1.15 m) 6. How large a stone can Sam move using a force of 80 N and a plank 2.5 m long with a fulcrum located 30 cm from the stone? (59.8 kg) Worksheet 4: Torque and Angular Momentum 1. Why do the two forces result in equal but opposite torques? 7. What is the moment of inertia, and how does it differ between an ordinary bicycle tire and the one which was used in the classroom demonstration? S Molesky @ Notre Dame 3
8. Using the conservation of angular momentum, explain why a gyroscope will tend to keep rotating in the same direction and may thus used for keeping a space ship on course. 9. Suppose the ice cap located at the South Pole of the earth melted and the water was distributed uniformly over the earth's oceans. Would the earth's rotational velocity increase, decrease, or stay the same? Explain. 10. A cloud of interstellar gas is rotating. Because the gravitational force pulls the gas particles together, the cloud shrinks, and, under the right conditions, a star may ultimately be formed. Would the angular velocity of the star be less than, equal to, or greater than, the angular velocity of the rotating gas? Justify your answer. Worksheet 5: Kepler s Laws Solar Data radius of the sun = 696.0 x 10 6 m mass of sun = 1.991 x 10 30 kg mass of moon = 7.35 x 10 22 kg radius of the moon = 1.74 x 10 6 m average radius of moon's orbit = 3.87 x 10 8 m Radius of Mass of Mean Distance Planet(m) Planet(kg) to Sun (m) mercury 2.43 x 10 6 3.2 x 10 23 5.80 x 10 10 venus 6.073 x 10 6 4.88 x 10 24 1.081 x 10 11 earth 6.37 x 10 6 5.98 x 10 24 1.4957 x 10 11 mars 3.38 x 10 6 6.42 x 10 23 2.278 x 10 11 jupiter 69.8 x 10 6 1.901x 10 27 7.781 x 10 11 saturn 58.2 x 10 6 5.68 x 10 26 1.427 x 10 12 uranus 23.5 x 10 6 8.68 x 10 25 2.870 x 10 12 neptune 22.7 x 10 6 1.03 x 10 26 4.500 x 10 12 pluto 1.15 x 10 6 1.2 x 10 22 5.9 x 10 12 1. A small ball tied to a string and twirled at constant speed in a circular path obeys Kepler's second law of planetary orbits. (The line joining the sun and a planet traces equal areas in equal intervals of time). If the string is broken and the ball is allowed to fly away, will it continue to obey the law? Explain. 2. The comet Hale-Bopp is travelling around the sun in a highly elliptical orbit. Compare the speed of the comet when it is closest to the sun to when it is farthest from the sun. 3. A comet travels with a speed of 190 km/s when it is 3.4 x 10 5 km from the sun. How fast will it be moving when it is 3.4 x 10 7 km from the sun? (19 km/s) 4. A comet travels at 120 km/s when it is closest to the sun, 1.5 x 10 7 km away. How fast will it be traveling when at its farthest point, 1.5 x 10 9 km away from the sun? (12 km/s) 5. The earth is traveling with a speed of 29.8 km/s when it is 1.4957 x 10 11 m from the sun. How fast will it be moving when it is 1.51 x 10 11 m from the sun? (29.7 km/s) 6. Pluto is traveling with a speed of 4.7 km/s when it is 5.9 x 10 12 m from the sun. How fast will it be moving when it is 5.5 x 10 12 m from the sun? (4.9 km/s) 7. Use Kepler's laws and the period of the moon (29.5 days) to determine the period of an artificial satellite orbiting near the earth's surface. (take the radius to be R E) (1.50 h) 8. If a satellite of the sun has an orbital radius of twice that of the earth, how does its period of revolution compare with that of the earth? (2.83 a). 9. The asteroid Icarus, though only a few hundred meters across, orbits the sun like the other planets. Its period is about 410 days. What is its average distance from the sun? (1.62 x 10 11 m) 10. Use Kepler's third law to estimate the orbital radius of a geostationary satellite (a satellite which orbits the earth, and remains directly above a fixed point on the S Molesky @ Notre Dame 4
