AP Physics B Free Response Solutions. (0 points) A sailboat at rest on a calm lake has its anchor dropped a distance of 4.0 m below the surface of the water. The anchor is suspended by a rope of negligible mass and volume. The mass of the anchor is 50 kg, and its volume is 6.5 x 0-3 m 3. The density of water is 000 kg/m 3. (a) On the dot below that represents the anchor, draw and label the forces (not components) that act on the anchor. (3 pts) T F b (b) Calculate the magnitude of the buoyant force acting on the anchor. If you need to draw anything other than what you have shown in part (a) to assist in your solution, use the space below. DO NOT add anything to the figure in part (a). ( pts) 3 Fb ρ flvob g 000 6.5 0 0 6.5 N (c) Calculate the tension in the rope. If you need to draw anything other than what you have shown in part (a) to assist in your solution, use the space below. DO NOT add anything to the figure in part (a). ( pts) y F 0 T + Fb W T W F 50 0 6.5 437. 5 N b (d) The bottom of the boat is at a depth d below the surface of the water. Suppose the anchor is lifted back into the boat so that the bottom of the boat is at a new depth d below the surface of the water. How does d compare to d? ( pt) d < d d d _X d > d Justify your answer. ( pts) W There is no longer a buoyant force acting on the anchor, so its apparent weight increases. This increased weight in the boat causes it to sink a little further. Alternatively, the total weight of the boat and anchor is fixed, so the total amount of water displaced must stay constant. When the anchor is pulled out of the water, the boat must displace more water to compensate.
. (5 points) A 0 kg box on a horizontal frictionless surface is moving to the right at a speed of 4.0 m/s. The box hits and remains attached to one end of a spring of negligible mass whose other end is attached to a wall. As a result, the spring compresses a maximum distance of 0.50 m, and the box then oscillates back and forth. (a) i. The spring does work on the box from the moment the box first hits the spring to the moment the spring first reaches its maximum compression. Indicate whether the work done by the spring is positive, negative, or zero. ( pt) Positive _X Negative Zero Justify your answer. ( pt) The box s displacement is to the right, but the spring force is to the left. When the force and displacement are in opposite directions, the work is negative. Alternatively, the kinetic energy is decreasing, and work equals K f K i. ii. Calculate the magnitude of the work described in part i. ( pts) W K mv 0 4.0 60 J (b) Calculate the spring constant of the spring. ( pts) W 60 k N/m x 0.50 W kx 80 (c) Calculate the magnitude of the maximum acceleration of the box. ( pts) F 640 F kx 80 0.50 640 N a 3 m/s m 0 (d) Calculate the frequency of the oscillation of the box. ( pts) f k 80 8.7 π m π 0 π Hz Correct units on parts (a)-ii, (b), (c), and (d): ( pt)
(e) Let x 0 be the point where the box makes contact with the spring, with positive x directed toward the right. i. On the axes below, sketch the kinetic energy K of the oscillating box as a function of position x for the range x 0.50 m to x +0.50 m. ( pts) ii. On the axes below, sketch the acceleration a of the oscillating box as a function of position x for the range x 0.50 m to x +0.50 m. ( pts) F kx a Acceleration is negative when x is positive and vice versa. m m
3. (0 points) A student is asked to experimentally determine the index of refraction of the semicircular block of transparent plastic shown in the figure above. The student aims a red laser beam of wavelength λ 63 nm at the center of the flat side of the block, as shown. The ray is refracted from air into the plastic and strikes the semicircular side of the block perpendicularly. The student uses a protractor to aim the laser at several different angles of incidence θ i between 0 and 90 and to measure the angles of refraction θ r. The student s data are given in the table below. θ i 0 5 30 45 60 75 θ r 0 0 30 37 44 sin θ i 0 0.6 0.50 0.7 0.87 0.97 sin θ r 0 0.7 0.36 0.50 0.60 0.70 (a) On the grid below, plot data that will allow the index of refraction of the plastic to be calculated from a straight line that represents the data. Clearly label the axes, including the scales. ( pts) sin θ i.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0 sin θ r
(b) On your graph, draw a straight line that best represents the data. Use the slope of the line to determine the index of refraction of the plastic. (4 pts) n θ θ sin i n sin r where n for air sinθi n sinθ r y mx + b So m (slope) n Calculate slope using points (0,0) and (0.7,.0): m 0 0.7 0 0.7.43 The index of refraction of the plastic is.43. (c) The student now wants to confirm the result obtained in part (b) by using the critical angle for the plastic. Describe one experimental method the student can use to measure the critical angle. Indicate how the index of refraction can be determined from this measurement. (4 pts) The laser should be directed from the other side (plastic to air) at various angles until it no longer refracts, which is the critical angle. sin sin n θ c where n for air n θ c n n sinθ c
4. (0 points) A 0.30 kg ball is in a cup of negligible mass attached to a block of mass M that is on a table. A string passing over a light pulley connects the block to a.5 kg object, as shown above. The system is released from rest, the block accelerates to the right, and after moving 0.95 m the block collides with a bumper near the end of the table. The ball continues to move and lands on the floor at a position.4 m below and.8 m horizontally from where it leaves the cup. Assume all friction is negligible. (a) Calculate the speed of the ball just after the block hits the bumper and the ball leaves the cup. (3 pts) a v o v t x.8 0?? y.4 0 0 y v t 0 t + y a at.4 0 0.69 s x.8 v.6 m/s t 0.69 (b) Calculate the magnitude of the acceleration of the block as it moves across the table. ( pts) v v + a x 0 v.6 a 3. 6 m/s x 0.95 (c) Calculate the mass M of the block. (3 pts) F ( M + mball + mob )a mob g ( M + mball + mob )a mob g.5 0 M mball mob 0.30.5 4. kg a 3.6 (d) If the mass of the ball is increased, the horizontal distance it travels before hitting the floor will decrease. Explain why this will happen. ( pts) The projectile motion part is independent of the mass of the ball. However, increasing the mass of the ball would slow the acceleration of the system after it is released from rest, resulting in the block hitting the bumper at a lower speed, causing the ball to leave the cup at a lower speed.
