Chapter 1 Problems 4. Each of the following equations was given by a student during an examination: 1 2 1
2
mv = mv 0 + mgh 2
2
v = v 0 + at 2 ma = v 2 Do a dimensional analysis of each equation and explain why the equation can’t be correct. 5. Newton’s law of universal gravitation is represented by Mm
F =G 2 r
where F is the gravitational force, M and m are masses, and r is a length. Force has the SI units kg ∙ m/s2. What are the SI units of the proportionality constant G? Section 1.4 Uncertainty in Measurement and Significant Figures 7. How many significant figures are there in (a) 78.9 ± 0.2, (b) 3.788 × 109, (c) 2.46 × 10–6, (d) 0.0032? 11. A farmer measures the perimeter of a rectangular field. The length of each long side of the rectangle is found to be 38.44 m, and the length of each short side is found to be 19.5 m. What is the perimeter of the field? Section 1.5 Conversion of Units 15. A fathom is a unit of length, usually reserved for measuring the depth of water. A fathom is approximately 6 ft in length. Take the distance from Earth to the Moon to be 250 000 miles, and use the given approximation to find the distance in fathoms. 17. A rectangular building lot measures 100 ft by 150 ft. Determine the area of this lot in square meters (m2). 21. The speed of light is about 3.00 × 108 m/s. Convert this figure to miles per hour. 23. The amount of water in reservoirs is often measured in acre‐
ft. One acre‐ft is a volume that covers an area of one acre to a depth of one foot. An acre is 43 560 ft2. Find the volume in SI units of a reservoir containing 25.0 acre‐ft of water. 26. (a) Find a conversion factor to convert from miles per hour to kilometers per hour. (b) For a while, federal law mandated that the maximum highway speed would be 55 mi/h. Use the conversion factor from part (a) to find the speed in kilometers per hour. (c) The maximum highway speed has been raised to 65 mi/h in some places. In kilometers per hour, how much of an increase is this over the 55‐mi/h limit? Section 1.7 Coordinate Systems 35. A point is located in a polar coordinate system by the coordinates r = 2.5 m and θ = 35°. Find the x‐ and y‐coordinates of this point, assuming that the two coordinate systems have the same origin. 38. Two points in a rectangular coordinate system have the coordinates (5.0, 3.0) and (–3.0, 4.0), where the units are centimeters. Determine the distance between these points. Section 1.8 Trigonometry 39. For the triangle shown in Figure P1.39, what are (a) the length of the unknown side, (b) the tangent of θ, and (c) the sine of φ? Figure P1.39 40. A ladder 9.00 m long leans against the side of a building. If the ladder is inclined at an angle of 75.0° to the horizontal, what is the horizontal distance from the bottom of the ladder to the building? 42. A right triangle has a hypotenuse of length 3.00 m, and one of its angles is 30.0°. What are the lengths of (a) the side opposite the 30.0° angle and (b) the side adjacent to the 30.0° angle? 43. In Figure P1.43, find (a) the side opposite θ, (b) the side adjacent to φ, (c) cos θ, (d) sin φ, and (e) tan φ. Figure P1.43 46. A surveyor measures the distance across a straight river by the following method: Starting directly across from a tree on the opposite bank, he walks 100 m along the riverbank to establish a baseline. Then he sights across to the tree. The angle from his baseline to the tree is 35.0°. How wide is the river? Chapter 2 Problems Section 2.1 Displacement Section Section 2.2 Velocity 1. A person travels by car from one city to another with different constant speeds between pairs of cities. She drives for 30.0 min at 80.0 km/h, 12.0 min at 100 km/h, and 45.0 min at 40.0 km/h and spends 15.0 min eating lunch and buying gas. (a) Determine the average speed for the trip. (b) Determine the distance between the initial and final cities along the route. 2. (a) Sand dunes on a desert island move as sand is swept up the windward side to settle in the leeward side. Such “walking” dunes have been known to travel 20 feet in a year and can travel as much as 100 feet per year in particularly windy times. Calculate the average speed in each case in m/s. (b) Fingernails grow at the rate of drifting continents, about 10 mm/yr. Approximately how long did it take for North America to separate from Europe, a distance of about 3 000 mi? 3. Two boats start together and race across a 60‐km‐wide lake and back. Boat A goes across at 60 km/h and returns at 60 km/h. Boat B goes across at 30 km/h, and its crew, realizing how far behind it is getting, returns at 90 km/h. Turnaround times are negligible, and the boat that completes the round trip first wins. (a) Which boat wins and by how much? (Or is it a tie?) (b) What is the average velocity of the winning boat? 5. A motorist drives north for 35.0 minutes at 85.0 km/h and then stops for 15.0 minutes. He then continues north, traveling 130 km in 2.00 h. (a) What is his total displacement? (b) What is his average velocity? 6. A graph of position versus time for a certain particle moving along the x‐axis is shown in Figure P2.6. Find the average velocity in the time intervals from (a) 0 to 2.00 s, (b) 0 to 4.00 s, (c) 2.00 s to 4.00 s, (d) 4.00 s to 7.00 s, and (e) 0 to 8.00 s. Figure P2.6 (Problems 6 and 15) 11. A person takes a trip, driving with a constant speed of 89.5 km/h, except for a 22.0‐min rest stop. If the person’s average speed is 77.8 km/h, how much time is spent on the trip and how far does the person travel? 12. A tortoise can run with a speed of 0.10 m/s, and a hare can run 20 times as fast. In a race, they both start at the same time, but the hare stops to rest for 2.0 minutes. The tortoise wins by a shell (20 cm). (a) How long does the race take? (b) What is the length of the race? 15. A graph of position versus time for a certain particle moving along the x‐axis is shown in Figure P2.6. Find the instantaneous velocity at the instants (a) t = 1.00 s, (b) t = 3.00 s, (c) t = 4.50 s, and (d) t = 7.50 s. Section 2.3 Acceleration 22. The velocity vs. time graph for an object moving along a straight path is shown in Figure P2.22. (a) Find the average acceleration of the object during the time intervals 0 to 5.0 s, 5.0 s to 15 s, and 0 to 20 s. (b) Find the instantaneous acceleration at 2.0 s, 10 s, and 18 s. Figure P2.22 Section 2.5 One‐Dimensional Motion with Constant Acceleration 26. A truck covers 40.0 m in 8.50 s while smoothly slowing down to a final speed of 2.80 m/s. (a) Find the truck’s original speed. (b) Find its acceleration. 30. A truck on a straight road starts from rest and accelerates at 2.0 m/s2 until it reaches a speed of 20 m/s. Then the truck travels for 20 s at constant speed until the brakes are applied, stopping the truck in a uniform manner in an additional 5.0 s. (a) How long is the truck in motion? (b) What is the average velocity of the truck during the motion described? 