equator), given that a satellite in near-earth orbit has a period of approximately 90 minutes. (4.04 x 10 7 m) 11. Pluto makes one complete orbit of the sun in 248.4 a. What is the average radius of Pluto's orbit in metres? (5.91 x 10 12 m) 12. Use Kepler's third law to determine at what height above the earth a geostationary satellite must orbit if it is to remain at the same place relative to earth? (For earth orbits, is the Kepler constant) (3.6 x 10 7 m) 13. Use the earth's average radius about the sun, 1.4957 x 10 11 m,and the length of a year in seconds to determine the Kepler constant, system. (2.98 x 10-19 s 2 /m 3 ), for the solar Worksheet 6: Newton's Gravitational Law 1. State Newton's law of gravitation. Explain what is meant by the statement that G is a universal constant. 2. Calculate the force of attraction of a 75.0 kg boy separated by: a) 0.500 m from a 65.0 kg girl. (1.30 x 10-6 N) b) Is the attraction between the two less on the moon? 3. The gravitational force between two objects that are 2.1 x 10-1 m apart is 3.2 x 10-6 N. If the mass of one object is 5.5 x 10 1 kg, what is the mass of the other object? (39 kg) 4. Complete the sentences: a) When the mass of one of the objects is doubled the gravitational force. b) When the distance between two objects is doubled the gravitational force will be. c) When the mass of one of the objects is tripled the gravitational force. d) When the distance between two objects is tripled the gravitational force will be. e) The more influential factor in determining gravitational attraction is. 5. What is the gravitational attraction between the earth and the moon? (1.90 x 10 20 N) T R 2 3 6. What is the radius of a planet where a g = 13.2 m/s 2 if the mass of the planet is 4.96 x 10 24 kg. (5.01 x 10 6 m) 7. The gravitational attraction between two identical spheres is 1.00 x 10-8 N. What will happen to the gravitational attraction if the distance between the spheres is tripled and the masses are each doubled? ( 4.44 x 10-9 N) 8. Find the weight of a body whose mass is 35.0 kg (343 N) 9. If your mass is 50.0 kg, what is your weight on the surface of Jupiter where a g = 26 m/s 2? (1.3 x 10 3 N) 10. On a certain planet, the acceleration due to gravity is only 1/7 th its value on Earth's surface. What would be the weight of an object of mass 4.9 kg on such a planet? (6.9 N) Worksheet 7: Gravitational Field Intensity 1. The mass of Mars is 6.37 x 10 23 kg and its radius is 3.43 x 10 6 m. Calculate the acceleration due to gravity at the surface of Mars. (3.61 m/s 2 ) 2. Sketch a graph showing the relationship between gravitational field intensity (vertical axis) and the distance from an attracting body. 3. Why is the gravitational field at the earth's south pole greater than the gravitational field at the north pole?) 4. A 3.0 x 10 2 kg satellite is in orbit about a star. If the orbital radius is 3.0 x 10 10 m and the gravitational force on the satellite is 4.5 x 10 1 N, find a) the gravitational field intensity of the star at this point. (0.15 N/kg) b) the mass of the star (2.0 x 10 30 kg) 5. The planet Jupiter has a mass of 1.9 x 10 27 kg and an average radius of 7.2 x 10 7 m. Determine a) the gravitational field intensity at the planet's surface. (24 N/kg) b) the weight of a 70.0 kg boy on Jupiter. ( 1.7 x 10 3 N) 6. An object (mass = 525 kg) is 3.0 x 10 3 km above the earth's surface. This object is falling toward the earth because of the earth's gravitational force on it. What is the rate of acceleration at this distance? (4.54 m/s 2 ) 7. Calculate the gravitational field strength at a point in space where the weight of the object is 7.22 x 10 2 N and its mass is 1.10 x 10 2 kg. (6.56 m/s 2 ) S Molesky @ Notre Dame 5
