ANGULAR VELOCITY. 1. A wheel rotates three times. Through what angular displacement [in radians] did this wheel rotate?
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1 PHYSICS HOMEWORK #61 ANGULAR VELOCITY 1. A wheel rotates three times. Through what angular displacement [in radians] did this wheel rotate? 2. Through what angular displacement does the earth turn in 24 hours? 3. According to the tachometer in a car the engine is rotating at 3500 RPM [revolutions per minute]. Through what angle will this engine rotate each minute? 4. The Milky Way Galaxy rotates once each 200,000,000 years. Through what angle will this galaxy rotate over a period of 2,000 years? 5. A spinning wheel rotates 47 times in 12 seconds. What is the angular velocity of the wheel? 6. What is the angular velocity of the Earth s orbit around the sun? 7. Jupiter rotates once on its axis every 10 hours. What is Jupiter s angular velocity? 8. An ice skater going into a spin rotates with an angular velocity of 27 Rad/sec. How many revolutions will the skater make each second? 9. A motor is rotating with an angular velocity of 285 Rad/sec. What is the angular velocity of this motor in RPM s? 10. A wheel, which has a radius of 14 cm., is rotating with an angular velocity of 25 Rad/sec. What will be the linear velocity of a point on the outer edge of the wheel? 11. An automobile is moving with a velocity of 22 m/s. The tires on the car have a diameter of 78 cm. What will be the angular velocity of the wheels on this car? 12. What is the angular velocity of the Earth s rotation? 13. What will be the linear velocity of someone standing at the equator of the Earth, assuming that the diameter of the Earth is 7926 miles? 14. Suppose that a wheel, which is rotating with an angular velocity of 14 Rad/sec., is accelerated to an angular velocity of 22 Rad/sec. over a period of 5.0 seconds. What is the rate of acceleration of this wheel? 15. Suppose that the angular velocity of a spinning ball fell from 27 Rad/sec to 6.0 Rad/sec over a period of 2.3 seconds. What is the angular acceleration of this ball? 16. A disc is accelerated from rest at a rate of 4.3 Rad/sec2 for a period of 12 seconds. What will be the final angular velocity of this disc? 17. A wheel rotates with a constant angular velocity of 34 Rad/sec for a period of 12 seconds. What will be the final angular displacement of this wheel? Answers to opposite side: x 10-8 Rad/s kg m kg m kg m x kg m Nm Rad/s Rad/s 2 26a Nm b kg m 2 26c Rad/s 2 d Rad/s e m/s f Rad/s g. 206 Rad h J i J 26j kg m/s k Watts l Rad/s m. 266 Rad n J
2 PHYSICS HOMEWORK #62 MOMENT OF INERTIA 18. What angular acceleration would be required in order to stop the Earth from rotating over a period of 30.0 minutes? 19. What is the moment of inertia of a wheel shaped like a ring, which has a radius of 34 cm and a mass of 4.5 kg.? 20. What is the moment of inertia of a 2.5 kg. sphere which has a radius of 6.0 cm.? 21. What is the moment of inertia of a disc which has a mass of 4.4 kg and a radius of 45 cm.? 22. The Earth has a mass of 5.98 x kg and an average radius of 6.37 x 10 6 meters. Assuming that the Earth is a uniform sphere [which it is not!] what is the moment of inertia of the Earth? 23. A force of 80.0 Newtons is tangentially applied to the outer edge of a wheel which has a radius of 30.0 cm. How much torque is being applied? 24. A torque of 100. Nm is applied to a sphere which has a moment of inertia of 400 kg m 2. What will be the resulting angular acceleration of the sphere? 25. A torque of 30.0 Nm is applied to a ring which has a moment of inertia of 2.81 kgm 2. What will be the resulting angular acceleration of this ring? 26. Consider a wheel, which is shaped like a ring, has a mass of 8.2 kg and a diameter of 74 cm. A tangential force of 50.0 N is applied to the outer edge of the wheel for 5.0 seconds. The wheel is initially at rest. a. What is the magnitude of the torque being applied to the wheel? b. What is the moment of inertia of this wheel? c. What will be the resulting angular acceleration? d. What will be the angular velocity of the wheel after 5.0 seconds? e. What will be the linear velocity of a point on the outer edge of the wheel after 5.0 seconds? f. What will be the average angular velocity of this wheel during this 5.0 second interval? g. What will be the total angular displacement of this wheel during the 5.0 second interval? h. What is the total work done