33. A driver in a car traveling at a speed of 60 mi/h sees a deer 100 m away on the road. Calculate the minimum constant acceleration that is necessary for the car to stop without hitting the deer (assuming that the deer does not move in the meantime). 35. A train is traveling down a straight track at 20 m/s when the engineer applies the brakes, resulting in an acceleration of –1.0 m/s2 as long as the train is in motion. How far does the train move during a 40‐s time interval starting at the instant the brakes are applied? 37. A car starts from rest and travels for 5.0 s with a uniform acceleration of +1.5 m/s2. The driver then applies the brakes, causing a uniform acceleration of –2.0 m/s2. If the brakes are applied for 3.0 s, (a) how fast is the car going at the end of the braking period, and (b) how far has the car gone? Section 2.6 Freely Falling Objects 43. A ball is thrown vertically upward with a speed of 25.0 m/s. (a) How high does it rise? (b) How long does it take to reach its highest point? (c) How long does the ball take to hit the ground after it reaches its highest point? (d) What is its velocity when it returns to the level from which it started? 47. A small mailbag is released from a helicopter that is descending steadily at 1.50 m/s. After 2.00 s, (a) what is the speed of the mailbag, and (b) how far is it below the helicopter? (c) What are your answers to parts (a) and (b) if the helicopter is rising steadily at 1.50 m/s? 49. A model rocket is launched straight upward with an initial speed of 50.0 m/s. It accelerates with a constant upward acceleration of 2.00 m/s2 until its engines stop at an altitude of 150 m. (a) What is the maximum height reached by the rocket? (b) How long after lift‐off does the rocket reach its maximum height? (c) How long is the rocket in the air? Chapter 3 Problems Section 3.1 Vectors and Their Properties 2. An airplane flies 200 km due west from city A to city B and then 300 km in the direction of 30.0° north of west from city B to city C. (a) In straight‐line distance, how far is city C from city A? (b) Relative to city A, in what direction is city C? 4. A jogger runs 100 m due west, then changes direction for the second leg of the run. At the end of the run, she is 175 m away from the starting point at an angle of 15.0° north of west. What were the direction and length of her second displacement? Use graphical techniques. 5. A plane flies from base camp to lake A, a distance of 280 km at a direction of 20.0° north of east. After dropping off supplies, the plane flies to lake B, which is 190 km and 30.0° west of north from lake A. Graphically determine the distance and direction from lake B to the base camp. r
8. Each of the displacement vectors A r
and B shown in Figure P3.8 has a magnitude of 3.00 m. Graphically find (a) r
r
r
r
r
r
r
A + B , (b) A – B , (c) B – A , and (d) A – r
2 B . Figure P3.8 Section 3.2 Components of a Vector 13. A vector has an x‐component of –25.0 units and a y‐component of 40.0 units. Find the magnitude and direction of the vector. 17. A commuter airplane starts from an airport and takes the route shown in Figure P3.17. The plane first flies to city A, located 175 km away in a direction 30.0° north of east. Next, it flies for 150 km 20.0° west of north, to city B. Finally, the plane flies 190 km due west, to city C. Find the location of city C relative to the location of the starting point. edge with a speed of 18.0 m/s. The cliff is 50.0 m above a flat, horizontal beach, as shown in Figure P3.24. How long after being released does the stone strike the beach below the cliff? With what speed and angle of impact does the stone land? Figure P3.17 18. The helicopter view in Figure P3.18 shows two people pulling on a stubborn mule. Find (a) the single force that is equivalent to the two forces shown and (b) the force that a third person would have to exert on the mule to make the net force equal to zero. The forces are measured in units of newtons (N). Figure P3.24 Figure P3.18 24. A student stands at the edge of a cliff and throws a stone horizontally over the 28. An artillery shell is fired with an initial velocity of 300 m/s at 55.0° above the horizontal. To clear an avalanche, it explodes on a mountainside 42.0 s after firing. What are the x‐ and y‐coordinates of the shell where it explodes, relative to its firing point? 29. A brick is thrown upward from the top of a building at an angle of 25° to the horizontal and with an initial speed of 15 m/s. If the brick is in flight for 3.0 s, how tall is the building? Chapter 4 Problems Section 4.1 Forces Section 4.2 Newton’s First Law Section 4.3 Newton’s Second Law Section 4.4 Newton’s Third Law 1. A 6.0‐kg object undergoes an acceleration of 2.0 m/s2. (a) What is the magnitude of the resultant force acting on it? (b) If this same force is applied to a 4.0‐
kg object, what acceleration is produced? 6. A freight train has a mass of 1.5 × 107 kg. If the locomotive can exert a constant pull of 7.5 × 105 N, how long does it take to increase the speed of the train from rest to 80 km/h? 8. A 5.0‐g bullet leaves the muzzle of a rifle with a speed of 320 m/s. What force (assumed constant) is exerted on the bullet while it is traveling down the 0.82‐m‐long barrel of the rifle? 12. Two forces are applied to a car in an effort to move it, as shown in Figure P4.12. (a) What is the resultant of these two forces? (b) If the car has a mass of 3 000 kg, what acceleration does it have? Ignore friction. Figure P4.12 13. After falling from rest from a height of 30 m, a 0.50‐kg ball rebounds upward, reaching a height of 20 m. If the contact between ball and ground lasted 2.0 ms, what average force was exerted on the ball? Section 4.5 Applications of Newton’s Laws 15. Find the tension in each cable supporting the 600‐N cat burglar in Figure P4.15. Figure P4.15 Figure P4.19 16. Find the tension in the two wires that support the 100‐N light fixture in Figure P4.16. Figure P4.16 19. Two blocks are fastened to the ceiling of an elevator as in Figure P4.19. The elevator accelerates upward at 2.00 m/s2. Find the tension in each rope. tension does the bird produce in the wire? Ignore the weight of the wire. 25. A 2 000‐kg car is slowed down uniformly from 20.0 m/s to 5.00 m/s in 4.00 s. (a) What average force acted on the car during that time, and (b) how far did the car travel during that time? 26. Two packing crates of masses 10.0 kg and 5.00 kg are connected by a light string that passes over a frictionless pulley as in Figure P4.26. The 5.00‐kg crate lies on a smooth incline of angle 40.0°. Find the acceleration of the 5.00‐kg crate and the tension in the string. the average value of the coefficient of kinetic friction between puck and ice? (c) How far does the puck travel during the 5.00‐s interval? 41. The coefficient of static friction between the 3.00‐kg crate and the 35.0° incline of Figure P4.41 is 0.300. What r
