PROBLEMS SOLVED IN CLASS

Chapter 2.

1. A person walks first at a constant speed of 5.00 m/s along a straight line from point A to point B and then back along the line from B to A at a constant speed of 3.00 m/s. (a) What is her average speed over the entire trip? (b) Her average velocity over the entire trip?

2. Halliday/Resnick/Walker 2.12

3. A particle moves along the x axis according to the equation x = 2.0 + 3.0 t - t² , where x is in meters and t in seconds. At t = 3 s, find (a) the position, (b) the velocity, (c) the acceleration of the particle. Is the velocity ever zero?

4. Given the acceleration versus time graph, sketch the graph of its velocity and position as functions of time if x = 0 and v = 0 at t = 0.

5. Halliday/Resnick/Walker 2.36

6. Halliday/Resnick/Walker 2.38

7. A student throws a set of keys vertically upward to a sorority sister, who is in a window 4.00 m above. The keys are caught 1.50 s later by the sister’s outstretched hand. (a) With what initial velocity were the keys thrown? (b) What was the velocity of the keys just before they were caught? (c) How high did the keys go?

8. Halliday/Resnick/Walker 2.61

Chapter 3.

1. Halliday/Resnick/Walker 3.2

2. A particle undergoes three consecutive displacements:

d₁ = (1.5 i + 3.0 j - 1.2 k) cm

d₂ = (2.3 i -1.4 j - 3.6 k) cm

d₃ = (-1.3 i + 1.5 j) cm

Find the components of the resultant displacement and its magnitude.

3. Halliday/Resnick/Walker 3.24

Chapter 4.

1. Halliday/Resnick/Walker 4.11

2. A rescue plane drops a package of emergency rations to a stranded party of explorers. If the plane is traveling horizontally at 40 m/s at a height of 100 m above the ground, where does the package strike the ground relative to the point at which it is released? What are the horizontal and vertical components of the velocity of the package just before it hits the ground?

3. Halliday/Resnick/Walker 4.33

4. A skier leaves the ramp of a ski jump with a velocity of 10 m/s, 15^o above the horizontal. The slope is inclined at 50^o, and air resistance is negligible. Find (a) the distance from the ramp where the jumper lands and (b) the velocity components just before the landing.

5. The orbit of the Moon around the Earth is approximately circular, with a mean radius of 3.84x10⁸ m. It takes 27.3 days for the Moon to complete one revolution around the Earth. Find (a) the mean orbital speed of the Moon and (b) its centripetal acceleration. Repeat the calculations for a geosynchronous satellite orbiting at 22,300 miles. Repeat calculation (b) for the Space Station that is orbiting the Earth at an average of 220 miles above the Earth’s surface and with an average speed of 17200 miles/hour.

6. Halliday/Resnick/Walker 4.54

7. Halliday/Resnick/Walker 4.57

8. Halliday/Resnick/Walker Problem Supplement 4.92

Chapters 5 and 6.

1. A hockey puck with a mass of 0.30 kg slides on the horizontal surface of an ice rink. Two forces act on the puck. Force F₁ has a magnitude of 5.0 N, and force F₂ has a magnitude of 8.0 N. Determine the acceleration of the puck.

2. A traffic light weighing 125 N hangs from a cable tied to two other cables fastened to a support. The upper cables make angles of 37^o and 53^o with the horizontal. Find the tension in the three cables.

3. A sled loaded with vacationing students (total weight W) slides down a long, snow-covered slope. The hill slopes at a constant angle α, and it is frictionless. What is the sled’s acceleration? What is the normal force?

4. Atwood’s machine. It consists of two unequal masses hung vertically over a frictionless pulley of negligible mass. Calculate the magnitude of the acceleration of the two masses and the tension in the string. Assume m₁ = 1.3 kg and m₂ = 2.8 kg.

5. Halliday/Resnick/Walker 5.34

6. A sled loaded with vacationing students (total weight W) slides down a long, snow-covered slope. The hill slopes at a constant angle α, and the coefficient of kinetic friction between the sled and snow is μ_k. Derive an expression for the acceleration in terms of g. α, μ_k, and W.

7. Halliday/Resnick/Walker 6.15

8. Two blocks with masses of 4.00 kg and 8.00 kg are connected by a string and slide down a 30^o inclined plane. The coefficients of kinetic friction are μ_k = 0.2 between the 4 kg block and the plane, a nd μ_k = 0.3 between the 8 kg block and the plane. a) Calculate the acceleration of each block. b) Calculate the tension in the string. Repeat the calculations but with μ_k = 0.1 between the 8 kg and the plane.

