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Physics 105 Quick-Review Problem statements, answers, formula summaries Rev. 4/08 Note: Problem numbers refer to a previous textbook Underlined problems will probably be worked in class Contents: • Linear Kinematics: 4-28, 4-29 • Newton’s Second Law: 5-38, 5-43, 5-45 • More 2nd Law, Friction: 6-20, 6-22, 6-27, 6-39 • Kinetic Energy and Work: 7-16, 7-17, 7-22 • PE, Energy Conservation: 8-4, 8-19, 8-23, 8-36 • Mass Ctr., Momentum: 9-3, 9-4, 9-22, 9-39, 9-40 • Impulse and Collisions: 10-2, 10-7, 10-30, 10-35 1 Answers to Physics 105 Review Problems for Physics 106 Students (12/2003, rev 5/2006, rev 4/2008) 4-28: a) 12.0 m above release point b) vx = unchanged = 19.15 m/s, vy = 4.81 m/s c) No, since vy is positive 4-29: a) 10.9 m. b) 22.7 m c) 16.5 m/s d) – 62.5 degrees 5-38: a) 0.74 m/s2 , b) 7.27 m/s2 5-43: a) 0.735 m/s2 , b) down, c) T = 20.8 N 5-45: a) -1.18 m b) -0.674 s c) v = 3.5 m/s 6-20: max fs = 21.7 N a) fs = 16.6 N up b) fs = 19.64 N up c) fk = 14.8 N (slides) 6-22: Mass of block B = 3.3 kg 6-27: the block slides on the slab a) a(block) = -6.08 m/s2 b) a(slab) = 0.98 m/s2 6-39: 21 m 7-16: a) 267 N b) -401 J c) +401 J d) zero e) Zero 7-17: a) 11,642 J b) -10,584 J c) 1058 J d) 5.4 m/s 7-22: a) 0.29 J b) -1.8 J c) 3.5 m/s 8-4: a) zero b) mgh/2 c) mgh d) +mgh/2 e) mgh f) It would double as m -> 2m 8-19: 39.2 J b) 2.0 m vertical c) 4.0 m d) 39.2 J 8-23: a) 4.85 m/s b) 2.42 m/s 8-36: a) 4.8 N b) x = 1.5 m and x = 14 m c) 3.4 m/s 9-3: a) xcm = 1.07 m b) ycm = 1.33 m 9-4: xcm = 0, ycm = -0.2L 9-22: Dp = 4.9 kgm/s 9-39: v = 14 m/s at 225o CCW from x-axis 9-40: one chunk has v = 0, the other has v = 4 m/s 10-2: 6.2 x 104 N 10-7: a) 42 kgm/ s b) 2100 N 10-30: 7.35 cm 10-35: a) 1.9 m/s b) yes c) v = - 5.6 m/s No. 2 Summary: Linear Kinematics Chapter 4 3 Problem 4- 28P: You throw a ball toward a wall with a speed of 25.0 m/s and at an angle of 40.0° above the horizontal (Fig. 4-35). The wall is 22.0 m from the release point of the ball. (a) How far above the release point does the ball hit the wall? (b) What are the horizontal and vertical components of its velocity as it hits the wall? (c) When it hits, has it passed the highest point on its trajectory? Problem 4- 29E*: A ball is shot from the ground into the air. At a height of 9.1 m, its velocity is observed to be v = 7.6 i + 6.1 j in meters per second ( i horizontal, upward j). (a) To what maximum height does the ball rise? (b) What total horizontal distance does the ball travel? What are (c) the magnitude and (d) the direction of the ball's velocity just before it hits the ground? 4 Force and Motion – I 5 PROBLEM 5-38: A worker drags a crate across a factory floor by pulling on a rope tied to the crate (Fig. 5-38 ). The worker exerts a force of 450 N on the rope, which is inclined at 38° to the horizontal, and the floor exerts a horizontal force of 125 N that opposes the motion. Calculate the magnitude of the acceleration of the crate if (a) its mass is 310 kg and (b) its weight is 310 N. Problem 5 - 43P: A block of mass m1 = 3.70 kg on a frictionless inclined plane of angle 30.0° is connected by a cord over a massless, frictionless pulley to a second block of mass m2 = 2.30 kg hanging vertically (Fig. 5-41). What are (a) the magnitude of the acceleration of each block and (b) the direction of the acceleration of the hanging block? (c) What is the tension in the cord? PROBLEM 5-45: A block is projected up a frictionless inclined plane with initial speed v0 = 3.50 m/s. The angle of incline is q = 32.0°. (a) How far up the plane does it go? (b) How long does it take to get there? (c) What is its speed when it gets back to the bottom? (Can use second law + kinematics, or energy.) 6 Force and Motion – II (add friction) 7 PROBLEM 6-20: A force P, parallel to a surface inclined 15° above the horizontal, acts on a 45 N block, as shown in Fig. 6-30 . The coefficients of friction for the block and surface are ms = 0.50 and mk = 0.34. If the block is initially at rest, determine the magnitude and direction of the frictional force acting on the block for magnitudes of P of (a) 5.0 N, (b) 8.0 N, and (c) 15 N. Problem 6-22P: In the figure, two blocks are connected over a pulley. The mass of block A is 10 kg and the coefficient of kinetic friction between A and the incline is 0.20. Angle q of the incline is 30°. Block A slides down the incline at constant speed. What is the mass of block B? 8 PROBLEM 6-27: A 40 kg slab rests on a frictionless floor. A 10 kg block rests on top of the slab (Fig. 6-35 ). The coefficient of static friction ms between the block and the slab is 0.60, whereas their kinetic friction coefficient mk is 0.40. The 10 kg block is pulled by a horizontal force with a magnitude of 100 N. What are the resulting accelerations of (a) the block and (b) the slab? Problem 6-39P: What is the smallest radius of an unbanked (flat) track around which a bicyclist can travel if her speed is 29 km/h and the coefficient of static friction between tires and track is 0.32? 