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Homework #20: Review with springs Problem 15.54 DNA spring A molecule of DNA (deoxyribonucleic acid) is 2.17µm long. The ends of the molecule become singly ionized – negative on one end, positive on the other. The helical molecule acts as a spring and compresses 1% on becoming charged. Determine the effective spring constant of the molecule. [k=2.25x10-9N/m] Homework #19: Relativistic Kinematics Problem 22.4&3&1 Speed of light Figure P22.4 shows an apparatus used to measure the speed distribution of gas molecules. It consists of two slotted rotating disks separated by a distance s, with the slots displaced by the angle 2. Suppose the speed of light is measured by sending a light beam toward the right disk of this apparatus. A)Show that a light beam will be seen in the detector (that is, will make it through both slots) T2 only if its speed is given by c=sT/2, where T is the angular speed of the disks and 2 is measured in radians. B)What is the measured speed of light if the distance between the two slotted rotating disks in 2.5m, the slot in the second disk is displaced 1/60 of one degree from the slot in the first disk, and the disks are rotating at 5555 rev/s? [- ,c=2.999x108m/s] Albert A. Michelson very carefully measured the speed of light using an alternative version of the technique developed by Fizeau. Figure P22.3 shows the approach he used. Light was reflected from one face of a rotating eight-sided mirror toward a stationary mirror 35km away. At certain rates of rotation, the returning beam of light was directed toward the eye of an observer as shown. A)What minimum angular speed must the rotating mirror have in order that side A will have rotated to position B, causing the light to be reflected to the eye? B)What is the next highest angular speed that will enable the source of light to be seen? T T [T=536rev/s,T=1.07x103rev/s] During the Apollo XI Moon landing, a highly reflecting screen was erected on the Moon’s surface. When the Moon is directly overhead, the speed of light may be found by measuring the time it takes a laser beam to travel from Earth, reflect from the screen, and return to Earth. If this interval is measured to be 2.51s, what is the measured speed of light? Take the center-to-center distance from Earth to Moon to be 3.84x108m, and do not neglect the sizes of the Earth and Moon. [v=2.995x108m/s] Problem 26.12 Super Train A supertrain of proper len gth 100m travels at a speed of .95c as it passes through a tunnel having proper length 50m. As seen by a trackside observer, is the train ever completely within the tunnel? If so, by how much? If not, by how much? [fits with 19m to spare] Problem 26.8 Spaceship length A friend in a spaceship travels past you at a high speed. He tells you that his ship is 20m long and that the identical ship you are sitting in is 19m long. According to your observations, a)how long is your ship, b)how long is his ship, and c)what is the speed of your friend’s ship? [L=20m, L=19m, v=.31c] Problem 26.4 Heart Rate An astronaut at rest on Earth has a heartbeat rate of 70 beats/min. When the astronaut is traveling in a spaceship at .9c, what will this rate be as measured by a)an observer also in the ship and b)an observer at rest on the Earth? [R=70beats/min, R=30beats/min] Problem 26.42 Radioactive decay A radioactive nucleus moves with a speed of v relative to a laboratory observer. The nucleus emits an electron in the positive x direction with a speed of .7c relative to the decaying nucleus and a speed of .85c in the +x direction relative to the laboratory observer. What is the value of v? [v=.37c] Problem 26.52&26.7 Muon lifetime The muon is an unstable particle that spontaneously decays into an electron and two neutrinos. If the number of muons at t=0 is No, the number of at time t is given by N=Noe-t/J where J is the mean lifetime, equal to 2.2µs. J Suppose that the muons move at a speed of .95c and that there are 5x104 muons at t=0. A)What is the observed lifetime of the muons? B)How many muons remain J after traveling a distance of 3km? [J=7.05µs, N=1.12x104] A muon formed high in the Earth’s atmosphere travels at speed v=.99c for a distance of 4.6km before it decays into an electron, a neutrino, and an antineutrino (µ- 6 e- + < + <bar) a)How long does the muon live, as measured in its reference ) frame? B)How far does the muon