2014 Free Response

Question 1

1 10 20 30 2 40 50 Photogates 3 60 70 4 80 90 5 100 0 Mech. 1. 110 120 cm In an experiment, a student wishes to use a spring to accelerate a cart along a horizontal, level track. The spring is attached to the left end of the track, as shown in the figure above, and produces a nonlinear restoring force of magnitude FS = As2 + Bs, where s is the distance the spring is compressed, in meters. A measuring tape, marked in centimeters, is attached to the side of the track. The student places five photogates on the track at the locations shown.

(d) i. Compare the speed of the cart measured by photogate 1 to the predicted value of the speed of the cart just after it loses contact with the spring. List a physical source of error that could account for the difference. ii. From the measured speed values of the cart as it rolls down the track, give a physical explanation for any trend you observe.

2 points For stating that the measured initial speed of the cart is greater than the predicted value For correctly identifymg a source of error regarding the initial speed of the cart Examples: The student com ressed the s rin more than was determined. This would lead to more potential energy in the spring and greater kinetic energy for the cart. The cart would therefore move faster than predicted. The table is not level, sloping downward would result in a greater measured speed. The constants A and B for the sorina are not accurate. The true values are larger than what is given. This would lead to smaller predicted potential energy of the spring and a smaller predicted value for the kinetic energy of the cart. Therefore, the cart would move faster than predicted ii. 2 points For correctly identifying the trend For a correct physical explanation for the cart slowing down Examples: Friction in the axles and air resistance against the cart are slowing it down. theaaGk.is.AQ.LP%ie.Gfly-Iey.el and the cart is going uphill. This is slowing down the cart.

Question 2

1 Mech. 2. Side View Rough Wall Smooth Ramp Top View A small block of mass m starts from rest at the top of a frictionless ramp, which is at a height h above a horizontal tabletop, as shown in the side view above. The block slides down the smooth ramp and reaches point P with a speed . After the block reaches point P at the bottom of the ramp, it slides on the tabletop guided by a circular vertical wall with radius R, as shown in the top view. The tabletop has negligible friction, and the coefficient of kinetic friction between the block and the circular wall is g .

(e) Derive an expression for v(t) , the speed of the block as a function of time t after passing point P on the track.

(e) 3 points For substituting dv/dt for a into the answer from part (d), substituting dv/dt for a and the friction force for Fnet into Newton's second law dv du or m— dt For Substituting for Ff produces the same relationship as the first equation above. For separation of variables and using correct limits —du = Adv = Integrate the equation to solve for v . gt R + gvot 1 + ,uvot/R

Question 3

m/20 Top View Mech. 3. Side View A large circular disk of mass m and radius R is initially stationary on a horizontal icy surface. A person of mass m/2 stands on the edge of the disk. Without slipping on the disk, the person throws a large stone of mass m/20 horizontally at initial speed from a height h above the ice in a radial direction, as shown in the figures above. The coefficient of friction between the disk and the ice is g . All velocities are measured relative to the ground. The time it takes to throw the stone is negligible. Express all algebraic answers in terms of m, R, , h, g , and fundamental constants, as appropriate.

(b) Assuming that the disk is free to slide on the ice, derive an expression for the speed of the disk and person immediately after the stone is thrown.

(b) 3 points For a statement of onservation of moment or Newton's third law For substituting the momentum of the stone into a correct expression for conservation of momentum For substituting the momentum of the person-disk system into a correct expression for conservation of momentum 0 = nilD1 + ni2V2 (Do) + ni+#-v 3 —mv = 2 mvo 20 Note: Since the question asks for speed, the negative sign is not needed. There is no penalty for including it.

Top View The person now stands on a similar disk of mass m and radius R that has a fixed pole through its center so that it can only rotate on the ice. The person throws the same stone horizontally in a tangential direction at initial speed , as shown in the figure above. The rotational inertia of the disk is mR 2. (d) Derive an expression for the angular speed (D of the disk immediately after the stone is thrown.