5. (0 points) In a certain process, 300 J of energy is added to an ideal gas by heating. During the same process, 00 J of work is done on the gas. (a) Determine the change in the internal energy of the gas. ( pt) From the first law of thermodynamics, U Q + W 300 + 00 5300 J (b) Indicate whether each of the following properties of the gas increases, decreases, or remains the same during the process. i. Volume ( pt) Increases _X Decreases Remains the same Justify your answer. ( pt) Since the work is positive and does work on the gas, it must compress it. ii. Temperature ( pt) _X Increases Decreases Remains the same Justify your answer. ( pt) W P V Since U is positive, T must also be positive since U is proportional to T. iii. Pressure ( pt) _X Increases Decreases Remains the same Justify your answer. ( pt) From the ideal gas law, both a decrease in volume and in increase in temperature cause an increase in pressure. PV nrt Suppose that in a different process 800 J of work is done on the ideal gas at a constant temperature. (c) Determine the change in internal energy of the gas during the process. ( pt) U 0 since it is an isothermal process ( T 0) (d) Which of the following correctly describes the energy transfer by heating, if any, between the gas and its surroundings? ( pt) Energy is transferred into the gas. There is no energy transfer by heating. Justify your answer. ( pt) _X Energy is transferred out of the gas. Since U 0, the first law of thermodynamics becomes Q W. Since work is done on the gas, W is positive, so Q must be negative, meaning that energy is transferred out of the gas.
6. (5 points) Two long, straight horizontal wires are near each other and parallel, with one directly above the other as shown in the figure. Wire X is fixed in place and connected to a battery (not shown) so that it carries a current of 65 A. Wire Y, which is part of a second circuit, is free to move vertically and is suspended at rest by the magnetic force between the wires. The mass per length of wire Y is 5.6 x 0-3 kg/m. Neglect effects from the parts of the circuits that are not shown. (a) Calculate the magnitude of the magnetic field produced by wire X at the position of wire Y. ( pts) B µ 0 I π r 7 0 65 5. 0 0.05 4 T (b) i. Calculate the magnitude of the current in wire Y. ( pts) y 3 mg 5.6 0 0 F 0 ILB mg I 07. 7 A 4 LB 5. 0 ii. Indicate the direction of the current in wire Y. ( pt) To the left _X To the right Neither left nor right, since there is no current Two parallel currents attract when they are in the same direction. (c) Now wire Y is moved to a new position that is closer to wire X, but wire Y is still below wire X and is still carrying the same current as determined in part (b). Wire Y is released from rest. Describe the initial motion of wire Y. Justify your answer. (3 pts) Wire Y will accelerate upward. The downward gravitational force on it does not change, but the upward magnetic force due to the magnetic field of the current in Wire X is stronger, so the net force, and thus the acceleration, is upward. (d) Suppose wire Y is moved to a position 0.05 m above wire X. What changes in current, if any, must occur to maintain equilibrium? ( pt) The magnitude of the current must remain the same since it is the same distance from Wire X, but the direction of the current must be reversed, since two parallel currents repel when they are in opposite directions.
(e) With wire Y still above wire X, the circuit connected to wire Y is removed. Wire Y, which is. m long, is then moved vertically up and away from wire X at a constant speed of 3.0 m/s. i. Calculate the magnitude of the induced emf in wire Y when the wires are 0.050 m apart. ( pts) B µ 0 I π r 7 0 65.6 0 0.050 4 T BLv 4 4 ε.6 0. 3.0 9.4 0 V ii. Indicate which end of wire Y is at a higher electric potential. (3 pts) The left end _X The right end Neither end, since they are at the same electric potential Justify your answer. Since the wire is disconnected from its circuit, it now simply acts as a conductor. Above Wire X, the magnetic field is directed out of the page, as determined by the first right-hand rule. Since Wire Y is moving upward in this magnetic field, the right side accumulates a net positive charge and the left side accumulates a net negative charge, as determined by the second right-hand rule. This means that the right end of the wire is at higher electric potential than the left end. Correct units on parts (a), (b)-i, and (e)-i: ( pt)
7. (0 points) The energy-level diagram for an isolated hypothetical atom is shown above. (a) A collection of such atoms with electrons in the n 3 state undergo transitions in which the atoms only emit photons, and the electrons eventually end in the n state. On the diagram above, draw arrows to indicate all possible transitions, given the starting and ending states for the electrons. ( pts) (b) Calculate the longest wavelength of photons that the atom can emit during the transitions identified in part (a). (3 pts) The longest wavelength corresponds to the transition with the lowest energy change, which is.5 ev as it goes from n 3 to n. hc hc 40 ev nm E λ 55 nm λ E.5 ev (c) What is the ionization energy of an atom in the ground state? ( pt) The ionization energy required to go from n to n is ev. (d) Photons of energy.0 ev are incident on the atom. What effect can this have on an electron in the n state? Justify your answer. ( pts) This will have no effect because there is no energy level that is exactly.0 ev above the ground state. (e) Photons of energy 4.0 ev are incident on the atom. What effect can this have on an electron in the n state? Justify your answer. ( pts) Since this is ev more than the ionization energy, an electron will be ejected with ev of kinetic energy.