9. A car rounds a banked curve. The radius of curvature of the road is R, the banking angle is θ and the coefficient of static friction is μ_S. (a) Determine the range of speeds the car can have without slipping up or down the road. (b) Find the minimum value for μ_S such that the minimum speed is zero. (c) What is the range of speeds possible if R = 100 m, θ = 10^o, and μ_S = 0.1 (slippery conditions)?

10. A pilot of mass m in a jet aircraft executes a loop-the-loop maneuver. In this flying pattern, the aircraft moves in a vertical circle of radius 2.70 km at a constant speed of 225 m/s. Determine the force exerted by the seat on the pilot at (a) the bottom of the loop and (b) the top of the loop. Express the answers in terms of the weight of the pilot, mg.

Chapters 7 and 8.

1. A 6.0-kg block initially at rest is pulled to the right along a horizontal, frictionless surface by a constant, horizontal force of 12 N. Find the speed of the block after it has moved 3.0 m.

2. Find the final speed of the block described in the previous problem (Example 1) if the surface is rough and the coefficient of kinetic friction is 0.15.

3. Halliday/Resnick/Walker 7-44

4. Halliday/Resnick/Walker 7-17

5. Halliday/Resnick/Walker 7-24

6. Halliday/Resnick/Walker 7-22

7. An elevator has a mass of 1000 kg and carries a maximum load of 800 kg. A constant frictionless force of 4000 N retards its motion upward. (a) What must be the minimum power delivered by the motor to lift the elevator at a constant speed of 3.0 m/s? (b) What power must the motor deliver at any instant if it is designed to provide an upward acceleration of 1.0 m/s²?

8. A compact car has a mass of 800 kg and its efficiency is rated at 18%. Find the amount of gas used to accelerate the car from rest to 60 mi/hr (27 m/s). The energy equivalent of 1 gal of gasoline is 1.3x10⁸ J.

9. The simple pendulum consists of a sphere of mass m attached to a light cord of length L. The sphere is released from rest when the cord makes an angle θ_o with the vertical, and the pivot at P is frictionless. (A) Find the speed of the sphere when it is at the lowest point, b. (B) How much work does the tension T do? (C) What is the value of T at b?

10. A car in an amusement park ride rolls without friction around the track. It starts from rest at point A at a height h above the bottom of the loop. A) What is h_min (in terms of R) such that the car moves around the loop without falling off at the top? B) If h = 3.5 R and R = 30 m, compute the speed, a_rad,and a_tan of the passengers at C.

11. Halliday/Resnick/Walker 8-25

12. A small rock with a mass of 0.1 kg is released from rest at point A. When it reaches point B at the bottom of the hemispherical bowl, the rock is observed to have a speed of 1.8 m/s. Calculate the work done by friction on the rock when it moves from A to B.

13. A 10.0 kg-block is released from point A. The track is frictionless except for the portion BC, of length 6.0 m. the block travels down the track, hits a spring of force constant k=2250 N/m, and compress it 0.300 m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between surface BC and block.

14. Halliday/Resnick/Walker 8-58

Chapters 9 and 10.

1. A water molecule consists of an oxygen atom with two hydrogen atoms bound to it. The angle between the two bonds is 106^o. If the bonds are 0.100 nm long, where is the c.m. of the molecule?

2. Halliday/Resnick/Walker 9-4

3. Halliday/Resnick/Walker 9-6

4. James and Ramon are standing on the smooth surface of a frozen pond. Ramon has mass 60 kg and James 90 kg. They’re 20 m apart and hold the ends of a light rope that is stretched between them. Midway between the two men, a glass of soda sits on the ice. James pulls on the rope. What happens? When James has moved 4 m, how far has Ramon moved?

5. Halliday/Resnick/Walker 9-18

6. Halliday/Resnick/Walker 9-29

7. Halliday/Resnick/Walker 9-34

8. Halliday/Resnick/Walker 9-40

9. A rocket moving in free space has a speed of 3000 m/s relative to Earth. Its engines are turned on, and fuel is ejected in a direction opposite the rocket’s motion at a speed of 5000 m/s relative to the rocket. (a) What is the speed of the rocket relative to Earth once its mass is reduced to one half its mass before ignition? (b) What is the thrust on the rocket if it burns fuel at the rate of 50 kg/s?

10. Halliday/Resnick/Walker 10-10

11. Halliday/Resnick/Walker S10-2

12. Halliday/Resnick/Walker 10-40

13. A block of mass m₁ = 1.60 kg, initially moving to the right with a speed of 4.0 m/s on a frictionless horizontal track collides with a spring attached to a second block of mass m₂ = 2.10 kg moving to the left with a speed of 2.50 m/s. The spring has a force constant of 600 N/m. (a) At the instant when m₁ is moving to the right with a speed of 3.0 m/s, determine the speed of m₂. (b) Determine the distance the spring is compressed at that instant.