on the wheel during the 5.0 second interval? i. What will be the final kinetic energy of this wheel at the end of the 5.0 second interval? j. What will be the angular momentum of this wheel at the end of the 5.0 second interval? k. What is the power being applied to the wheel when t = 5.0 seconds? Suppose that the wheel has an initial angular velocity of 12.0 Rad/s before the external torque has been applied. l. What will be the angular velocity of the wheel at the end of 5.0 seconds? m. What will be the total angular displacement of this wheel during the 5.0 second period? n. How much work will be done on this wheel during the 5.0 second period? Answers to opposite side: Rad Rad 3. 22,000 Rad x 10-5 Rad Rad/s x 10-7 Rad/s x 10-4 Rad/s Rev RPM m/s Rad/s x 10-5 Rad/s miles/s Rad/s Rad/s Rad/s Rad
3 PHYSICS HOMEWORK #63 ANGULAR ACCELERATION 27. A uniform disc, which has a mass of 3.50 kg and a radius of 28.0 cm, is accelerated from rest by a tangential force of 28.0 N applied to the outer edge of the disc. a. What is the moment of inertia of this disc? b. What is the magnitude of the torque being applied to this disc? c. What will be the angular acceleration of this disc? d. What will be the angular velocity of this disc after 8.00 seconds? e. What will be the linear velocity of a point on the outer edge of the disc at the end of 8.00 seconds? f. What will be the angular displacement of this disc during the 8.00 seconds period? g. What will be the linear displacement of a point on the outer edge of this disc during the 8.00 second period? h. How much work was done on this disc during the 8.0 seconds time period? i. What will be the final angular momentum of this disc? j. What is the final angular kinetic energy of this disc? 28. Consider a ball [shaped like a solid sphere], which has a mass of 4.80 kg and a radius of 18.0 cm, and which is rolling along a horizontal surface with a velocity of 7.20 m/s when it encounters a hill, rolls to the bottom of the hill and through a loop-the-loop as shown to the right. The hill has a height of h = 5.60 m and the loop has a radius of 2.20 m. a. What is the total energy of this ball as it rolls along the top of the hill? b. What will be the total energy of this ball when it reaches point B? c. What will be the velocity of this ball when it reaches point B? d. What will be the total energy of this ball when it reaches point A? e. What will be the kinetic energy of the ball when it reaches point A? f. What will be the velocity of this ball when it reaches point A? g. Is this ball going fast enough to make it through the loop? 29. A disc, which has a mass of 12.0 kg and a radius of 65.0 cm., sits at the top of an inclined plane, which is 8.40 meters long and 1.50 meters high. At t = 0 the disc is released and is allowed to roll to the bottom of the incline without slipping. a. What is the GPE of this disc as it sits at the top of the incline? b. What will be the total kinetic energy of this disc as it reaches the bottom of the incline? c. What will be the linear velocity of this disc when it reaches the bottom of the inclined plane? d. What would the linear velocity be if the disc is replaced by a sphere? e. What would the velocity be if the object was a ring? 30. Consider a uniform sphere, which has a mass of 4.80 kg and a radius of 22.0 cm. A tangential force of 11.2 N is applied to the outer edge of this sphere: a. What is the moment of inertia of this sphere? b. What is the magnitude of the applied torque? c. What will be the resulting angular acceleration? d. That will be the angular velocity of this sphere after 3.50 seconds? e. What will be the total angular displacement of this sphere after 3.50 seconds? f. What will be the total work done on the sphere by the tangential force? g. What will be the angular kinetic energy of this sphere after 3.50 seconds? h. What will be the magnitude of the angular momentum of this sphere after 3.50 seconds? Answers to opposite side: 31a Rad/s b. 156 J c J d. 219 J e m f sec 31g m/s 2 h Rad/s 2 i kg m 2 j N m k. 219 J 32a Rad/s 2 b sec 32c Rad d J e J 33a kg m 2 /s b. counter-clockwise c. 105 kg m 2 /s 33d. 101 Rad/s e J