travel, as measured in its frame? [)t=2.2µs, d=.65km] Homework #18: Galilean kinematics Problem 3.35 Rain A car travels due east with a speed of 50km/h. Rain is falling vertically with respect to the Earth. The traces of the rain on the side windows of the car make an angle of 60o with the vertical. Find the velocity of the rain with respect to a)the car and b)the Earth. [v=57.7km/h at 60o west of vertical, v=28.9km/h downward] Problem 3.30 Train A science student is riding on a flatcar of a train traveling along a straight horizontal track at a constant speed of 10m/s. The student throws a ball along a path that she judges to make an initial angle of 60o with the horizontal and to be in line with the track. The student’s professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does the ball rise? [h=15.3m] Problem 3.26 River Crossing A river flows due east at 1.5m/s. A boat crosses the river from the south shore to the north shore by maintaining a constant velocity of 10m/s due north relative to the water. A)What is the velocity of the boat relative to shore? B)If the river is 300m wide, how far downstream has the boat moved by the time it reaches the north shore? [v=10.1m/s, x=45m] Problem 3.24&28, 26.1 Jets A jet airliner moving initially at 300m/hr due east enters a region where the wind is blowing at 100mph in a direction 30o north of east. What is the new velocity of the aircraft relative to the ground? [v=390mph at 7.4o N of E relative to ground] The pilot of another aircraft wishes to fly due west in a 50km/h wind blowing toward the south. If the speed of the aircraft relative to the air is 200km/h, a)in what direction should the aircraft head, and b)what will be its speed relative to the 2 ground? [2=14.5o N of W, v=194km/h] Two airplanes fly paths I and II, specified in Figure 26.5a. Both planes have air speeds of 100m/s and fly a distance L=200km. The wind blows at 20m/s in the direction shown in the figure. Find a)the time of flight to each city, b)the time to return, and c)the difference in total flight times. [t=2.04x103s, t=2.04x103s, )t=90s] Problem 3.29 Car How long does it take an automobile traveling in the left lane at 60km/h to overtake another car that is traveling in the right lane at 40km/h, when the cars’ front bumpers are initially 100m apart? [t=18s] Problem 4.26 Elevator Two blocks are fastened to the ceiling of an elevator, as in Figure P4.26. The elevator accelerates upward at 2m/s2 from rest. If after 4s a passenger cuts the rope between C and D, and D is .3m from the floor, a)how long before the 10kg hits the floor, and b)with respect to the elevator, how fast is it going when it hits. Homework #17: Rotational kinematics Problem 7.8 Dentist drill A dentist’s drill starts from rest. After 3.2s of constant angular acceleration it turns at a rate of 2.51x104rev/min. A)Find the drill’s angular acceleration. B)Determine the angle (in radians) through which the drill rotates " during this period. ["=8.22x102rad/s2, 2=4.21x103rad] Problem 7.42 Bicycle Wheel A rotating bicycle wheel has an angular speed of 3rad/s at some instant of time. It is then given an angular acceleration of 1.5rad/s2. A chalk line drawn on the wheel is horizontal at t=0. A)What angle does this line make with its original direction 2 at t=2s? B)What is the angular speed of the wheel at t=2s? [2=516o or 156o, T=6rad/s] Problem 8.72 Ball-in-cup A common physics demonstration (Figure P8.72) consists of a ball resting at the end of a board of length R that is elevated at an angle 2 with the horizontal. A light cup is attached to the board at rc so that it will catch the ball when the support stick is suddenly removed. A)Show that the ball will lag behind the falling board when 2<35.3o, and b)the ball will fall into the cup when the board is R 2 supported at this limiting angle and the cup is placed at rc=2R/(3cos2). Problem 8.32 Landing Gear An airliner lands with a speed of 50m/s. Each wheel of the plane has a radius of 1.25m and a moment of inertia of 110kg m2. At touchdown, the wheels begin to spin under the action of friction. Each wheel supports a weight of 1.4x104N, and the wheels attain the angular speed of rolling without slipping in .48s. What is the coefficient of kinetic friction between the wheels and the runway? Assume that the speed of the plane is constant. [µ=.524] Problem 8.28 Fishing Reel A cylindrical fishing reel has a