317K PHY October 12, 2006 HOMEWORK # 4 DUE on October 30, 2006 by 8:00pm Instructions: Use a SCANTRON ANSWER SHEET to mark answers. Provide your NAME (last, first), UT EID, and UNIQUE NUMBER of the course. Sign and date it. The answer sheets must be placed in the 317K box (by bulletin board outside PAI 2.48) by 8:00pm on Monday, October 30, 2006. Late homework will not be graded. Answer the following 56 questions: 1. A radian is about: A) 25 B) 37 C) 45 D) 57 E) 90 2. One revolution per minute is about: A) 0.0524 rad/s B) 0.105 rad/s C) 0.95 rad/s D) 1.57 rad/s E) 6.28 rad/s 3. If a wheel turning at a constant rate completes 100 revolutions in 10 s its angular speed is: A) 0.31 rad/s B) 0.63 rad/s C) 10 rad/s D) 31 rad/s E) 63 rad/s 4. A flywheel, is initially rotating at 20 rad/s and has a constant angular acceleration. After 9.0 s it has rotated through 450 rad. Its angular acceleration is: A) 3.3 rad/s2 B) 4.4 rad/s2 C) 5.6 rad/s2 D) 6.7 rad/s2 E) 11 rad/s2
Page 1
5. A flywheel rotating at 12 rev/s is brought to rest in 6 s. The magnitude of the average angular acceleration in rad/s2 of the wheel during this process is: A) 1/ B) 2 C) 4 D) 4 E) 72 6. A wheel initially has an angular velocity of 36 rad/s but after 6.0s its angular velocity is 24 rad/s. If its angular acceleration is constant the value is: A) 2.0 rad/s2 B) 2.0 rad/s2 C) 3.0 rad/s2 D) 3.0 rad/s2 E) 6.0 rad/s2 7. A wheel starts from rest and has an angular acceleration that is given by (t) = (6 rad/s4)t2. The angle through which it turns in time t is given by: A) [(1/8)t4] rad B) [(1/4)t4] rad C) [(1/2)t4] rad D) (t4) rad E) 12 rad 8. The angular velocity vector of a spinning body points out of the page. If the angular acceleration vector points into the page then: A) the body's rotation is slowing down B) the body's rotation is speeding up C) the body is starting to turn in the opposite direction D) the axis of rotation is changing orientation E) none of the above 9. For a wheel spinning with constant angular acceleration on an axis through its center, the ratio of the speed of a point on the outer rim to the speed of a point halfway between the center and the rim is: A) 1 B) 2 C) 1/2 D) 4 E) 1/4
Page 2
10. Four identical particles, each with mass m, are arranged in the x, y plane as shown. They are connected by light sticks to form a rigid body. If m = 2.0 kg and a = 1.0 m, the rotational inertia of this array about the y-axis is:
A) B) C) D) E)
4.0 kg m2 12 kg m2 9.6 kg m2 4.8 kg m2 none of these
11. A and B are two solid cylinders made of aluminum. Their dimensions are shown. The ratio of the rotational inertia of B to that of A about the common axis XX' is:
A) B) C) D) E)
2 4 8 16 32
12. The rotational inertia of a solid uniform sphere about a diameter is (2/5)MR2, where M is its mass and R is its radius. If the sphere is pivoted about an axis that is tangent to its surface, its rotational inertia is: A) MR2 B) (2/5)MR2 C) (3/5)MR2 D) (5/2)MR2 E) (7/5)MR2
Page 3
13. A rod is pivoted about its center. A 5-N force is applied 4 m from the pivot and another 5-N force is applied 2 m from the pivot, as shown. The magnitude of the total torque about the pivot (in Nm) is:
A) B) C) D) E)
0 5 8.7 15 26
14. A uniform disk, a thin hoop, and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis through its center. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown. Rank the objects according the their angular velocities after a given time t, least to greatest.
