1999 Free Response

Question 3

As shown above, a uniform disk is mounted to an axle and is free to rotate without friction. A thin uniform rod is rigidly attached to the disk so that it will rotate with the disk A block is attached to the end of the rod. Properties of the disk, rod, and block are as follows. Rod mass = 2m Block Sfring Disk 2 Disk: Rod: Block: = 3m radius mass mass = m, length = R, moment of inertia about center ID = 2R, moment ofinertia about one end IR 4 2 The system is held in equilibrium with the rod at an angle 90 to the vertical, as shown above, by a horizontal string of negligible mass with one end attached to the disk and the other to a wall. Express your answers to the following in terms of m, R, 90, and g. (a) Determine the tension in the string.

For indicating that the net torquc is zero, or that the clockwise and counterclockwise torques are equal For a correct expression for the torque exerted by the rod = mgRsin 00 For a correct expression for the torque exerted by the block = 4mgRsinOo For a correct expression for the torque exerted by the string For adding the counterclockwise torques and setting the sum equal to the clockwise torque (this point not awarded for just one torque) TR 4mgRsin 00 + mgRsin 00 T = 5mg smoo

As the disk rotates, the rod passes the horizontal position shown above. (c) Determine the linear speed of the mass at the end of the rod for the instant the rod is in the horizontal position.

For indicating that energy is conserved For indicating that the potential energy of two bodies (the rod and the block) changes For the correct expressions for these two potential energies AU = mgR cos 90 + 2mg(2R)cos 00 For indicating the correct kinetic energy when the rod is horizontal 2 Equating the kinetic and potential energies, and solving for the angular speed mgRcosOo +4mgRcosOo 12g cosOo 13R For using the relationship bcmeen linear and angular speed, and substituting mand the correct radius, 2R v = or 3gR cos 00 12gcosO 0 (210-4 1311 13

2000 Free Response

Question 2

RI, Il Mech 3. A pulley of radius RI and rotational inertia Il is mounted on an axle with negligible friction. A light cord passing over the pulley has two blocks of mass m attached to either end, as shown above. Assume that the cord does not slip on the pulley. Determine the answers to parts (a) and (b) in terms of m, RI, Il, and fundamental constants.

(b) One block is now removed from the right and hung on the left. When the system is released from rest, the three blocks on the left accelerate downward with an acceleration -e. Determine the following. . The tension T3 in the section of cord supporting the three blocks on the left ii. The tension Tl in the section of cord supporting the single block on the right iii. The rotational inertia Il of the pulley

i. 2 points For correct substitutions into Newton's second law 344) 3mg — T3 = For a correct solution for T3 T3 2mg

ii. 2 points EF - ma For correct substitutions into Newton' s second law For a correct solution for T;

iii. 4 points For Ila For a = For = (T3 — TI)RI For correct substitutions into t = Ila and solution for Il 2mg 3 mg RI IJ 11 = 2,nR12 SQlgtiQt! Use conservation of energy, AE = AK + AV O For AK -AU For AK = — mv2 + (3m)v2 + 1102 , where = — 2 2 3v For = mgh — 3mgh —2mgh , where h 2g For correct substitutions and solution for Il = 2mRi2

2001 Free Response

Question 2

(c) Suppose that the injection of the satellite into orbit is less than perfect. For an injection velocity that differs from the desired value in each of the following ways, sketch the resulting orbit on the figure. (J is the center of Jupiter, the dashed circle is the desired orbit, and P is the injection point.) Also, describe the resulting orbit qualitatively but specifically. i. When the satellite is at the desired altitude over the equator, its velocity vector has the correct direction, but the speed is slightly faster than the correct speed for a circular orbit of that radius. ii. When the satellite is at the desired altitude over the equator, its velocity vector has the correct direction, but the speed is slightly slower than the correct speed for a circular orbit of that radius.

2. (c) i. 3 points For stating that the orbit is an ellipse For diagram with orbit drawn completely outside the circle with point of contact only at point P and major axis along PJ. Partial credit of 1 point awarded for any path or orbit completely outside the circle. No points were awarded in any part of path or orbit was inside the circle. 1 point 2 points

2. (c) ii. 3 points Distribution of Points 1 point 2 points For stating that the orbit is an ellipse For diagram with orbit drawn completely inside the circle with point of contact only at point P and major axis along PJ. Partial credit of 1 point awarded for any path or orbit completely inside the circle. No points were awarded if any part of path or orbit was outside the circle.

Question 3

radius r Experiment A Mech 3. A light string that is attached to a large block of mass 4m passes over a pulley with negligible rotational inertia and is wrapped around a vertical pole of radius r, as shown in Experiment A above. The system is released from rest, and as the block descends the string unwinds and the vertical pole with its attached apparatus rotates. The apparatus consists of a horizontal rod of length 2L, with a small block of mass m attached at each end. The rotational inertia of the pole and the rod are negligible.

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3. (b) 6 points For a correct expression of Newton's 2nd law F = ma For correct substitutions into Newton 's law 4m —T = 4ma For a correct formula for torque T = la Tr la-Tr la From Newton 's 2nd law equation above: T = 4mg — 4ma Substituting into the torque equation: la = 4mg — 4ma For substituting the expression for I from part (a) into Newton's law 2mL2a = 4mg — 4ma For the expression a = a Substituting this expression into the previous equation: 2mL2 a = 4mg — 4ma For the correct answer 2gr2 L2 +2r2

(c) When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare with the value 4mgD ? Check the appropriate space below. Greater than 4mgD Equal to 4mgD Less than 4mgD

3. (c) 3 points For correctly checking the space in front of "Equal to 4mgD' For correct justification, such as ' 'The kinetic energy gained by the two smaller blocks comes from the decrease in the potential energy of the 4m block." OR "Total energy is conserved '

radius r 4m Experiment B The system is now reset. The string is rewound around the pole to bring the large block back to its original location. The small blocks are detached from the rod and then suspended from each end of the rod, using strings of length L The system is again released from rest so that as the large block descends and the apparatus rotates, the small blocks swing outward, as shown in Experiment B above. This time the downward acceleration of the block decreases with time after the system is released. (d) When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare to that in part (c) ? Check the appropriate space below. Grea ter Less

3. (d) 3 points For correctly checking the space in front of "Less" For correct justification, such as "The small blocks rise and in otential ener . The total energy available is still 4mgD. Therefore the kinetic energy must be less than in part (c)."

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