moment of inertia of I=6.8x10-4kg m2 and a radius of 4cm. A friction clutch in the reel exerts a restraining torque of 1.3nM if a fish pulls on the line. The fisherman gets a bit, and the reel begins to spin with an angular acceleration of 66rad/s2. A)What is the force of the fish on the line? B)How much line unwinds in .5s? [F=34N, s=33cm] Problem 8.33 Falling Spool A light string is wrapped around a solid cylindrical spool of radius .5m and mass .5kg. A 5kg mass is hung from the string, causing the spool to rotate and the string to unwind (Figure P8.33). Assume that the system starts from rest and no slippage takes place between the string and the spool. By direct application of Newton’s second law, determine the angular speed of the spool after T the mass has dropped 4m. [T=17.3rad/s] Problem 7.46 Sander A highspeed sander has a disk 6cm in radius that rotates about its axis at a constant rate of 1200rpm. Determine a)the angular speed of the disk in radians per second, b)the linear speed of a point 2cm from the disk’s center, c)the centripetal acceleration of the point on the rim, and d)the total distance (angular and linear) T traveled by the point on the rim in 2s. [T=126rad/s, v=2.51m/s, a=947m/s2, 2=251rad & s=15.1m] Homework #16: Circular kinematics Problem 7.20 Race car A race car starts from rest on a circular track of radius 400m. The car’s speed increases at the constant rate of .5m/s2. At the point at which 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. [v=14.14m/s, s=200m, t=28.3] Problem 7.10 Washing machine The tub of a washer goes into its spin-dry cycle, starting from rest and reaching an angular speed of 4rev/s in 8s. 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 12s. Through how many revolutions does the tub turn during this 20s interval? Assume constant angular acceleration while it is starting and stopping. 2 [2=50rev] Problem 7.19 & 7.15 Bug & Turntable a)What is the tangential acceleration of a bug on the rim of a 10in diameter disk if the disk moves from rest to an angular speed of 78rpm in 3s? 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, radial acceleration, and total acceleration? [a=.345m/s2, v=1.04m/s, at=.345m/s2, ar=.940m/s2, atot=1m/s2 The turntable of a record player rotates initially at 33rev/min and takes 20s to come to rest. A)What is the angular acceleration of the turntable, assuming it is uniform? B)How many rotations does the turntable make before coming to rest? C)If the radius of the turntable is .14m, what is the initial linear speed of a bug riding " on the rim? ["=-.17rad/s2, 2=34.6rad, v=.48m/s] Problem 7.30 Water Pail A pail of water is rotated in a vertical circle of radius 1m (the approximate length of a person’s arm). What must be the minimum speed of the pail at the top of the circle if no water is to spill out? [v=3.13m/s] Problem 7.13 Rolling Coin A coin with a diameter of 2.4cm is dropped onto a horizontal surface. The coin starts out with an initial angular speed of 18rad/s and rolls in a straight line without slipping. If the rotation slows with an angular acceleration of magnitutde 1.9rad/s2, how far does the coin roll before coming to rest? [s=1.02m] Problem 8.61 Tethered Model Airplane A model airplane whose mass is .75kg is tethered by a wire so that it flies in a circle 30m in radius. The airplane engine provides a net thrust of .8N perpendicular to the tethering wire. A)Find the torque the net thrust produces about the center of the circle. B)Find the angular acceleration of the airplane when it is in level flight, c)Find the linear acceleration of the airplane J tangent to its flight path. [J=24Nm, "=.0356rad/s2] Homework #15: Orbital kinematics etc Problem 15.55 Proton Projectile Protons are projected with an initial speed of vo=9550m/s into a region in which a uniform electric field E=720N/C is present (Figure P15.55) The electric force is equal to the electric field E times the charge of the proton. The protons are to hit a target that lies a horizontal distance of 1.27mm from the point at which the protons are launched. Find a)the two projection angles 2 that will result 2 in a hit and b)the total duration of flight for each of these two trajectories. [2=36.9o or 53.1o], t=1.66x10-7s or 2.21x10-7s] Problem 7.58 Planetary Conjunction Two planets X and Y travel counterclockwise in circular orbits about a star, as in Figure P7.58. The radii of their