2002 Free Response

Question 2

Bumper Mech 2. The cart shown above is made of a block of mass m and four solid rubber tires each of mass m/4 and radius r. Each tire may be considered to be a disk. (A disk has rotational inertia L ML2 , where M is the mass and L is the radius of the disk.) The cart is released from rest and rolls without slipping from the top of an inclined plane of height h. Express all algebraic answers in terms of the given quantities and fundamental constants.


(b) 7 points For an indication of the conservation of mechanical energy AU = —AK ; or equivalent top bottom ; For correct expressions for energies at the top K to = 0; =mgh+4 —mgh = 2mgh For a correct expression for potential energy at the bottom and for recognizing that kinetic energy at the bottom is the sum of translational and rotational kinetic ener •es bottom For a correct expression for translational kinetic energy at the bottom trans = —(2m)D2 = mu 2 K 2 For a correct expression for rotational kinetic energy at the bottom 2 Kroc -—10 2 For recognition of the relationship between translational and rotational velocity Substituting these expressions to determine total kinetic energy at the bottom 2 211 bottom = RID + _ 22 top equal to the kinetic energy at the bottom ettmg potentla energy at —mu = 2mgh 4 For the correct solution for v 8 -gh 5

Question 3

(c) Suppose that the object is released from rest at the origin. Determine the speed of the particle at x = 2 m. In the laboratory, you are given a glider of mass 0.5 kg on an air track. The glider is acted on by the force determined in part (b). Your goal is to determine experimentally the validity of your theoretical calculation in part (c). (d) From the list below, select the additional equipment you will need from the laboratory to do your experiment by checking the line next to each item. If you need more than one of an item, place the number you need on the line. Meterstick Balance Stopwatch Wood block Photo gate timer String Spring Set of objects of different masses (e) Briefly outline the procedure you will use, being explicit about what measurements you need to make in order to determine the speed. You may include a labeled diagram of your setup if it will clarify your procedure.

(d) 2 points For indicating items of equipment consistent with the procedure described in part (e) (at least two of the items if more than two were used) Note: If part (e) was not attempted, only 1 point maximum was awarded. Unreasonable indications, such as all the items being checked, were not awarded any points. (e) 3 points For a complete description of any correct procedure. Partial credit was awarded for less complete descriptions. The following were common examples. Other examples, though rarely cited, could receive partial or full credit. 1 Using photogates Place the photogates near x = 2 m and a small distance apart (such as a glider length). Measure the distance between the photogates. Measure the time the glider takes to travel between the photogates. Obtain the speed from distance/time. time to travel 2 m was used. 2. Using a spring The spring constant k of the spring must be known, or if not, then measured. Set up the spring at x = 2 m so that it is compressed when struck by the glider. Measure the distance of maximum compression Xm. The velocity can then be determined from the equation = 3. Treating the glider as a projectile Adjust the starting point so that x = 2 m is at end of the track. Thus the glider leaves the track at this point and becomes a projectile. The height of the track determines the time interval t that the glider is in the air. The horizontal distance x from the end of the track to the point where the glider hits the ground is measured and then the velocity is computed from x/t.

2003 Free Response

Question 2

0M Mech. 2. An ideal spring is hung from the ceiling and a pan of mass M is suspended from the end of the spring, stretching it a distance D as shown above. A piece of clay, also of mass M, is then dropped from a height H onto the pan and sticks to it. Express all algebraic answers in terms of the given quantities and fundamental constants.


For use of the correct equation for the period of a mass on a spring k For recognition that m = 2M For correct calculation of k using the force equation for the initial stretching of the spring Mg g = kD, giving k = For correct answer substituting for m and k

Question 3

12m 12m Figure 1 Figure 2 Mech. 3. Some physics students build a catapult, as shown above. The supporting plafform is fixed firmly to the ground. The projectile, of mass 10 kg, is placed in cup A at one end of the rotating arm. A counterweight bucket B that is to be loaded with various masses greater than 10 kg is located at the other end of the arm. The arm is released from the horizontal position, shown in Figure 1, and begins rotating. There is a mechanism (not shown) that stops the arm in the vertical position, allowing the projectile to be launched with a horizontal velocity as shown in Figure 2.

(b) The students assume that the mass of the rotating arm, the cup, and the counterweight bucket can be neglected. With this assumption, they develop a theoretical model for x as a function of the counterweight mass using the relationship x = vrt , where v x is the horizontal velocity of the projectile as it leaves the cup and t is the time after launch. . How many seconds after leaving the cup will the projectile strike the ground? ii. Derive the equation that describes the gravitational potential energy of the system relative to the ground when in the position shown in Figure 1, assuming the mass in the counterweight bucket is M. iii. Derive the equation for the velocity of the projectile as it leaves the cup, as shown in Figure 2.