(d) 4 points For a statement of onservation of total angular momentum Li2L L = mrt) for linear motion L = 10 for rotation For substituting the angular momentum of the stone into a correct expression of conservation of angular momentum For substituting the angular momentum of the person into a correct expression of conservation of angular momentum For substituting the angular momentum of the disk into a correct expression of conservation of an ar momentum 0 = m rv +1 + Ipop 20 m 20 m 2 20 R 2 2 2 = mR m

2015 Free Response

Question 1

MotiorvSensor Mech.l. x A block of mass m is projected up from the bottom of an inclined ramp with an initial velocity of magnitude vo . The ramp has negligible friction and makes an angle 9 with the horizontal. A motion sensor aimed down the ramp is mounted at the top of the incline so that the positive direction is down the ramp. The block starts a distance D from the motion sensor, as shown above. The block slides partway up the ramp, stops before reaching the sensor, and then slides back down.

(b) Derive an expression for the position xmin of the block when it is closest to the motion sensor. Express your answer in terms of m, D, vo , 9 , and physical constants, as appropriate.

(b) 2 points Using an equation that can be solved for the closest position to the sensor v; V12 + 2ad For substitution into a For setting v2 to z 0 = v; +2(gsin0) x —D 2g sin 9 ent with part (a) Ing D for the Initial position

(c) On the axes provided below, sketch graphs of position x, velocity v, and acceleration a as functions of time t for the motion of the block while it goes up and back down the ramp. Explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate.

4 points For a position graph that is a parabola that does not cross the t-axis and has a vertex that does not touch the t-axls For a velocity graph that is a straight line and crosses the t-axis For an acceleration graph that is a horiz 3/10 For a set of graphs that are consistent 1 point 1 point 1 point 1 point

Question 3

Pivot Point Mech.3. A uniform, thin rod of length L and mass M is allowed to pivot about its end, as shown in the figure above. (a) Using integral calculus, derive the rotational inertia for the rod around its end to show that it is ML2 3 .

3 points Writing an integral to derive the rotational inertia of the rod r dm For a correct expression for dm 1 = MIL, M = AL, dm=mr For using the correct limits of integration or a correct constant of integration Ir2dr For correctly evaluating the integral above, leading to the answer ML 3 Ir = —I(L3 - O) = —ML2

(b) Derive an expression for the velocity of the free end of the rod at position B. Express your answer in terms of M, L, and physical constants, as appropriate.

(b) 4 points For using any expression of conservation of energy Kl + Ugl = + For a correct energy expression relating gravitational potential energy to rotational kinetic energy 1 1022 mghl = 2 For correctly substituting L/2 for the change in height Mg(L/2) = 31ML2 For with r = L to solve for the velocity of the end of the rod 2 MgL_ 1 —ML2 v' 2 6 3gL

ii. Describe two ways in which the effects of air resistance could be reduced.

For one example that directly decreases the effect of air resistance For another example that directly decreases the effect of air resistance Som x I in I Do the experiment in a vacuum Use shorter rod lengths Use more massive (or denser) rods Use a more aerodynamic shape for the rods

2016 Free Response

Question 2

2M Mech.2. 3M A block of mass 2M rests on a horizontal, frictionless table and is attached to a relaxed spring, as shown in the figure above. The spring is nonlinear and exerts a force F (x) = —BX , where B is a positive constant and x is the displacement from equilibrium for the spring. A block of mass 3M and initial speed vo is moving to the left as shown.

(d) Derive an expression for the maximum distance D that the spring is compressed.

(d) 4 points For a correct expression of the conservation of energy AK + AU system system final 10 —MIG - 10 ate the spring force equation —Bx3dr Bx3dx For using the correct limits of integration or an appropriate constant of integration —Mvä - 10 2 —Mvo - 10 Bx3 dr For an answer consistent with the speed from (b) or the kinetic energy from part (c) 18Mv02 5B

ii. Which of the following correctly describes the magnitude of the net force on each of the two blocks when the spring is at maximum compression? The magnitude of the net force is greater on the block of mass 2M. The magnitude of the net force is greater on the block of mass 3M. The magnitude of the net force on each block has the same nonzero value. The magnitude of the net force on each block is zero.

(e) ii. 2 points The magnitude of the net force is greater on the block of mass 3M. If the incorrect selection is made, no points are earned for the justification. For an indication that both blocks will have the same acceleration For a correct justification for why the net force is greater on the block of mass 3M Example: Because the blocks stick together, both blocks must have the same acceleration. Because the block of mass 3M has more mass, the net force on it must be greater than the net force on the block of mass 2M.

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