orbits is in the ratio 3:1. At some time, they are aligned, as in Figure P7.58a, making a straight line with the star. Five years later, planet X has rotated through 90o, as in Figure P7.58b. Where is planet Y at this time? [Y has turned through 1.3rev] Problem 7.59 Tidals A spacecraft in the shape of a long cylinder has a length of 100m and its mass with occupants is 1000kg. It has strayed too close to a 1m radius black hole having a mass 100 times that of the Sun (Figure P7.59). If the nose of the spacecraft points toward the center of the black hole, and if distance between the nose of the spacecraft and the black hole’s center is 10km, a)determine the total force on the spacecraft. B)What is the difference in the force per kilogram of mass felt by the occupants in the nose of the ship and those in the rear of the ship farthest from the black hole? [F=1.31x1017N, )F/m=2.62x1012N/kg] Problem 7.38 Moon Satellite A satellite is in a circular orbit just above the surface of the Moon. (See Table 7.3). What are the satellite’s a)acceleration and b)speed? C)What is the period of the satellite orbit? [a=1.62m/s2, v=1.68x103, t=1.81h] Problem 7.49 Humped Bridge A car moves at speed v across a bridge made in the shape of a circular arc of radius r. A)Find an expression for the normal force acting on the car when it is at the top of the arc. B)At what minimum speed will the normal force become zero (causing occupants of the car to seem weightless) if r=30m? [N=mg - mv2/r, v=17.1m/s] Homework assignment #14: 2-dimensional kinematics (projectiles) 1. Problem 3.41 (projectile: initial horizontal) The determined coyote is out once more to try to capture the elusive roadrunner. The coyote wears a new pair of Acme power roller skates, which provide a constant horizontal acceleration of 15m/s2, as shown in Figure P3.41. The coyote starts off at rest 70m from the edge of the cliff at the instant the roadrunner zips by in the direction of the cliff. A)If the roadrunner moves with constant speed, determine the minimum speed he must have in order to reach the cliff before the coyote. B)If the cliff is 100m above the base of a canyon, determine where the coyote lands in the canyon. (Assume that his skates are still in operation when he is in “flight” and that his horizontal component of acceleration remains constant at 15m/s2). [v=23m/s], x=360m] 2. Problem 3.22 (projectile: initial angled up) A firefighter, 50m away from a burning building, directs a stream of water from a ground level fire hose at an angle of 30o above the horizontal. If the speed of the stream as it leaves the hose is 40m/s, at what height will the stream of water strike the building? [y=18.6m] 3. Problem 3.21 (projectile: initial angled down) A car is parked on a cliff overlooking the ocean on an incline that makes an angle of 24o below the horizontal. The negligent driver leaves the car in neutral, and the emergency brakes are defective. The car rolls from rest down the incline with a constant acceleration of 4m/s2 for a distance of 50m to the edge of the cliff. The cliff is 30m above the ocean. Find a)the car’s position relative to the base of the cliff when the car lands in the ocean, and b)the length of time the car is in the air. [x=32.5m, t=1.78s] 4. Problem 3.60 (projectile: angle vs. time) When baseball outfielders throw the ball, they usually allow it to take one bounce on the theory that the ball arrives sooner this way. Suppose that after the bounce, the ball rebounds at the same angle 2 as it had when released (Figure P3.60) but loses half its speed. A)Assuming the ball is always thrown with the same initial speed, at what angle 2 should the ball be thrown in order to go the same distance D with one bounce (blue path) as one thrown upward at 45o with no bounce (green path)? B)Determine the ratio of the times for the one-bounce and no-bounce throws. 