ii. (3 points) For determining the potential energy of both the load in the counterweight bucket and the projectile For the correct value of the potential energy of the bucket load For the correct value of the potential energy of the projectile =Ub+U = 29.4M +294 (or 30M+300using g=10nv/s2)

iii. (5 points) For a valid statement or equation indicating conservation of energy init For the correct final potential energy of the bucket load For the correct final potential energy of the projectile = 9.8M +1470 For having terms for the final kinetic energy of both the bucket load and the projectile K p and OR For using one of the following relationships to write all expressions in terms of Db OR o=Vx/12 Substituting into the conservation of energy equation above and solving for vx : 29.4M + 294 = 9.8M +1470+ + (M/72)vx2 = (19.6M 72)) V (or (20M — + (M 72)) using g = 10 m/s2 ) 1 point 1 point 1 point 1 point 1 point

2004 Free Response

Question 1

900 Lake Mech. 1. A rope of length L is attached to a support at point C. A person of mass ml sits on a ledge at position A holding the other end of the rope so that it is horizontal and taut, as shown above. The person then drops off the ledge and swings down on the rope toward position B on a lower ledge where an object of mass nt2 is at rest. At position B the person grabs hold of the object and simultaneously lets go of the rope. The person and object then land together in the lake at point D, which is a vertical distance L below position B. Air resistance and the mass of the rope are negligible. Derive expressions for each of the following in terms of ml , m2, L, and g.

(b) The tension in the rope just before the collision with the object

(b) 4 points For any Indication that there are two forces acting on the person For an indication that the acceleration of the person is cenü-ipetal, i.e. equal to v rorv L For a correct application of Newton's second law that includes the two forces (tension T and weight) and a non-zero acceleration 2 T - mig = For substitution of the expression for VB from part (a) and L for the radius ml (2gL) + ntlg = 2n11g + ntlg

(d) The ratio of the kinetic energy of the person-object system before the collision to the kinetic energy after the collision

(d) 2 points For correct expressions for the kinetic energy before and after the collision, using the answers to parts (a) and (c) before = 5 ml VB = (2gL) = 1 1 Kafter = —(ntl + n12)Vafter = -j(ntl + 7112) 2 ml 2gL = 2 nil (ml + 7112) (nil + m2)2 2 For constructing the ratio Kbefore K after from valid expressions for kinetic energy, in terms of the required quantities. The ratio does not need to be simplified, but if it is the algebra needs to be correct. Kb nil gL (nil + (ml + ml

(e) The total horizontal displacement x of the person from position A until the person and object land in the water at point D.

For a correct expression relating the distance fallen, L, to the time it takes to fall from ointBto the water: L = lgt2 For indicating that the horizontal displacement from B to D is the answer to part (c) multiplied by the time XBD = Daftert For correctly solving the first equation for t and substituting two quantities into the second equation (this must yield an expression in terms of the required tities) XBD = Daftert = ml ml + nt2 ml + nt2 For indicating that the total horizontal displacement from A to D is XBD plus L Xtot XBD L = ml + nt2 (37111 + m2)L nil + 1112

Question 3

—H Pivot Mech. 3. A uniform rod of mass M and length L is attached to a pivot of negligible friction as shown above. The pivot is located at a distance from the left end of the rod. Express all answers in terms of the given quantities and fundamental constants. (a) Calculate the rotational inertia of the rod about the pivot. (b) The rod is then released from rest from the horizontal position shown above. Calculate the linear speed of the bottom end of the rod when the rod passes through the vertical. 3 Pivot (c) The rod is brought to rest in the vertical position shown above and hangs freely. It is then displaced slightly from this position. Calculate the period of oscillation as it swings.

Alternate Solution For an statement of the oarallel axis theorem I = lcm + mr2 where r is the distance from the center of mass to the pivot point For a correct value of the center of mass inertia (calculated or remembered) 1 2 cm — 12 For indicating that r = L/6 2 1 —ML2 + M 12 6 For the correct answer 2 9

(b) 7 points For any indication of For correctly calculating the change in potential energy of the rod (or the work done on it) For example: AU = Mghcm = Mg For writing a conservation equation that includes a rotational kinetic energy re ardless of whether the potential energy is correct) 2 _ MgL 1 2 6 For any indication that o is linear speed divided by a distance (regardless of whether the correct distance is used) Substituting and solving for v : 6 2 2 MgL 18 3g 6 ML2 - L 4 For the correct answer 3

(c) 4 points For an equation for the period of a physical pendulum mgd For substitution of the inertia from part (a) For indicating that the distance d is the distance from the pivot to the center of mass, i.e. d 6 ML2 9 MgL/6 For the correct answer

(c) (continued) Alternate solution For an equation relating the angular acceleration to the torque and inertia d29 T dt2 — I For substituting the Inertia from part (a) and the torque as a function of 9 sin 9 ML2/9 dt2 For usmg the approximation sine = 9 d29 dt2 d29 Taking 9 = ksinmt , the second derivative is — —m k sinmt dt2 Substituting into the differential equation and solving for m : —m ksinmt = — —k sin mt Usmg the relationship between T and o : T = 27t/m For the correct answer 3

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