minimum force F must be applied to the crate perpendicular to the incline to prevent the crate from sliding down the incline? Figure P4.26 Section 4.6 Forces of Friction 37. A 1 000‐N crate is being pushed across a level floor at a constant speed by a r
force F of 300 N at an angle of 20.0° below the horizontal, as shown in Figure P4.37a. (a) What is the coefficient of kinetic friction between the crate and the floor? (b) If the 300‐N force is instead pulling the block at an angle of 20.0° above the horizontal, as shown in Figure P4.37b, what will be the acceleration of the crate? Assume that the coefficient of friction is the same as that found in (a). Figure P4.37 38. A hockey puck is hit on a frozen lake and starts moving with a speed of 12.0 m/s. Five seconds later, its speed is 6.00 m/s. (a) What is its average acceleration? (b) What is Figure P4.41 47. A 3.00‐kg block starts from rest at the top of a 30.0° incline and slides 2.00 m down the incline in 1.50 s. Find (a) the acceleration of the block, (b) the coefficient of kinetic friction between the block and the incline, (c) the frictional force acting on the block, and (d) the speed of the block after it has slid 2.00 m. 58. (a) What is the minimum force of friction required to hold the system of Figure P4.58 in equilibrium? (b) What coefficient of static friction between the 100‐
N block and the table ensures equilibrium? (c) If the coefficient of kinetic friction between the 100‐N block and the table is 0.250, what hanging weight should replace the 50.0‐N weight to allow the system to move at a constant speed once it is set in motion? Figure P4.58 Chapter 5 Problems Section 5.1 Work 1. A weight lifter lifts a 350‐N set of weights from ground level to a position over his head, a vertical distance of 2.00 m. How much work does the weight lifter do, assuming he moves the weights at constant speed? 2. If a man lifts a 20.0‐kg bucket from a well and does 6.00 kJ of work, how deep is the well? Assume that the speed of the bucket remains constant as it is lifted. 6. A horizontal force of 150 N is used to push a 40.0‐kg packing crate a distance of 6.00 m on a rough horizontal surface. If the crate moves at constant speed, find (a) the work done by the 150‐N force and (b) the coefficient of kinetic friction between the crate and surface. 7. A sledge loaded with bricks has a total mass of 18.0 kg and is pulled at constant speed by a rope inclined at 20.0° above the horizontal. The sledge moves a distance of 20.0 m on a horizontal surface. The coefficient of kinetic friction between the sledge and surface is 0.500. (a) What is the tension in the rope? (b) How much work is done by the rope on the sledge? (c) What is the mechanical energy lost due to friction? Section 5.2 Kinetic Energy and the Work–Energy Theorem 9. A mechanic pushes a 2.50 × 103‐kg car from rest to a speed of v, doing 5 000 J of work in the process. During this time, the car moves 25.0 m. Neglecting friction between car and road, find (a) v and (b) the horizontal force exerted on the car. 10. A 7.00‐kg bowling ball moves at 3.00 m/s. How fast must a 2.45‐g Ping‐Pong ball move so that the two balls have the same kinetic energy? 12. A crate of mass 10.0 kg is pulled up a rough incline with an initial speed of 1.50 m/s. The pulling force is 100 N parallel to the incline, which makes an angle of 20.0° with the horizontal. The coefficient of kinetic friction is 0.400, and the crate is pulled 5.00 m. (a) How much work is done by gravity? (b) How much mechanical energy is lost due to friction? (c) How much work is done by the 100‐N force? (d) What is the change in kinetic energy of the crate? (e) What is the speed of the crate after being pulled 5.00 m? 15. A 2.0‐g bullet leaves the barrel of a gun at a speed of 300 m/s. (a) Find its kinetic energy. (b) Find the average force exerted by the expanding gases on the bullet as it moves the length of the 50‐cm‐
long barrel. 30. A bead of mass m = 5.00 kg is released from point and slides on the frictionless track shown in Figure P5.30. Determine (a) the bead’s speed at points and and (b) the net work done by the force of gravity in moving the bead from to . Figure P5.30 Figure P5.32 (Problems 32 and 89) 33. The launching mechanism of a toy gun consists of a spring of unknown spring constant, as shown in Figure P5.33a. If the spring is compressed a distance of 0.120 m and the gun fired vertically as shown, the gun can launch a 20.0‐g projectile from rest to a maximum height of 20.0 m above the starting point of the projectile. Neglecting all resistive forces, determine (a) the spring constant and (b) the speed of the projectile as it moves through the equilibrium position of the spring (where x = 0), as shown in Figure P5.33b. Figure P5.33 41. A 2.1 × 103‐kg car starts from rest at the top of a 5.0‐m‐long driveway that is inclined at 20° with the horizontal. If an average friction force of 4.0 × 103 N impedes the motion, find the speed of the car at the bottom of the driveway. 43. Starting from rest, a 10.0‐kg block slides 3.00 m down to the bottom of a frictionless ramp inclined 30.0° from the floor. The block then slides an additional 5.00 m along the floor before coming to a stop. Determine (a) the speed of the block at the bottom of the ramp, (b) the coefficient of kinetic friction between block and floor, and (c) the mechanical energy lost due to friction. Section 5.6 Power 51. The electric motor of a model train accelerates the train from rest to 0.620 m/s in 21.0 ms. The total mass of the train is 875 g. Find the average power delivered to the train during its acceleration. 52. An electric scooter has a battery capable of supplying 120 Wh of energy. [Note that an energy of 1 Wh = (1 J/s)(3600 s) = 3600 J] If frictional forces and other losses account for 60.0% of the energy usage, what change in altitude can a rider achieve when driving in hilly terrain if the rider and scooter have a combined weight of 890 N? 54. A 650‐kg elevator starts from rest and moves upwards for 3.00 s with constant acceleration until it reaches its cruising speed, 1.75 m/s. (a) What is the average power of the elevator motor during this period? (b) How does this amount of power compare with its power during an upward trip with constant speed? Section 5.7 Work Done by a Varying Force The force acting on 55. a particle varies as in Figure P5.55. Find the work done by the force as the particle moves (a) from x = 0 to x = 8.00 m, (b) from x = 8.00 m to x = 10.0 m, and (c) from x = 0 to x = 10.0 m. Figure P5.55 56. An object is subject to a force Fx that varies with position as in Figure P5.56. Find the work done by the force on the object as it moves (a) from x = 0 to x = 5.00 m, (b) from x = 5.00 m to x = 10.0 m, and (c) from x = 10.0 m to x = 15.0 m. (d) What is the total work done by the force over the distance x = 0 to x = 15.0 m? Figure P5.56 Chapter 6 Problems Section 6.1 Momentum and Impulse 1. A ball of mass 0.150 kg is dropped from rest from a height of 1.25 m. It rebounds from the floor to reach a height of 0.960 m. What impulse was given to the ball by the floor? 