2 [2=26.6o, t26/t45=.939] 5. Problems 3.37 and 3.50 (range) A rocket is launched at an angle of 53o above the horizontal with an initial speed of 100m/s. It moves for 3s along its initial line of motion with an accelration of 30m/s2. At this time its engines fail and the rocket proceeds to move as a free body. Find a)the maximum altitude reached by the rocket, b)its total time of flight, and c)its horizontal range. [H=1521.5m, T=36s, R=4045m] A projectile is fired with an initial speed of vo at an angle of 2o to the horizontal, as in Figure 3.11. When it reaches its peak, it has (x,y) coordinates given by (R/2,h), and when it strikes the ground, its coordinates are (R,0), where R is called the horizontal range. A)Show that it reaches a maximum height, h, given by h=(vo2sin22o)/2g. B)show that its horizontal range is given by 2 R=(vo2sin222o)/g. Homework assignment #13: 1-dimensional kinematics (inclined plane) Problem 4.4 and 4.22 (trains) A freight train has a mass of 1.5x107kg. If the locomotive can exert a constant pull of 7.5x105N, how long does it take to increase the speed of the train from rest to 80km/h. [t=7.4min] Another train has a mass of 5.22x106kg and is moving at 90km/h. The engineer applies the brakes, which results in a net backward force of 1.87x106N on the train. The brakes are held on for 30s. A)What is the new speed of the train? B)How far does it travel during this period? [v=14.3m/s, x=589m] Problem 4.23 (mass sliding down inclined plane) A 2kg mass starts from rest and slides down an inclined plane 80cm long in .5s. What net force is acting on the mass along the incline? [F=13N] Problem 4.53 (mass sliding down inclined plane) A 3kg block starts from rest at the top of a 30o incline and slides 2m down the incline in 1.5s. Find a)the acceleration of the block, b)the coefficient of kinetic friction between it and the incline, c)the frictional force acting on the block, and d)the speed of the block after it has slid 2m. [a=1.78m/s2, f=9.37N, µ=.368, v=2.67] Problem 4.40 (mass sliding down inclined plane with string tension) Masses m1=4kg and m2=9kg are connected by a light string that passes over a frictionless pulley. As shown in Figure P4.40, m1 is held at rest on the floor and m2 rests on a fixed incline of 2=40o. The masses are released from rest, and m2 slides 1m down the incline in 4s. Determine a)the acceleration of each mass, b)the tension in the string, and c)the coefficient of kinetic friction between m2 and the incline. [a=.125m/s2, T=39.7N, µ=.235] Problem 4.24 (towed wagon on a hill) A 40kg wagon is towed up a hill inclined at 18.5o with respect to the horizontal. The tow rope is parallel to the incline and has a tension of 140N in it. Assume that the wagon starts from rest at the bottom of the hill, and neglect friction. How fast is the wagon going after moving 80m up the hill? [v=7.9m/s] Homework assignment #12: 1-dimensional kinematics (horizontal/vertical) *Problem 2.52 (Averaged velocity) In order to pass a physical education class at a university, a student must run 1mi in 12min. After running for 10min, she stil has 500yd to go. If her maximum acceleration is .15m/s2, can she make it? Why? [She can make it, a=.032m/s2] Problem 2.53 (Distance, time, and acceleration) In Mostar, Bosnia, the ultimate test of a young man’s courage once was to jump off a 400-year-old bridge (now destroyed) into the River Neretva, 23m below the bridge. A)How long did the jump last? B)How fast was the diver traveling on impact with the river? C)If the speed of sound in air is 340m/s, how long after the diver took off did the spectator on the bridge hear the splash? [t=2.17s, v=21.3m/s, t=2.24s] *Problem 2.36 (Distance, time, and acceleration with meeting point) A hockey player is standing on his skates on a frozen pond when an opposing player, moving with a uniform speed of 12m/s, skates by with the puck. After 3s, the first player makes up his mind to chase his opponent. If he accelerates uniformly at 4m/s2, a)how long does it take him to catch his opponent, and b)how far has he traveled in this time? (Assume the player with the puck remains in motion at constant speed.) [t=8.2s, )x=134m] Problem 2.22 (Distance, velocity, and acceleration during launch) Jules Verne in 1865 proposed sending men to the Moon by firing a space capsule from a 220m-long cannon with final velocity of 10.97km/s. What would have been the unrealistically large acceleration experience by the space travelers during launch? Compare your answer with freefall acceleration, the maximum acceleration a person can withstand before he/she blacks out, and the maximum before a person perishes (do a little research on this). [a=2.79x104g] Problem 2.57 (Distance, velocity, and acceleration during bouncing) A hard rubber ball, released at chest height, falls to the pavement and bounces back to nearly the same height. When it is in contact with the pavement, the lower side of the ball