14. Halliday/Resnick/Walker 10-26

15. A 1500-kg car traveling east with a speed of 25.0 m/s collides with a 2500-kg van traveling north at a speed of 20.0 m/s. Find the direction and magnitude of the velocity of the wreckage after the collision, assuming that the vehicles undergo a perfectly inelastic collision.

16. Halliday/Resnick/Walker 10-54

Chapter 11.

1. Halliday/Resnick/Walker 11-4

2. Halliday/Resnick/Walker 11-16

3. The tub of a washer goes into its spin cycle, starting from rest and reaching an angular speed of 5.00 rev/s in 8.00 s. At this point the person doing the laundry opens the lid, and a safety switch turns off the washer. The tub slows to rest in 12.0 s. Through how many revolutions does the tub turn? Assume constant angular acceleration while the machine is starting and stopping.

4. Four pulleys are connected by two belts. Pulley A (r_A = 15 cm) is the drive pulley, and it rotates at 10 rad/s. Pulley B (r_B = 10 cm) is connected by belt 1 to pulley A. Pulley B’ (r_B’ = 5 cm) is concentric with pulley b and is rigidly attached to it. Pulley C (r_C = 25 cm) is connected by belt 2 to pulley B’. Calculate (a) the linear speed of a point on belt 1, (b) the angular speed of pulley B, (c) the angular speed of pulley B’, (d) the linear speed of a point on belt 2, and (e) the angular speed of a pulley C.

5. Halliday/Resnick/Walker 11-32

6. Halliday/Resnick/Walker 11-42

7. Halliday/Resnick/Walker 11-36

8. Halliday/Resnick/Walker 11-46

9. Find the net torque, on the wheel, about the axle through O if a = 10 cm and b = 25 cm.

10. A wheel of radius R, mass M, and rotational inertia I is mounted on a frictionless, horizontal axle. A light cord wrapped around the wheel supports an object of mass m. Calculate the linear acceleration of the object, the angular acceleration of the wheel, and the tension in the cord.

11. Halliday/Resnick/Walker 11-54

12. Halliday/Resnick/Walker 11-63

13. Consider two masses connected by a string that passes over a pulley having a rotational inertia of I about its axis of rotation. The string does not slip on the pulley, and the system is released from rest. Use conservation of energy to find the linear speeds of the masses after m₂ descends through a distance h and the angular speed of the pulley at this time.

Chapter 12.

1. Two vectors lying in the xy plane are given by A = 2i + 3j and B = -i + 2j. Find AxB, and verify explicitly that AxB = -BxA.

2. A particle is located at the vector position r = i + 3j m and the force acting on it is F = (3i + 2j)N. What is the torque about (a) the origin and (b) the point having coordinates (0,6)m.

3. Determine the angular momentum of the system shown, composed of two masses, m₁ and m₂, connected by a light cord that passes over a pulley of radius R and moment of inertia I about its axle.

4. A uniform solid sphere of radius R = .5 m and mass 15 kg rotates about the z-axis through its center. Find the magnitude of its angular momentum when the angular speed is 3 rad/s.

5. A projectile of mass m and velocity v_o is fired at a solid cylinder of mass M and radius R. The cylinder is initially at rest and is mounted on a fixed horizontal axle that runs through the center of mass. Find the angular speed of the system after the projectile strikes and adheres to the surface of the cylinder.

6. The merry-go-round. A horizontal platform in the shape of a circular disk rotates in a horizontal plane about a frictionless vertical axle. The platform has a mass M = 100 kg and radius R = 2m. A student whose mass is m = 60 kg walks slowly from the rim of the platform toward the center. If the angular speed of the system is 2 rad/s when the student is at the rim, (a) calculate the angular speed when the student has reached a point 0.5 m from the center, (b) calculate the initial and final rotational energies of the system.

Chapter 13.

1. A ball with radius r₁ = 0.60 m and mass 1.00 kg is attached by a uniform rod (mass = 2.00 kg) 0.400 m long to a second ball with radius r₂ = 0.080 m and mass 4.00 kg. Where is the center of gravity of the system?

2. Two people carry a heavy electric motor by placing it on a board 2.00 m long that weighs 200 N, with its center of gravity at its center. One person lifts at one end with a force of 700 n and the other lifts the opposite end with a force of 400 N. What is the weight of the motor, and where is its c.g.?

3. The horizontal beam in the figure weighs 150 N, and its c.g. is at its center. Find a) the tension in the upper cable, b) the horizontal and vertical components of the force exerted on the beam at the wall.