2. A tennis player receives a shot with the ball (0.060 0 kg) traveling horizontally at 50.0 m/s and returns the shot with the ball traveling horizontally at 40.0 m/s in the opposite direction. (a) What is the impulse delivered to the ball by the racquet? (b) What work does the racquet do on the ball? 4. A 0.10‐kg ball is thrown straight up into the air with an initial speed of 15 m/s. Find the momentum of the ball (a) at its maximum height and (b) halfway to its maximum height. 8. A 75.0‐kg stuntman jumps from a balcony and falls 25.0 m before colliding with a pile of mattresses. If the mattresses are compressed 1.00 m before he is brought to rest, what is the average force exerted by the mattresses on the stuntman? 11. The force shown in the force vs. time diagram in Figure P6.11 acts on a 1.5‐kg object. Find (a) the impulse of the force, (b) the final velocity of the object if it is initially at rest, and (c) the final velocity of the object if it is initially moving along the x‐axis with a velocity of –2.0 m/s. Figure P6.11 12. A force of magnitude Fx acting in the x‐direction on a 2.00‐kg particle varies in time as shown in Figure P6.12. Find (a) the impulse of the force, (b) the final velocity of the particle if it is initially at rest, and (c) the final velocity of the particle if it is initially moving along the x‐axis with a velocity of –2.00 m/s. Figure P6.12 15. The front 1.20 m of a 1 400‐kg car is designed as a “crumple zone” that collapses to absorb the shock of a collision. If a car traveling 25.0 m/s stops uniformly in 1.20 m, (a) how long does the collision last, (b) what is the magnitude of the average force on the car, and (c) what is the acceleration of the car? Express the acceleration as a multiple of the acceleration of gravity. Section 6.2 Conservation of Momentum 18. A 730‐N man stands in the middle of a frozen pond of radius 5.0 m. He is unable to get to the other side because of a lack of friction between his shoes and the ice. To overcome this difficulty, he throws his 1.2‐
kg physics textbook horizontally toward the north shore at a speed of 5.0 m/s. How long does it take him to reach the south shore? 20. A rifle with a weight of 30 N fires a 5.0‐g bullet with a speed of 300 m/s. (a) Find the recoil speed of the rifle. (b) If a 700‐
N man holds the rifle firmly against his shoulder, find the recoil speed of the man and rifle. 22. A 65.0‐kg person throws a 0.045 0‐kg snowball forward with a ground speed of 30.0 m/s. A second person, with a mass of 60.0 kg, catches the snowball. Both people are on skates. The first person is initially moving forward with a speed of 2.50 m/s, and the second person is initially at rest. What are the velocities of the two people after the snowball is exchanged? Disregard friction between the skates and the ice. Section 6.3 Collisions Section 6.4 Glancing Collisions 27. A railroad car of mass 2.00 × 104 kg moving at 3.00 m/s collides and couples with two coupled railroad cars, each of the same mass as the single car and moving in the same direction at 1.20 m/s. (a) What is the speed of the three coupled cars after the collision? (b) How much kinetic energy is lost in the collision? 30. An 8.00‐g bullet is fired into a 250‐g block that is initially at rest at the edge of a table of height 1.00 m (Fig. P6.30). The bullet remains in the block, and after the impact the block lands 2.00 m from the bottom of the table. Determine the initial speed of the bullet. Figure P6.30 32. A 1 200‐kg car traveling initially with a speed of 25.0 m/s in an easterly direction crashes into the rear end of a 9 000‐kg truck moving in the same direction at 20.0 m/s (Fig. P6.32). The velocity of the car right after the collision is 18.0 m/s to the east. (a) What is the velocity of the truck right after the collision? (b) How much mechanical energy is lost in the collision? Account for this loss in energy. Figure P6.34 Figure P6.32 34. (a) Three carts of masses 4.0 kg, 10 kg, and 3.0 kg move on a frictionless horizontal track with speeds of 5.0 m/s, 3.0 m/s, and 4.0 m/s, as shown in Figure P6.34. The carts stick together after colliding. Find the final velocity of the three carts. (b) Does your answer require that all carts collide and stick together at the same time? 40. A billiard ball rolling across a table at 1.50 m/s makes a head‐on elastic collision with an identical ball. Find the speed of each ball after the collision (a) when the second ball is initially at rest, (b) when the second ball is moving toward the first at a speed of 1.00 m/s, and (c) when the second ball is moving away from the first at a speed of 1.00 m/s. 43. A 2 000‐kg car moving east at 10.0 m/s collides with a 3 000‐kg car moving north. The cars stick together and move as a unit after the collision, at an angle of 40.0° north of east and a speed of 5.22 m/s. Find the speed of the 3 000‐kg car before the collision. Chapter 7 Problems Section 7.1 Angular Speed and Angular Acceleration 1. The tires on a new compact car have a diameter of 2.0 ft and are warranted for 60 000 miles. (a) Determine the angle (in radians) through which one of these tires will rotate during the warranty period. (b) How many revolutions of the tire are equivalent to your answer in (a)? 2. A wheel has a radius of 4.1 m. How far (path length) does a point on the circumference travel if the wheel is rotated through angles of 30°, 30 rad, and 30 rev, respectively? 4. A potter’s wheel moves from rest to an angular speed of 0.20 rev/s in 30 s. Find its angular acceleration in radians per second per second. Section 7.2 Rotational Motion under Constant Angular Acceleration Section 7.3 Relations between Angular and Linear Quantities 5. A dentist’s drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of 2.51 × 104 rev/min. (a) Find the drill’s angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period. 7. A machine part rotates at an angular speed of 0.60 rad/s; its speed is then increased to 2.2 rad/s at an angular acceleration of 0.70 rad/s2. Find the angle through which the part rotates before reaching this final speed. 9. The diameters of the main rotor and tail rotor of a single‐engine helicopter are 7.60 m and 1.02 m, respectively. The respective rotational speeds are 450 rev/min and 4 138 rev/min. Calculate the speeds of the tips of both rotors. Compare these speeds with the speed of sound, 343 m/s. 12. A coin with a diameter of 2.40 cm is dropped on edge onto a horizontal surface. The coin starts out with an initial angular speed of 18.0 rad/s and rolls in a straight line without slipping. If the rotation slows with an angular acceleration of magnitude 1.90 rad/s2, how far does the coin roll before coming to rest? 13. A rotating wheel requires 3.00 s to rotate 37.0 revolutions. Its angular velocity at the end of the 3.00‐s interval is 98.0 rad/s. What is the constant angular acceleration of the wheel? Section 7.4 Centripetal Acceleration 14. It has been suggested that rotating cylinders about 10 mi long and 5.0 mi in diameter be placed in space and used as colonies. What angular speed must such a cylinder have so that the centripetal acceleration at its surface equals the free‐fall acceleration on Earth? 17. (a) What is the tangential acceleration of a bug on the rim of a 10‐in.‐