4 PHYSICS HOMEWORK #64 ANGULAR MOMENTUM CONSERVATION 31. A sphere, which has a mass of 2.10 kg and a radius of 35.0 cm., is rolling along a horizontal surface with a velocity of 12.2 m/s when the sphere encounters an inclined plane which meets the horizontal at an angle of 22.0 o. a. What is the angular velocity of this sphere? b. What is the linear kinetic energy of this sphere? c. What is the angular kinetic energy of this sphere? d. What is the total kinetic energy of this sphere? e. How far up this incline will the sphere roll before it comes to a halt? f. How long will it take for this sphere to come to a halt? g. What will be the linear acceleration of this sphere as it rolls up the incline? h. What will be the angular acceleration of this sphere as it rolls up the incline? i. What is the moment of inertia of this sphere? j. What will be the net torque acting on this sphere as it rolls up the incline? k. What will be the gravitational potential energy of this sphere just as it comes to a halt on the incline? 32. A bicycle wheel is mounted as in the lab and as shown to the right. This wheel has a mass of 6.55 kg, a radius of R = 38.0 cm and is in the shape of a ring. A mass M = 1.85 kg is attached to the end of a string which is wrapped around an inner hub which has a radius r = 5.40 cm. Initially, the mass M is a distance h = 72.0 cm above the floor. [Assume friction is negligible!] a. What will be the resulting angular acceleration of this wheel? b. How long will it take for the mass M to reach the floor? c. What will be the total angular displacement of the wheel during the time in which the mass M is falling to the floor? d. How much work was done on the wheel by the external torque as the mass M falls to the floor? e. What will be the angular kinetic energy of this wheel just as the mass M reaches the floor? 33. Consider a gyroscope which consists of a large wheel [ring] which has a mass of 8.8 kg and a radius of 27 cm. This wheel does 13 complete rotations each second and is rotating such that an observer from the left side sees the wheel rotating counterclockwise as shown to the right. When viewed from in front the top edge of the wheel is moving away from you. a. What will be the direction and magnitude of the angular momentum of this wheel? b. Suppose that a single upward force F is applied to the right end of the axle as shown. What will be the resulting direction of precession? Suppose that you are holding this same wheel in your hands while you are sitting on a chair capable of rotating freely. Your body has a mass of 72 kg and a radius of gyration [average radius enabling you to treat an irregular body as a ring!] of 12 cm. When observed from above the wheel is rotating clockwise. [All of the above data still apply!] You suddenly turn the wheel over during a time interval of 0.35 seconds. View from Left Side c. What will be the change in angular momentum of the wheel as you turn it over? d. What will be your final angular velocity after inverting the wheel? e. How much work did you do in inverting the wheel? Answers to opposite side: 27a kg m 2 b N m c Rad/sec 2 d. 458 Rad/sec 27e. 128 m/s f Rad g. 513 m h. 14,300 J i kg m 2 /s j. 14,300 J 28a. 438 J 28b. 438 J c m/sec d. 438 J e. 248 J f m/sec g. yes, 8.59 m/s > 4.64 m/s 29a. 176 J b. 176 J c m/s d m/s e m/s 30a kg m 2 b N m 30c Rad/s2 d Rad/s e. 162 Rad f. 400 J g. 400 J h kg m 2 /s
5 PHYSICS HOMEWORK #65 ANGULAR MOMENTUM CONSERVATION 34. A star, which has a mass of 4.60 x kg and a radius of 9.30 x 10 8 m, rotates on its axis once every 16.0 days. a. What is the moment of inertia of this star? b. What is the angular velocity of this star? c. What will be the linear velocity of a sunspot that is located on the equator of this star? d. What is the angular momentum of this star? As this star finally consumes most of its Hydrogen fuel it starts to contract and heat up until it gets hot enough to start burning Helium as a fuel. The burning of the Helium fuel is very intense and the star quickly becomes a red giant as it expands to a new radius of 4.65 x 10 9 m. e. What will be the angular momentum of this star after it has expanded into a red giant? f. What will be the angular velocity of the star after it has expanded into a red giant? g. What will be the new rotational period of this star after it has become a red giant? After a few million years the Helium fuel is all used up and the star again begins to contract, becomes extremely hot and then undergoes a massive supernova explosion. After the explosion the remaining neutron star has a radius of only 1200 km. [Assume that the mass of the star remains unchanged.] h. What will be the angular momentum of this star after it has collapsed into a neutron star? i. What will be the moment of inertia of this star after it has collapsed into a neutron star? j. What will be the angular velocity of this star after it has collapsed? k. What will be the rotational period of this star after it has collapsed? 