A) B) C) D) E)
disk, hoop, sphere hoop, disk, sphere hoop, sphere, disk sphere, hoop, disk sphere, disk, hoop
15. A disk with a rotational inertia of 5.0 kg m2 and a radius of 0.25 m rotates on a frictionless fixed axis perpendicular to the disk and through its center. A force of 8.0 N is applied tangentially to the rim. If the disk starts at rest, then after it has turned through half a revolution its angular velocity is: A) 0.57 rad/s B) 0.64 rad/s C) 0.80 rad/s D) 1.6 rad/s E) 3.2 rad/s
Page 4
16. A circular saw is powered by a motor. When the saw is used to cut wood, the wood exerts a torque of 0.80 N m on the saw blade. If the blade rotates with a constant angular velocity of 20 rad/s the work done on the blade by the motor in 1.0 min is: A) 0 B) 480 J C) 960 J D) 1400 J E) 1800 J 17. A wheel rolls without slipping along a horizontal road as shown. The velocity of the center of the wheel is represented by . Point P is painted on the rim of the wheel. The instantaneous velocity of point P when it is in contact with the road is:
A) B) C) D) E)
zero
18. Two wheels roll side-by-side without sliding, at the same speed. The radius of wheel 2 is twice the radius of wheel 1. The angular velocity of wheel 2 is: A) twice the angular velocity of wheel 1 B) the same as the angular velocity of wheel 1 C) half the angular velocity of wheel 1 D) more than twice the angular velocity of wheel 1 E) less than half the angular velocity of wheel 1 19. A thin-walled hollow tube rolls without sliding along the floor. The ratio of its translational kinetic energy to its rotational kinetic energy (about an axis through its center of mass) is: A) 1 B) 2 C) 3 D) 1/2 E) 1/3
Page 5
20. A hoop, a uniform disk, and a uniform sphere, all with the same mass and outer radius, start with the same speed and roll without sliding up identical inclines. Rank the objects according to how high they go, least to greatest. A) hoop, disk, sphere B) disk, hoop, sphere C) sphere, hoop, disk D) sphere, disk, hoop E) hoop, sphere, disk 21. The fundamental dimensions of angular momentum are (with M indicating mass, L indicating length, and T indicating time): A) ML/T B) M/(L2T2) C) M2/T D) ML2/T2 E) none of these 22. A newtonsecond is a unit of: A) work B) angular momentum C) power D) linear momentum E) none of these 23. Two objects are moving in the x,,y plane as shown. The magnitude of their total angular momentum (about the origin O) is:
A) B) C) D) E)
zero 6 kg m2/s 12 kg m2/s 30 kg m2/s 78 kg m2/s
Page 6
24. A single force, represented by the arrow in the figures, acts on a particle P. Rank each of the orientations of the force shown below according to the magnitude of the time rate of change of the particle's angular momentum about the point O, least to greatest.
A) B) C) D) E)
1, 2, 3, 4 1 and 2 tie, then 3, 4 1 and 2 tie, then 4, 3 1 and 2 tie, then 3 and 4 tie All are the same
25. An ice skater with rotational inertia I0 is spinning with angular speed 0. She pulls her arms in, thereby increasing her angular speed to 4 0. Her rotational inertia is then: A) I0 B) I0 /2 C) 2 I0 D) I0 /4 E) 4 I0 26. A playground merry-go-round has a radius R and a rotational inertia I. When the merry-go-round is at rest, a child with mass m runs with speed v along a line tangent to the rim and jumps on. The angular velocity of the merry-go-round is then: A) mv/I B) v/R C) mRv/I D) 2mRv/I E) mRv/(mR2 + I) 27. A uniform sphere of radius R rotates about a diameter with angular momentum of magnitude L. Under the action of internal forces the sphere collapses to a uniform sphere of radius R/2. The magnitude of its new angular momentum is: A) L/4 B) L/2 C) L D) 2L E) 4L
Page 7
28. Two disks are mounted on low-friction bearings on a common shaft. The first disc has rotational inertia I and is spinning with angular velocity . The second disc has rotational inertia 2I and is spinning in the same direction as the first disc with angular velocity 2 as shown. The two disks are slowly forced toward each other along the shaft until they couple and have a final common angular velocity of:
A) 5 /3 B) 3 C) 7 3 D) E) 3 29. A block with mass M, on the end of string, a moves in a circle on a horizontal frictionless table as shown. As the string is slowly pulled through a small hole in the table:
A) B) C) D) E)
the angular momentum of M remains constant the angular momentum of M decreases the kinetic energy of M remains constant the kinetic energy of M decreases none of the above
30. A net torque applied to a rigid object always tends to produce: A) linear acceleration B) rotational equilibrium C) angular acceleration D) rotational inertia E) none of these
Page 8
31. The conditions that the sum of forces and the sum of the torques both vanish: A) hold for every solid body in equilibrium B) hold only for elastic solid bodies in equilibrium C) hold for every solid body D) are always sufficient to calculate the forces on a solid object in equilibrium E) are sufficient to calculate the forces on a solid object in equilibrium only if the object is elastic 32. A uniform plank XY is supported by two equal 120-N forces at X and Y, as shown. The support at X is then moved to Z (half-way to the plank's center). The supporting forces at Y and Z are then:
A) B) C) D) E)
FY = 240 N, FZ = 120 N FY = 200 N, FZ = 40 N FY = 40 N, FZ = 200 N FY = 80 N, FZ = 160 N FY = 160 N, FZ = 80 N
33. For an object in equilibrium the sum of the torques acting on it vanishes only if each torque is calculated about: A) the center of mass B) the center of gravity C) the geometrical center D) the point of application of the force E) the same point 34. The center of gravity coincides with the center of mass: A) always B) never C) if the center of mass is at the geometrical center of the body D) if the acceleration due to gravity is uniform over the body E) if the body has a uniform distribution of mass
Page 9
35. A 160-N child sits on a light swing and is pulled back and held with a horizontal force of 100 N. Assuming they are equal, the magnitude of the tension force of each of the two supporting ropes is: A) 60 N B) 94 N C) 120 N D) 190 N E) 260 N 36. A picture P of weight W is hung by two strings as shown. The magnitude of the tension force of each string is T. The total upward pull of the strings on the picture is:
A) B) C) D) E)
2W cos T sin T cos 2T sin 2T cos
37. An 800-N man stands halfway up a 5.0 m ladder of negligible weight. The base of the ladder is 3.0 m from the wall as shown. Assuming that the wall-ladder contact is frictionless, what is the force of the wall on the ladder? (Note: the ladder is stationary, since there is friction at the ladder-floor contact.)
A) B) C) D) E)
150 N 300 N 400 N 600 N 800 N
Page 10
38. A 240-N weight is hung from two ropes as shown. The tension in the horizontal rope has magnitude:
A) B) C) D) E)
0 656 N 480 N 416 N 176 N
39. A certain wire stretches 0.90 cm when outward forces with magnitude F are applied to each end. The same forces are applied to a wire of the same material but with three times the diameter and three times the length. The second wire stretches: A) 0.10 cm B) 0.30 cm C) 0.90 cm D) 2.7 cm E) 8.1 cm 40. Young's modulus is a proportionality constant that relates the force per unit area applied perpendicularly at the surface of an object to: A) the shear B) the fractional change in volume C) the fractional change in length D) the pressure E) the spring constant 41. The approximate value of g at an altitude above Earth equal to one Earth diameter is: A) 9.8 m/s2 B) 4.9 m/s2 C) 2.5 m/s2 D) 1.9 m/s2 E) 1.1 m/s2
Page 11