diameter disk if the disk moves from rest to an angular speed of 78 rev/min in 3.0 s? (b) When the disk is at its final speed, what is the tangential velocity of the bug? (c) One second after the bug starts from rest, what are its tangential acceleration, centripetal acceleration, and total acceleration? 18. A race car starts from rest on a circular track of radius 400 m. The car’s speed increases at the constant rate of 0.500 m/s2. At the point where the magnitudes of the centripetal and tangential accelerations are equal, determine (a) the speed of the race car, (b) the distance traveled, and (c) the elapsed time. 19. A 55.0‐kg ice‐skater is moving at 4.00 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius 0.800 m around the pole. (a) Determine the force exerted by the horizontal rope on her arms. (b) Compare this force with her weight. 21. A certain light truck can go around a flat curve having a radius of 150 m with a maximum speed of 32.0 m/s. With what maximum speed can it go around a curve having a radius of 75.0 m? 22. The cornering performance of an automobile is evaluated on a skid pad, where the maximum speed that a car can maintain around a circular path on a dry, flat surface is measured. Then the centripetal acceleration, also called the lateral acceleration, is calculated as a multiple of the free‐fall acceleration g. The main factors affecting the performance of the car are its tire characteristics and suspension system. A Dodge Viper GTS can negotiate a skid pad of radius 61.0 m at 86.5 km/h. Calculate its maximum lateral acceleration. 23. A 50.0‐kg child stands at the rim of a merry‐go‐round of radius 2.00 m, rotating with an angular speed of 3.00 rad/s. (a) What is the child’s centripetal acceleration? (b) What is the minimum force between her feet and the floor of the carousel that is required to keep her in the circular path? (c) What minimum coefficient of static friction is required? Is the answer you found reasonable? In other words, is she likely to stay on the merry‐go‐round? Chapter 8 Problems Section 8.1 Torque 1. If the torque required to loosen a nut that is holding a flat tire in place on a car has a magnitude of 40.0 N ∙ m, what minimum force must be exerted by the mechanic at the end of a 30.0‐cm lug wrench to accomplish the task? 2. A steel band exerts a horizontal force of 80.0 N on a tooth at point B in Figure P8.2. What is the torque on the root of the tooth about point A? Figure P8.3 4. Write the necessary equations of equilibrium of the object shown in Figure P8.4. Take the origin of the torque equation about an axis perpendicular to the page through the point O. Figure P8.2 3. Calculate the net torque (magnitude and direction) on the beam in Figure P8.3 about (a) an axis through O perpendicular to the page and (b) an axis through C perpendicular to the page. Figure P8.4 Section 8.2 Torque and the Two Conditions for Equilibrium Section 8.3 The Center of Gravity Section 8.4 Examples of Objects in Equilibrium 7. The arm in Figure P8.7 weighs 41.5 N. The force of gravity acting on the arm acts through point A. Determine the r
magnitudes of the tension force Ft in the r
deltoid muscle and the force Fs exerted by the shoulder on the humerus (upper‐arm bone) to hold the arm in the position shown. Figure P8.7 8. A water molecule consists of an oxygen atom with two hydrogen atoms bound to it as shown in Figure P8.8. The bonds are 0.100 nm in length, and the angle between the two bonds is 106°. Use the coordinate axes shown, and determine the location of the center of gravity of the molecule. Take the mass of an oxygen atom to be 16 times the mass of a hydrogen atom. Figure P8.8 9. A cook holds a 2.00‐kg carton of milk at arm’s length (Fig. r
P8.9). What force FB must be exerted by the biceps muscle? (Ignore the weight of the forearm.) Figure P8.9 17. A 500‐N uniform rectangular sign 4.00 m wide and 3.00 m high is suspended from a horizontal, 6.00‐m‐long, uniform, 100‐N rod as indicated in Figure P8.17. The left end of the rod is supported by a hinge, and the right end is supported by a thin cable making a 30.0° angle with the vertical. (a) Find the tension T in the cable. (b) Find the horizontal and vertical components of force exerted on the left end of the rod by the hinge. force exerted by the food being chewed r
against the jawbone, T is the force of r
tension in the masseter, and R is the force exerted by the socket on the mandible. Find r
r
T and R for a person who bites down on a piece of steak with a force of 50.0 N. Figure P8.19 Figure P8.17 19. The chewing muscle, the masseter, is one of the strongest in the human body. It is attached to the mandible (lower jawbone) as shown in Figure P8.19a. The jawbone is pivoted about a socket just in front of the auditory canal. The forces acting on the jawbone are equivalent to those acting on r
the curved bar in Figure P8.19b: Fc is the 20. A hungry 700‐N bear walks out on a beam in an attempt to retrieve some “goodies” hanging at the end (Fig. P8.20). The beam is uniform, weighs 200 N, and is 6.00 m long; the goodies weigh 80.0 N. (a) Draw a free‐body diagram of the beam. (b) When the bear is at x = 1.00 m, find the tension in the wire and the components of the reaction force at the hinge. (c) If the wire can withstand a maximum tension of 900 N, what is the maximum distance the bear can walk before the wire breaks? 22. A 20.0‐kg floodlight in a park is supported at the end of a horizontal beam of negligible mass that is hinged to a pole, as shown in Figure P8.22. A cable at an angle of 30.0° with the beam helps to support the light. Find (a) the tension in the cable and (b) the horizontal and vertical forces exerted on the beam by the pole. Figure P8.20 21. A uniform semicircular sign 1.00 m in diameter and of weight w is supported by two wires as shown in Figure P8.21. What is the tension in each of the wires supporting the sign? Figure P8.22 23. A uniform plank of length 2.00 m and mass 30.0 kg is supported by three ropes, as indicated by the blue vectors in Figure P8.23. Find the tension in each rope when a 700‐N person is 0.500 m from the left end. Figure P8.21 Figure P8.23 24. A 15.0‐m, 500‐N uniform ladder rests against a frictionless wall, making an angle of 60.0° with the horizontal. (a) Find the horizontal and vertical forces exerted on the base of the ladder by the Earth when an 800‐N firefighter is 4.00 m from the bottom. (b) If the ladder is just on the verge of slipping when the firefighter is 9.00 m up, what is the coefficient of static friction between ladder and ground? 