35. You are holding a sphere, which has a mass of 3.55 kg and a radius of 0.38 m, by an axle inserted vertical through its center as shown to the right. Initially, the sphere is rotating about its vertical axis 12 times every 5.0 seconds as shown. a. What is the moment of inertia of this sphere? b. What is the magnitude of the angular velocity of this sphere? c. What is the magnitude of the angular momentum of this sphere? d. What is the direction of this sphere s angular momentum? Assuming that you are sitting on a freely rotating chair, that your mass is 65.5 kg, and that your radius of gyration is 14.0 cm. Suddenly, you invert the sphere as shown in the diagram to the right. e. What will be the sphere s final angular momentum? f. What will be the direction of the sphere s final angular momentum? g. What will be the magnitude of your final angular momentum? h. What is the magnitude and direction of the impulse delivered to the sphere? i. How much work was done in inverting the sphere? 36. A uniform disk, which has a mass of 8.50 kg and a radius of 35.0 cm, is rotating on its axis such that it makes 8.0 complete rotations each second. a. What is the moment of inertia of this disk? b. What is the angular velocity of this disk? c. What is the angular momentum of this disk? d. What is the angular kinetic energy of this disk? A string is attached to the end of the axle as shown to the right. e. What is the direction of the angular velocity vector in the diagrams to the right? f. What will be the direction of the torque exerted on the disk by the gravitational force? Be specific! g. What will be the direction of the wheel s precession? Answers to this side: 34a x kgm 2 b x 10-6 rad/sec c x 10 3 m/sec d x kgm 2 /sec e. same as d 34f x 10-7 rad/sec g. 400 days h. same as e i x kgm 2 j rad/sec k sec 35a kgm 2 35b rad/sec c kgm 2 /sec d. up e kgm 2 /sec f. up g kgm 2 /sec h kgm 2 /sec i J 36a kgm 2 b rad/sec c kgm 2 /sec d. 658 J e. into page, left f. into page g. downward
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7 PHYSICS HOMEWORK #67 MOMENT OF INERTIA - INTEGRATION Moment of Inertia through Integration 1. Consider a disk which has a radius of R = 12.0 cm and which has a mass density which varies with the radius of the disk according to the function = ( r 2 ) kg/m 2. a. What is the total mass of this disk? b. What will be the moment of inertia of this disk through an axis through the center of mass and perpendicular to the plane of the disk? 2. Consider a rod which has a length of L = 56 cm and which has a mass density which varies with length according to the formula = ( r 2 ) kg. a. What is the total mass of this rod? b. Where is the center of mass of this rod? c. What is the moment of inertia of this rod about an axis perpendicular to the rod and through the lighter end of the rod? 3. Consider a rectangular plate x = 35.0 cm long and y = 15.0 cm wide, as shown to the right, where the surface mass density of the plate varies with the x coordinate according to = ( x 3 ) kg/m 2. a. What is the total mass of this plate? b. Where is the center of mass of this rectangular plate? c. What will be the moment of inertia of this plate about an axis along the left edge of the plate? d. What will be the moment of inertia of this plate about the bottom edge of the rectangular plate? 4. Consider a thick ring with an inner radius of r 1 = 6.0 cm and an outer radius of r 2 = 16 cm where the mass per unit area varies according to = ( r 2 ) kg/m 2. a. what is the total mass of this ring? b. What will be the moment of inertia of this disk through an axis perpendicular to the plane of the ring and through the center of mass? Moment of Inertia - Parallel axis Theorem/Plane Figure Theorem 5. Consider a uniform disk which has a radius of R = 35.0 cm and a total mass of M = 8.20 kg. a. What is the moment of inertia of this disk about an axis perpendicular to the plane of the disk and through the center of mass? b. What will be the moment of inertia of this disk about an axis perpendicular to the plane of the disk and through the outer edge of the disk? c. What will be the moment of inertia of this disk about an axis in the plane of