42. In the formula F = Gm1m2/r2, the quantity G: A) depends on the local value of g B) is used only when the Earth is one of the two masses C) is greatest at the surface of the Earth D) is a universal constant of nature E) is related to the Sun in the same way that g is related to the Earth 43. Earth exerts a gravitational force on the Moon, keeping it in its orbit. The reaction to this force, in the sense of Newton's third law, is: A) the centripetal force on the Moon B) the nearly circular orbit of the Moon C) the gravitational force exerted on Earth by the Moon D) the tides due to the Moon E) the apple hitting Newton on the head 44. Four particles, each with mass m, are arranged symmetrically about the origin on the x axis. A fifth particle, with mass M, is on the y axis. The direction of the gravitational force on M is:
A) B) C) D) E)
none of these directions
45. Venus has a mass of about 0.0558 times the mass of Earth and a diameter of about 0.381 times the diameter of Earth. The acceleration of a body falling near the surface of Venus is about: A) 0.21 m/s2 B) 1.4 m/s2 C) 2.8 m/s2 D) 3.8 m/s2 E) 25 m/s2
Page 12
46. To measure the mass of a planet with the same radius as Earth, an astronaut drops an object from rest (relative to the planet) from an altitude of one radius above the surface. When the object hits its speed is 4 times what it would be if the same experiment were carried out for Earth, ignoring any drag forces from the planets' atmospheres. In units of Earth masses, the mass of the planet is: A) 2 B) 4 C) 8 D) 16 E) 32 47. Suppose you have a pendulum clock which keeps correct time on Earth (acceleration due to gravity = 9.8 m/s2). Without changing the clock, you take it to the Moon (acceleration due to gravity = 1.6 m/s2). For every hour interval (on Earth) the Moon clock will record: A) (9.8/1.6) h B) 1 h C) 9.8 /1.6 h D) (1.6/9.8) h E) 1.6 / 9.8 h 48. The mass of a hypothetical planet is 1/100 that of Earth and its radius is 1/4 that of Earth. If a person weighs 600 N on Earth, what would he weigh on this planet? A) 24 N B) 48 N C) 96 N D) 192 N E) 600 N 49. Two particles, each of mass m, are a distance d apart. To bring a third particle, with mass 2m, from far away to a resting point midway between the two particles the work done by an external agent does work given by: A) 4Gm2/d B) 4Gm2/d C) 8Gm2/d D) 8Gm2/d E) zero
Page 13
50. The escape velocity at the surface of Earth is approximately 8 km/s. What is the mass, in units of Earth's mass, of a planet with twice the radius of Earth for which the escape speed is twice that for Earth? A) 2 B) 4 C) 8 D) 1/2 E) 1/4 51. A projectile is fired straight upward from Earth's surface with a speed that is half the escape speed. If R is the radius of Earth, the highest altitude reached, measured from the surface, is: A) R/4 B) R/3 C) R/2 D) R E) 2R 52. Each of the four corners of a square with edge a is occupied by a point mass m. There is a fifth mass, also m, at the center of the square. To remove the mass from the center to a point far away the work that must be done be an external agent is given by: A) 4Gm2/a B) 4Gm2/a C) 4 2Gm2 / a D) -4 2Gm2 / a E) 4Gm2/a2 53. A planet is in circular orbit around the Sun. Its distance from the Sun is four times the average distance of Earth from the Sun. The period of this planet, in Earth years, is: A) 4 B) 8 C) 16 D) 64 E) 2.52
Page 14
54. A small satellite is in elliptical orbit around Earth as shown. If L denotes the magnitude of its angular momentum and K denotes kinetic energy:
A) B) C) D) E)
L2 > L1 and K2 > K1 L2 > L1 and K2 = K1 L2 = L1 and K2 = K1 L2 < L1 and K2 = K1 L2 = L1 and K2 > K1
55. A spaceship is returning to Earth with its engine turned off. Consider only the gravitational field of Earth and let M be the mass of Earth, m be the mass of the spaceship, and R be the distance from the center of Earth. In moving from position 1 to position 2 the kinetic energy of the spaceship increases by: A) 1 1 GMm 2 - 2 R1 R2 B) 1 1 GMm 2 + 2 R1 R2 C) R -R GMm 1 2 2 R1 D) R - R2 GM m 1 R1 R2 E) R -R GMm 1 2 2 2 R1 R2
Page 15
56. A planet travels in an elliptical orbit about a star X as shown. The magnitude of the acceleration of the planet is:
A) B) C) D) E)
greatest at point Q greatest at point S greatest at point U greatest at point W the same at all points
Page 16