25. An 8.00‐m, 200‐N uniform ladder rests against a smooth wall. The coefficient of static friction between the ladder and the ground is 0.600, and the ladder makes a 50.0° angle with the ground. How far up the ladder can an 800‐N person climb before the ladder begins to slip? 26. A 1 200‐N uniform boom is supported by a cable perpendicular to the boom as in Figure P8.26. The boom is hinged at the bottom, and a 2 000‐N weight hangs from its top. Find the tension in the supporting cable and the components of the reaction force exerted on the boom by the hinge. Figure P8.26 28. One end of a uniform 4.0‐m‐long rod of weight w is supported by a cable. The other end rests against a wall, where it is held by friction. (See Fig. P8.28.) The coefficient of static friction between the wall and the rod is μs = 0.50. Determine the minimum distance x from point A at which an additional weight w (the same as the weight of the rod) can be hung without causing the rod to slip at point A. Figure P8.2
Chapter 10 Problems Section 10.1 Temperature and the Zeroth Law of Thermodynamics Section 10.2 Thermometers and Temperature Scales 1. For each of the following temperatures, find the equivalent temperature on the indicated scale: (a) –273.15°C on the Fahrenheit scale, (b) 98.6°F on the Celsius scale, and (c) 100 K on the Fahrenheit scale. 2. The pressure in a constant‐volume gas thermometer is 0.700 atm at 100°C and 0.512 atm at 0°C. (a) What is the temperature when the pressure is 0.0400 atm? (b) What is the pressure at 450°C? Section 10.3 Thermal Expansion of Solids and Liquids 10. A cylindrical brass sleeve is to be shrink‐fitted over a brass shaft whose diameter is 3.212 cm at 0°C. The diameter of the sleeve is 3.196 cm at 0°C. (a) To what temperature must the sleeve be heated before it will slip over the shaft? (b) Alternatively, to what temperature must the shaft be cooled before it will slip into the sleeve? 11. The New River Gorge bridge in West Virginia is a 518‐m‐long steel arch. How much will its length change between temperature extremes of –20°C and 35°C? 12. A grandfather clock is controlled by a swinging brass pendulum that is 1.3 m long at a temperature of 20°C. (a) What is the length of the pendulum rod when the temperature drops to 0.0°C? (b) If a pendulum’s period is given by T = 2π L / g , where L is its length, does the change in length of the rod cause the clock to run fast or slow? 14. A cube of solid aluminum has a volume of 1.00 m3 at 20°C. What temperature change is required to produce a 100‐cm3 increase in the volume of the cube? 15. A brass ring of diameter 10.00 cm at 20.0°C is heated and slipped over an aluminum rod of diameter 10.01 cm at 20.0°C. Assuming the average coefficients of linear expansion are constant, (a) to what temperature must the combination be cooled to separate the two metals? Is that temperature attainable? (b) What if the aluminum rod were 10.02 cm in diameter? 17. A gold ring has an inner diameter of 2.168 cm at a temperature of 15.0°C. Determine its inner diameter at 100°C (αgold = 1.42 × 10–5 °C–1). 21. An automobile fuel tank is filled to the brim with 45 L (12 gal) of gasoline at 10°C. Immediately afterward, the vehicle is parked in the sunlight, where the temperature is 35°C. How much gasoline overflows from the tank as a result of the expansion? (Neglect the expansion of the tank.) 23. The average coefficient of volume expansion for carbon tetrachloride is 5.81 × 10–4 (°C)–1. If a 50.0‐gal steel container is filled completely with carbon tetrachloride when the temperature is 10.0°C, how much will spill over when the temperature rises to 30.0°C? Section 10.4 Macroscopic Description of an Ideal Gas 27. One mole of oxygen gas is at a pressure of 6.00 atm and a temperature of 27.0°C. (a) If the gas is heated at constant volume until the pressure triples, what is the final temperature? (b) If the gas is heated so that both the pressure and volume are doubled, what is the final temperature? 28. Gas is contained in an 8.0‐L vessel at a temperature of 20°C and a pressure of 9.0 atm. (a) Determine the number of moles of gas in the vessel. (b) How many molecules are in the vessel? 29. (a) An ideal gas occupies a volume of 1.0 cm3 at 20°C and atmospheric pressure. Determine the number of molecules of gas in the container. (b) If the pressure of the 1.0‐cm3 volume is reduced to 1.0 × 10–11 Pa (an extremely good vacuum) while the temperature remains constant, how many moles of gas remain in the container? 30. A tank having a volume of 0.100 m3 contains helium gas at 150 atm. How many balloons can the tank blow up if each filled balloon is a sphere 0.300 m in diameter at an absolute pressure of 1.20 atm? 31. A cylinder with a movable piston contains gas at a temperature of 27.0°C, a volume of 1.50 m3, and an absolute pressure of 0.200 × 105 Pa. What will be its final temperature if the gas is compressed to 0.700 m3 and the absolute pressure increases to 0.800 × 105 Pa? Chapter 11 Problems Section 11.1 Heat and Internal Energy Section 11.2 Specific Heat 1. Water at the top of Niagara Falls has a temperature of 10.0°C. If it falls a distance of 50.0 m and all of its potential energy goes into heating the water, calculate the temperature of the water at the bottom of the falls. 3. Lake Erie contains roughly 4.00 × 1011 m3 of water. (a) How much energy is required to raise the temperature of that volume of water from 11.0°C to 12.0°C? (b) How many years would it take to supply this amount of energy by using the 1 000‐
MW exhaust energy of an electric power plant? 4. An aluminum rod is 20.0 cm long at 20°C and has a mass of 350 g. If 10 000 J of energy is added to the rod by heat, what is the change in length of the rod? 5. How many joules of energy are required to raise the temperature of 100 g of gold from 20.0°C to 100°C? 6. As part of an exercise routine, a 50.0‐
