(1) The figure shows a lever (which is a uniform bar, length d and mass M), hinged at the bottom and supported steadily by a rope. The rope is attached a distance d/4 from the hinge. The two angles are 50 as shown, and hence the rope is perpendicular to the lever. (a) Find the tension in the rope in terms of the given quantities. (b) If the rope were cut, allowing the lever to drop, find the angular velocity of the lever just before hitting the floor, if the hinge has no friction. Assume for this part that the lever has length 3.5 m and mass 46 kg. (2) Choose the graph below that represents the velocity vs. time for constant, nonzero acceleration in one dimension. 2
(3) A projectile is fired from ground level, with initial speed 5.0 m/s. If the projectile is fired at an angle of 50 from horizontal, neglecting air friction, find: (a) the time required before the projectile hits the ground. (b) the x and y components of the velocity once the projectile has traveled 0.5 m horizontally. (4) A force, F 1, of magnitude 2.0 N and directed due east, is exerted on an object. A second force exerted on the object is F 2 = 2.0 N, due north. What is the magnitude and direction of a third force, F 3, which must be exerted on the object so that the resultant force is zero? a. 1.4 N, 45 north of east b. 1.4 N, 45 south of west c. 2.8 N, 45 north of east d. 2.8 N, 45 south of west e. 4.0 N, 45 east of north f. 2.0 N, 45 north of east g. 2.0 N, 45 south of east (5) In which one of the following situations does a car have an acceleration vector pointing towards the west? a. The car travels westward at constant speed. b. The car travels eastward and speeds up. c. The car travels westward and slows down. d. The car travels eastward and slows down. e. The car starts up from rest heading toward the east. f. The car traveling south at constant speed starts to curve towards the east. 3
(6) The two weights in the illustration are connected with a massless rope, and acted upon by normal gravity. The pulley has low friction and negligible moment of inertia. There is no friction between the weights and the incline. M = 45 kg, and m = 25 kg. Find the acceleration of the weight m (with direction; is it up or down the incline?). M m 45 45 (7) A 0.011 kg dart is pushed downward into a dart gun, compressing the (massless) spring by 0.030 m. The spring constant is 2400 N/m. Once the dart is released and flies free of the gun, what is the maximum height attained, assuming that 0.33 J is lost to heat due to air friction during the upward climb of the dart? The gun points straight upward. (8) Jim sits at his desk chair, which rotates freely. He sits initially with arms folded, and the chair spinning. He then stretches both arms straight out to his sides. Find the true statement. (a) Jim s angular velocity will increase, while the angular velocity of the chair will decrease accordingly. (b) The angular velocity of the system will increase, proportional to the increase of the moment of inertia. (c) Since this is a closed system, the moment of inertia remains constant. (d) The moment of inertia decreases, due to the energy expended in extending his arms. (e) The angular velocity of the system decreases, but the angular momentum remains the same. (f) Since there is no external torque, the angular velocity of Jim and of the chair both remain constant. (g) The angular momentum decreases, due to conservation of energy. 4
(9) A spool, as shown, is composed of a large wheel, diameter D and mass 3.0 kg, glued to a small wheel, diameter D/2 and mass 5.2 kg. Both wheels are solid disks. The spool is supported by a frictionless axle at its center. (a) Find the moment of inertia of the spool, if D = 0.25 m. (b) A rope is wound around the smaller wheel of this spool, as shown. The rope is pulled with constant tension, causing the spool to go from rest to 350 r.p.m. in 0.90 s. Find the tension in the rope. (c) During this process, choose the correct statement: (i) The tangential velocity at the outer edge of the spool is constant. (ii) The tangential velocities at the edges of the two spools are equal to each other, and increasing with time. (iii) The tangential acceleration at the outer edge of the spool is zero. (iv) At the edge of the small spool the tangential acceleration is smaller, and the radial acceleration larger, than at the edge of the large spool. (v) At the edge of the small spool the tangential acceleration is larger, and the radial acceleration smaller, than at the edge of the large spool. (10) Which one of the following statements is true concerning an object executing simple harmonic motion? (a) Its velocity is never zero. (b) Its acceleration is never zero. (c) Its velocity and acceleration are simultaneously zero. (d) Its velocity is zero when its acceleration is a maximum. (e) Its maximum acceleration is equal to its maximum velocity. (f) Its maximum acceleration is equal to 9.8 m/s 2. 5
(11) An amusement park ride involves a car attached to a cable which takes its occupants through a vertical circular arc. The cable length is 11.0 m, and the car (with occupants) has mass 250 kg. If the instantaneous speed (v) at the bottom of the arc is 21 m/s, (a) Find the acceleration (with direction) of the car at the instant it is at the bottom of the arc. (b) For an 75 kg occupant of the car, sitting upright on a seat inside (with no seat-belt), find the normal force exerted by the car on the occupant, at the bottom of the arc. (c) If the car continues around in a vertical circle, draw a force diagram for the occupant (now upside-down) when the car is at the highest point, and derive the minimum velocity at the top which allows the occupant to stay in her seat. [This may require a different starting velocity at the bottom than specified in part (a), so don t use the part-(a) velocity as a starting point.] (12) On an icy road, a car of mass M and initial velocity v o rear-ends a truck, with mass 2M. The truck is initially stationary. The vehicles lock together after the collision, and slide without friction on the road. Choose the correct statement about this collision: (a) The velocity after the collision is v o. (b) The final velocity can be found by considering that the momentum and the kinetic energy are both conserved during the collision. (c) The final velocity after the collision will be the same as the center of mass velocity before the collision. (d) The kinetic energy will be conserved in the collision, but not the momentum. (e) The final velocity will be zero since all energy is dissipated in such a collision. (f) The velocity after the collision is v o /2. 6
(13) A guitar string is 75 cm long, and is tuned to vibrate in its fundamental mode at 220 Hz. What is the wave propagation velocity for this string? (14) For the string from problem (13), what is the frequency of the next highest mode? (15) Suppose that astronomers discover that a distant, spherical, planet has just formed in our solar system, with the same mass as the Earth, but twice the radius. The acceleration of gravity at the surface of this new planet will be, (a) twice that of Earth. (b) one half that of Earth. (c) one quarter that of Earth. (d) three times that of Earth. (e) the same as that of Earth. (f) zero. (g) impossible to determine without knowing the composition of the new planet. 7