the disk and through the center of mass of the disk? d. What will be the moment of inertia of this disk about an axis in the plane of the disk and tangent to the edge of the disk? e. What will be the moment of inertia of this disk about an axis perpendicular to the plane of the disk and through a point 155 cm from the center of mass? 6. Consider a rectangular plate which has a length of L = 35.0 cm., a width of w = 20.0 cm. and a uniformly distributed mass of m = 6.60 kg. a. What is the moment of inertia of this plate about an axis in the plane of the plate and tangent to the width of the plate? b. What will be the moment of inertia of this plate about an axis in the plane of the plate and tangent to the length of the plate? Answers to opposite side: 6c kgm 2 d kgm 2 e kgm 2 f kgm 2 g kgm 2 7a kgm 2 b kgm 2 c kgm 2 8a. 173 kg b , m c kgm 2 d kgm 2 8e kgm 2 f kgm 2 g kgm 2 h kgm 2 9a kg b kgm 2 c kgm 2 9d kgm 2 e kgm 2
8 PHYSICS HOMEWORK #68 MOMENT OF INERTIA - PLANE FIGURE/PARALLEL AXIS THEOREMS 6c. What will be the moment of inertia of this plate about an axis perpendicular to the plane of the plate and through one of the corners of the plate? d. What will be the moment of inertia of this plate about an axis perpendicular to the plate and through the center of mass of the plate? e. What will be the moment of inertia of this plate about an axis in the plane of the plate and through the center of mass parallel to the width of the plate? f. What will be the moment of inertia of this plate about an axis parallel to the plane of the plate, through the center of mass and parallel to the length L of the plate? g. What will be the moment of inertia about an axis perpendicular to the plane of the plate and through the center of mass of the plate? [Use your answers to e & f above to determine this moment of inertia!] 7. Consider a uniform sphere which has a radius of R = 42.0 cm and a total mass of M = 12.0 kg. a. What is the moment of inertia of this sphere about an axis through the center of mass? b. What will be the moment of inertia of this sphere about an axis tangent to the surface of the sphere? c. What will be the moment of inertia of this sphere about an axis tangent to the surface of the sphere but at a distance of 150 cm from the center of the sphere? 8. Consider a rectangular plate x = 82.0 cm long and y = 22.0 cm wide, as shown to the right, where the surface mass density of the plate varies with the x coordinate according to = ( x 2 ) kg/m 2. a. What is the total mass of this plate? b. What are the x and y coordinates of the center of mass of this rectangular plate? c. What will be the moment of inertia of this plate about an axis along the left edge of the plate? d. What will be the moment of inertia of this plate about the bottom edge of the rectangular plate? e. What will be the moment of inertia about an axis perpendicular to the plane of the plate and through the bottom left corner of the plate as shown in the diagram? f. What will be the moment of inertia of this plate about an axis perpendicular to the plane of the plate and through the center of mass? g. What will be the moment of inertia about an axis in the plane of the plate, through the center of mass and parallel to the x axis? h. What will be the moment of inertia about an axis in the plane of the plate, through the center of mass and parallel to the y axis? [Confirm that you can get this answer through both the parallel axis theorem and the plane figure theorem!] 9. Consider a disk which has a radius of R = 18.0 cm and which has a mass density which varies with the radius of the disk according to the function = ( r 3 ) kg/m 2. a. What is the total mass of this disk? b. What will be the moment of inertia of this disk about an axis through the center of mass and perpendicular to the plane of the disk? c. What will be the moment of inertia of this disk about an axis parallel to the plane of the disk and through the center of mass? d. What will be the moment of inertia of this disk about an axis parallel to the plane of the disk and tangent to the edge of the disk? e. What will be the moment of inertia of this disk about an axis perpendicular to the plane of the disk and through the edge of the disk? Answers to opposite side: 1a kg b kgm 2 2a kg b m c kgm 2 3a kg b , m c kgm 2 d kgm 2 4a kg b kgm 2 5a kgm 2 b kgm 2 c kgm 2 d kgm 2 e kgm 2 6a kgm 2 b kgm 2
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