kg person climbs 10.0 meters up a vertical rope. How many (food) Calories are expended in a single climb up the rope? (1 food Calorie = 103 calories) 8. The apparatus shown in Figure P11.8 was used by Joule to measure the mechanical equivalent of heat. Work is done on the water by a rotating paddle wheel, which is driven by two blocks falling at a constant speed. The temperature of the stirred water increases due to the friction between the water and the paddles. If the energy lost in the bearings and through the walls is neglected, then the loss in potential energy associated with the blocks equals the work done by the paddle wheel on the water. If each block has a mass of 1.50 kg and the insulated tank is filled with 200 g of water, what is the increase in temperature of the water after the blocks fall through a distance of 3.00 m? Figure P11.8 The falling weights rotate the paddles, causing the temperature of the water to increase. 9. A 5.00‐g lead bullet traveling at 300 m/s is stopped by a large tree. If half the kinetic energy of the bullet is transformed into internal energy and remains with the bullet while the other half is transmitted to the tree, what is the increase in temperature of the bullet? 10. A 1.5‐kg copper block is given an initial speed of 3.0 m/s on a rough horizontal surface. Because of friction, the block finally comes to rest. (a) If the block absorbs 85% of its initial kinetic energy as internal energy, calculate its increase in temperature. (b) What happens to the remaining energy? 11. A 200‐g aluminum cup contains 800 g of water in thermal equilibrium with the cup at 80°C. The combination of cup and water is cooled uniformly so that the temperature decreases by 1.5°C per minute. At what rate is energy being removed? Express your answer in watts. Section 11.3 Calorimetry 12. Lead pellets, each of mass 1.00 g, are heated to 200°C. How many pellets must be added to 500 g of water that is initially at 20.0°C to make the equilibrium temperature 25.0°C? Neglect any energy transfer to or from the container. 15. An aluminum cup contains 225 g of water and a 40‐g copper stirrer, all at 27°C. A 400‐g sample of silver at an initial temperature of 87°C is placed in the water. The stirrer is used to stir the mixture until it reaches its final equilibrium temperature of 32°C. Calculate the mass of the aluminum cup. 16. It is desired to cool iron parts from 500°F to 100°F by dropping them into water that is initially at 75°F. Assuming that all the heat from the iron is transferred to the water and that none of the water evaporates, how many kilograms of water are needed per kilogram of iron? 17. A 100‐g aluminum calorimeter contains 250 g of water. The two substances are in thermal equilibrium at 10°C. Two metallic blocks are placed in the water. One is a 50‐g piece of copper at 80°C. The other sample has a mass of 70 g and is originally at a temperature of 100°C. The entire system stabilizes at a final temperature of 20°C. Determine the specific heat of the unknown second sample. Section 11.4 Latent Heat and Phase Change 20. A 50‐g ice cube at 0°C is heated until 45 g has become water at 100°C and 5.0 g has become steam at 100°C. How much energy was added to accomplish the transformation? 21. A 100‐g cube of ice at 0°C is dropped into 1.0 kg of water that was originally at 80°C. What is the final temperature of the water after the ice has melted? 22. How much energy is required to change a 40‐g ice cube from ice at –10°C to steam at 110°C? 31. Steam at 100°C is added to ice at 0°C. (a) Find the amount of ice melted and the final temperature when the mass of steam is 10 g and the mass of ice is 50 g. (b) Repeat with steam of mass 1.0 g and ice of mass 50 g. Section 11.5 Energy Transfer 32. The average thermal conductivity of the walls (including windows) and roof of a house in Figure P11.32 is 4.8 × 10–4 kW/m ∙ °C, and their average thickness is 21.0 cm. The house is heated with natural gas, with a heat of combustion (energy released per cubic meter of gas burned) of 9300 kcal/m3. How many cubic meters of gas must be burned each day to maintain an inside temperature of 25.0°C if the outside temperature is 0.0°C? Disregard radiation and energy loss by heat through the ground. Figure P11.32 33. (a) Find the rate of energy flow through a copper block of cross‐sectional area 15 cm2 and length 8.0 cm when a temperature difference of 30°C is established across the block. Repeat the calculation, assuming that the material is (b) a block of stagnant air with the given dimensions; (c) a block of wood with the given dimensions. 35. A steam pipe is covered with 1.50‐cm‐
thick insulating material of thermal conductivity 0.200 cal/cm ∙ °C ∙ s. How much energy is lost every second when the steam is at 200°C and the surrounding air is at 20.0°C? The pipe has a circumference of 800 cm and a length of 50.0 m. Neglect losses through the ends of the pipe. Chapter 12 Problems Section 12.1 Work in Thermodynamic Processes 2. Sketch a PV diagram and find the work done by the gas during the following stages: (a) A gas is expanded from a volume of 1.0 L to 3.0 L at a constant pressure of 3.0 atm. (b) The gas is then cooled at constant volume until the pressure falls to 2.0 atm. (c) The gas is then compressed at a constant pressure of 2.0 atm from a volume of 3.0 L to 1.0 L. (Note: Be careful of signs.) (d) The gas is heated until its pressure increases from 2.0 atm to 3.0 atm at a constant volume. (e) Find the net work done during the complete cycle. 5. A gas expands from I to F along the three paths indicated in Figure P12.5. Calculate the work done on the gas along paths (a) IAF, (b) IF, and (c) IBF. Figure P12.5 (Problems 5 and 15) 9. One mole of an ideal gas initially at a temperature of Ti = 0°C undergoes an expansion at a constant pressure of 1.00 atm to four times its original volume. (a) Calculate the new temperature Tf of the gas. (b) Calculate the work done on the gas during the expansion. 10. (a) Determine the work done on a fluid that expands from i to f as indicated in Figure P12.10. (b) How much work is done on the fluid if it is compressed from f to i along the same path? Figure P12.10 Section 12.2 The First Law of Thermodynamics 13. A gas is compressed at a constant pressure of 0.800 atm from 9.00 L to 2.00 L. In the process, 400 J of energy leaves the gas by heat. (a) What is the work done on the gas? (b) What is the change in its internal energy? 16. A gas is taken through the cyclic process described by Figure P12.16. (a) Find the net energy transferred to the system by heat during one complete cycle. (b) If the cycle is reversed—that is, the process follows the path ACBA—what is the net energy transferred by heat per cycle? Figure P12.22 Figure P12.16 (Problems 16 and 18) 22. One mole of gas initially at a pressure of 2.00 atm and a volume of 0.300 L has an internal energy equal to 91.0 J. In its final state, the gas is at a pressure of 1.50 atm and a volume of 0.800 L, and its internal energy equals 180 J. For the paths IAF, IBF, and IF in Figure P12.22, calculate (a) the work done on the gas and (b) the net energy transferred to the gas by heat in the process. Section 12.3 Heat Engines and the Second Law of Thermodynamics 23. A heat engine operates between two reservoirs at temperatures of 20°C and 300°C. What is the maximum efficiency possible for this engine? 