8. The gravitational field at a point 1.5 x 10 3 km above a planet is 7.5 N/kg. If the radius of the planet is 6.0 x 10 6 m, what is the mass of the planet? (6.32 x 10 24 kg) 9. Determine the mass of the earth using the gravitational field value, at the earth's surface, of 9.81 m/s 2, and the approximate radius of the earth of 6.37 x 10 6 m. ( 5.97 x 10 24 kg) 10. At what distance from the Earth's surface is the gravitational field strength 7.33 N/kg? (1.01 x 10 6 m) 11. At a distance of X earth radii from the surface of the earth, the acceleration due to gravity is 1/2 the value at the surface of the earth. What is the value of X? ( 9.01 x 10 6 m 0.42 radii above the earth) 12. If the gravitational field is 9.81 N/kg at the surface, how far from the earth's surface will the field have one third the value at the surface? (4.67 x 10 7 m) 13. The gravitational field at the north pole is 9.832 N/kg and at the south pole it is 9.900 N/kg. Calculate the ratio of the north pole radius of the earth to the south pole radius. (1.003 : 1) Worksheet 8: Orbital Motion 1. Calculate the speed required for a satellite moving in a circular orbit 200 km above the earth's surface. (7.79 x 10 3 m/s) 2. Calculate the speed of a satellite moving in a stable circular orbit about the earth at a height of 3200 km. (6.46 x 10 3 m/s) 3. One of the moons of Jupiter discovered by Galileo has a rotational period of 1.44 x 10 6 s and it is 1.88 x 10 9 m from the surface of Jupiter on the average. From these data, determine the mass of Jupiter. (2.12 x 10 27 kg) 4. Determine the mass of the sun using the known value of the period of the earth and its distance from the sun of 1.496 x 10 11 m. (1.99 x 10 30 kg) 5. How long would a day be if the earth were rotating so fast That objects at the equator would appear weightless? (hint: they would be in orbit. (1.40 h) 6. Calculate the time of one revolution (length of a year) on Mars using the mass of Sun, and the radius of Mars orbit. (5.928 x 10 7 s or 1.878 a) 7. The moon's nearly circular orbit about the earth has a radius of about 3.87x 10 5 km and a period of 29.5 days. Determine the mass of the earth from this data. (5.28 x 10 24 kg) 8. The moon Titan orbits the planet Saturn with a period of 1.4 x 10 6 s. The average radius of this orbit is 1.2 x 10 9 m. a) What is Titan's centripetal acceleration? (2.4 x 10-2 m/s 2 ) b) Calculate Saturn's mass. (5.2 x 10 26 kg) 9. How fast would the earth be rotating if objects at the equator appeared to be weightless? (7.91 x 10 3 m/s) 10. Evaluate the constant for the solar system in the formation of Kepler's third law, using the information for earth and for then calculate it again using the information form Mars. ( k = 2.98 x 10-19 s 2 /m 3 ) Worksheet 9: Satellites k = 1. If you are required to place a satellite in orbit above the earth, what is the single most important factor you must achieve for your satellite? Does this mean all satellites require the same force to place them in orbit? 2. You are in command of a space ship in orbit around the earth. You want to catch up with a satellite which is several hundred kilometres ahead of you in the same orbit. Describe the maneuver(s) you would carry out in order to rendezvous with the satellite. 3. Where is the most economical place on the earth to Launch a satellite from. Explain with the aid of a diagram. 4. In which direction should a satellite be placed in orbit around the earth to use minimal energy? 5. It is often said that astronauts in orbit around the earth are weightless. Are they really weightless? 6. Why does a satellite burn up when its orbit decays? 7. State three uses for low orbit earth satellites. 8. Why are some satellites placed in a low-earth polar orbit? 9. What is different about the SOHO satellite and why was It placed in that orbit? 10. What is a geosynchronous orbit and what type of satellite will usually be placed in such an orbit? 11. What does GPS stand for? How many satellites does it use in total, and what type of orbit are they in? T R 2 planet 3 planet S Molesky @ Notre Dame 6