9 Kinetic Energy and Work 10 Problem 7-16: A 45 kg block of ice slides down a frictionless incline 1.5 m long and 0.91 m high. A worker pushes up against the ice, parallel to the incline, so that the block slides down at constant speed. (a) Find the magnitude of the worker's force. How much work is done on the block by (b) the worker's force, (c) the gravitational force on the block, (d) the normal force on the block from the surface of the incline, and (e) the net force on the block? PROBLEM 7-17: A helicopter lifts a 72 kg astronaut 15 m vertically from the ocean by means of a cable. The acceleration of the astronaut is g /10. How much work is done on the astronaut by (a) the force from the helicopter and (b) the gravitational force on her? What are (c) the kinetic energy and (d) the speed of the astronaut just before she reaches the helicopter? Problem 7 – 22P: A 250 g block is dropped onto a relaxed vertical spring that has a spring constant of k = 2.5 N/cm (see figure). The block becomes attached to the spring and compresses the spring 12 cm before momentarily stopping. While the spring is being compressed, what work is done on the block by (a) the gravitational force on it and (b) the spring force? (c) What is the speed of the block just before it hits the spring? (Assume that friction is negligible.) (d) If the speed at impact is doubled, what is the maximum compression of the spring? 11 Potential Energy and Energy Conservation Chapter 8 12 Problem 8 - 4E: In the figure, a frictionless roller coaster of mass m tops the first hill with speed v0. How much work does the gravitational force do on it from that point to (a) point A, (b) point B, and (c) point C? If the gravitational potential energy of the coaster–Earth system is taken to be zero at point C, what is its value when the coaster is at (d) point B and (e) point A? (f) If mass m were doubled, would the change in the gravitational potential energy of the system between points A and B increase, decrease, or remain the same? Problem 8 – 19P*: A 2.00 kg block is placed against a spring on a frictionless 30.0° incline (see figure). (The block is not attached to the spring.) The spring, whose spring constant k is 19.6 N/cm, is compressed 20.0 cm and then released. (a) What is the elastic potential energy of the compressed spring? (b) What is the change in the gravitational potential energy of the block–Earth system as the block moves from the release point to its highest point on the incline? (c) How far along the incline is the highest point from the release point? 13 Problem 8 –23P*: The string in the figure is L = 120 cm long, has a ball attached to one end, and is fixed at its other end. The distance d to the fixed peg at point P is 75.0 cm. When the initially stationary ball is released with the string horizontal as shown, it will swing along the dashed arc. What is its speed when it reaches (a) its lowest point and (b) its highest point after the string catches on the peg? Problem 8 – 36P: A conservative force F(x) acts on a 2.0 kg particle that moves along the x axis. The potential energy U(x) associated with F(x) is graphed in the figure. When the particle is at x = 2.0 m, its velocity is -1.5 m/s. (a) What are the magnitude and direction of F(x) at this position? (b) Between what limits of x does the particle move? (c) What is its speed at x = 7.0 m? 14 Systems of Particles SUMMARY: SYSTEMS OF PARTICLES 15 Problem 9 - 3: What are (a) the x coordinate and (b) the y coordinate of the center of mass of the three-particle system shown in the figure? (c) What happens to the center of mass as the mass of the topmost particle is gradually increased? Problem 9 - 4: Three thin rods, each of length L, are arranged in an inverted U, as shown in the figure. The two rods on the arms of the U each have mass M; the third rod has mass 3M. Where is the center of mass of the assembly? 16 Problem 9 – 22: A 0.70 kg ball is moving horizontally with a speed of 5.0 m/s when it strikes a vertical wall. The ball rebounds with a speed of 2.0 m/s. What is the magnitude of the change in linear momentum of the ball? Problem 9 – 39: A vessel at rest explodes, breaking into three pieces. Two pieces, having equal mass, fly off perpendicular to one another with the same speed of 30 m/s. The third piece has three times the mass of each other piece. What are the magnitude and direction of its velocity immediately after the explosion? Problem 9 – 40: An 8.0 kg body is traveling at 2.0 m/s with no external force acting on it. At a certain instant an internal explosion occurs, splitting the body into two chunks of 4.0 kg mass each. The explosion gives the chunks an additional 16 J of kinetic energy. Neither chunk leaves the line of original motion. Determine the speed and direction of motion of each of the chunks after the explosion. 17 Impulse and Collisions 18 PROBLEM 10-2: The National Transportation Safety Board is testing the crash-worthiness of a new car. The 2300 kg vehicle, moving at 15 m/s, is allowed to collide with a bridge abutment, which stops it in 0.56 s. What is the magnitude of the average force that acts on the car during the impact? Problem 10-7: A 1.2 kg ball drops vertically onto a floor, hitting with a speed of 25 m/s. It rebounds with an initial speed of 10 m/s. (a) What impulse acts on the ball during the contact? (b) If the ball is in contact with the floor for 0.020 s, what is the magnitude of the average force on the floor from the ball? 19 PROBLEM 10-30: A 10 g bullet moving directly upward at 1000 m/s strikes and passes through the mass center of a 5.0 kg block initially at rest . The bullet emerges from the block moving directly upward at 400 m/s. To what maximum height does the block then rise above its initial position? Problem 10-35: The blocks in the figure slide without friction. (a) What is the velocity of the 1.6 kg block after the collision? (b) Is the collison elastic? (c) Suppose the initial velocity of the 2.4 kg block is the reverse of what is shown. Can the velocity of the 1.6 kg block after the collision be in the direction shown? 20