24. A steam engine has a boiler that operates at 300°F, and the temperature of the exhaust is 150°F. Find the maximum efficiency of this engine. 25. The energy absorbed by an engine is three times greater than the work it performs. (a) What is its thermal efficiency? (b) What fraction of the energy absorbed is expelled to the cold reservoir? 27. One of the most efficient engines ever built is a coal‐fired steam turbine engine in the Ohio River valley, driving an electric generator as it operates between 1 870°C and 430°C. (a) What is its maximum theoretical efficiency? (b) Its actual efficiency is 42.0%. How much mechanical power does the engine deliver if it absorbs 1.40 × 105 J of energy each second from the hot reservoir? 29. An engine absorbs 1 700 J from a hot reservoir and expels 1 200 J to a cold reservoir in each cycle. (a) What is the engine’s efficiency? (b) How much work is done in each cycle? (c) What is the power output of the engine if each cycle lasts for 0.300 s? 31. In one cycle, a heat engine absorbs 500 J from a high‐temperature reservoir and expels 300 J to a low‐temperature reservoir. If the efficiency of this engine is 60% of the efficiency of a Carnot engine, what is the ratio of the low temperature to the high temperature in the Carnot engine? 32. A heat engine operates in a Carnot cycle between 80.0°C and 350°C. It absorbs 21 000 J of energy per cycle from the hot reservoir. The duration of each cycle is 1.00 s. (a) What is the mechanical power output of this engine? (b) How much energy does it expel in each cycle by heat? 33. A nuclear power plant has an electrical power output of 1 000 MW and operates with an efficiency of 33%. If excess energy is carried away from the plant by a river with a flow rate of 1.0 × 106 kg/s, what is the rise in temperature of the flowing water? Chapter 13 Problems Section 13.1 Hooke’s Law 1. A 0.40‐kg object is attached to a spring with force constant 160 N/m so that the object is allowed to move on a horizontal frictionless surface. The object is released from rest when the spring is compressed 0.15 m. Find (a) the force on the object and (b) its acceleration at that instant. 2. A load of 50 N attached to a spring hanging vertically stretches the spring 5.0 cm. The spring is now placed horizontally on a table and stretched 11 cm. (a) What force is required to stretch the spring by that amount? (b) Plot a graph of force (on the y‐axis) versus spring displacement from the equilibrium position along the x‐axis. Section 13.2 Elastic Potential Energy 7. A slingshot consists of a light leather cup containing a stone. The cup is pulled back against two parallel rubber bands. It takes a force of 15 N to stretch either one of these bands 1.0 cm. (a) What is the potential energy stored in the two bands together when a 50‐g stone is placed in the cup and pulled back 0.20 m from the equilibrium position? (b) With what speed does the stone leave the slingshot? 10. An automobile having a mass of 1 000 kg is driven into a brick wall in a safety test. The bumper behaves like a spring with constant 5.00 × 106 N/m and is compressed 3.16 cm as the car is brought to rest. What was the speed of the car before impact, assuming that no energy is lost in the collision with the wall? 12. A 1.50‐kg block at rest on a tabletop is attached to a horizontal spring having constant 19.6 N/m, as in Figure P13.12. The spring is initially unstretched. A constant 20.0‐N horizontal force is applied to the object, causing the spring to stretch. (a) Determine the speed of the block after it has moved 0.300 m from equilibrium if the surface between the block and tabletop is frictionless. (b) Answer part (a) if the coefficient of kinetic friction between block and tabletop is 0.200. Figure P13.12 13. A 10.0‐g bullet is fired into, and embeds itself in, a 2.00‐kg block attached to a spring with a force constant of 19.6 N/m and whose mass is negligible. How far is the spring compressed if the bullet has a speed of 300 m/s just before it strikes the block and the block slides on a frictionless surface? [Note: You must use conservation of momentum in this problem. Why?] Section 13.3 Comparing Simple Harmonic Motion with Uniform Circular Motion Section 13.4 Position, Velocity, and Acceleration as a Function of Time 15. A 0.40‐kg object connected to a light spring with a force constant of 19.6 N/m oscillates on a frictionless horizontal surface. If the spring is compressed 4.0 cm and released from rest, determine (a) the maximum speed of the object, (b) the speed of the object when the spring is compressed 1.5 cm, and (c) the speed of the object when the spring is stretched 1.5 cm. (d) For what value of x does the speed equal one‐half the maximum speed? 16. An object–spring system oscillates with an amplitude of 3.5 cm. If the spring constant is 250 N/m and the object has a mass of 0.50 kg, determine (a) the mechanical energy of the system, (b) the maximum speed of the object, and (c) the maximum acceleration of the object. 18. A 50.0‐g object is attached to a horizontal spring with a force constant of 10.0 N/m and released from rest with an amplitude of 25.0 cm. What is the velocity of the object when it is halfway to the equilibrium position if the surface is frictionless? 23. A spring stretches 3.9 cm when a 10‐
g object is hung from it. The object is replaced with a block of mass 25 g that oscillates in simple harmonic motion. Calculate the period of motion. 26. The motion of an object is described by the equation ⎛ πt ⎞
x = (0.30 m ) cos⎜ ⎟ ⎝3⎠
Find (a) the position of the object at t = 0 and t = 0.60 s, (b) the amplitude of the motion, (c) the frequency of the motion, and (d) the period of the motion. 27. A 2.00‐kg object on a frictionless horizontal track is attached to the end of a horizontal spring whose force constant is 5.00 N/m. The object is displaced 3.00 m to the right from its equilibrium position and then released, initiating simple harmonic motion. (a) What is the force (magnitude and direction) acting on the object 3.50 s after it is released? (b) How many times does the object oscillate in 3.50 s? Section 13.8 Frequency, Amplitude, and Wavelength 36. A cork on the surface of a pond bobs up and down two times per second on ripples having a wavelength of 8.50 cm. If the cork is 10.0 m from shore, how long does it take a ripple passing the cork to reach the shore? 37. A wave traveling in the positive x‐
direction has a frequency of 25.0 Hz, as in Figure P13.37. Find the (a) amplitude, (b) wavelength, (c) period, and (d) speed of the wave. Figure P13.37 39. If the frequency of oscillation of the wave emitted by an FM radio station is 88.0 MHz, determine (a) the wave’s period of vibration and (b) its wavelength. (Radio waves travel at the speed of light, 3.00 × 108 m/s.)