is temporarily flattened. Before this dent in the ball pops out, suppose that its maximum depth is on the order of one centimeter. Estimate the maximum acceleration of the ball. State your assumptions, the quantities you estimate, and the values you estimate for them. [a=1500m/s2] Problem 2.37 (Distance, velocity, and acceleration with initial upward velocity) A ball is thrown vertically upward with a speed of 25m/s. A)How high does it rise? B)How long does it take to reach its highest point? C)How long does it take to hit the ground after it reaches its highest point? D)What is its speed when it returns to the level from which it started? [y=31.9m, t=2.55s, t=2.55s, v=- 25m/s] *Problem 2.51 (Distance, velocity, and acceleration mixed with distance, time, and acceleration, initial downward velocity, and a meeting time) A mountain climber stands at the top of a 50m cliff that overhangs a calm pool of water. He throws two stones vertically downward 1s apart and observes that they cause a single splash. The first stone has an initial velocity of -2m. A)How long after release of the first stone will the two stones hit the water? [t=3s, vo=15m/s, v2=35m/s] Problem 2.30 (Distance, velocity, acceleration) Two cars are traveling along a straight line in the same direction, the ead car at 25m/s and the other car at 30m/s. At the moment the cars are 40m apart, the lead driver applies the brakes, causing her car to have an acceleration of 02m/s2. A)How long does it take for the lead car to stop? B)Assuming that the chasing car brakes at the same time as the lead car, what must be the casing car’s minimum negative acceleration so as not to hit the lead car? C)How long does it take for the chasing car to stop? [x1=156m, a2-2.30m/s2, t=13.1s] Homework assignment #11: Velocity kinematics *Problem 2.4 (Definition of velocity) A certain bacterium swims with a speed of 3.5microns/s. How long would it take this bacterium to swim across a petri dish having a diameter of 8.4cm? [t=6.67h] *Problem 2.6 (Average velocity from graph) A tennis player moves in a straight-line path as shown in Figure P2.6. Find her average velocities in the time intervals a)0 to 1s, b)0 to 4s, c)1 to 5s, and d)0 to 5s. [v=+4m/s, v=-.5m/s, v=-1m/s, v=0] Problem 2.8 (Definition of velocity - circular path) If the average speed of an orbiting space shuttle is 19,800mph, determine the time required for it to circle the Earth. Make sure you consider the fact that the shuttle is orbiting about 200mi above the Earth’s surface, and the Earth’s radius is 3963 miles. [t=1.32h] * Problem 2.12 (Velocity and meeting points) Runner A is initially 4mi west of a flagpole and is running with a constant velocity of 6mph due east. Runner B is initially 3mi east of the flagpole and is running with a constant velocity of 5mph due west. How far are the runners from the flagpole when they meet? [x=.18mi W of flagpole] Problem 2.10 (Velocity and meeting time) A tortoise can run with a speed of 10cm/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 2min. The tortoise wins by a shell (20cm). A)How long does the race take? B)What is the length of the race? [t=126s, x=12.6m] Problem 2.14 (Plotting position vs. time) A race car moves such that its position fits the relationship x=(5m/s) t + (.75m/s3 ) t3 where x is measured in meters and t in seconds. A)Plot a graph of position versus time. B)Determine the instantaneous velocity at t=4s using time intervals of .4s, .2s, and .1s. C)Compare the average velocity during the first 4s with the results of B). [-, v=41.03m/s, 41.008m/s, 41.002m/s, vavg=17m/s] *Problem 2.16 (Definition of acceleration) A car traveling initially at 7m/s accelerates at the rate of .8m/s2 for an interval of 2s. What is its velocity at the end of the acceleration? [v=8.6m/s] Problem 2.18 (Definition acceleration - bouncing off) A tennis ball with a speed of 109m/s is thrown perpendicularly at a wall. After striking the wall, the ball rebounds in the opposite direction with a speed of 8m/s. If the ball is in contact with the wall for .012s, what is the average acceleration of the ball while it is in contact with the wall? [aavg=-1.5x103m/s2] *Problem 2.20 (Acceleration from graph) The velocity-versus-time graph for an object moving along a straight path is shown in Figure P2.20. A)Find the average accelerations of this object during the time intervals 0 to 5s, 5s to 15s, and 0 to 20s. B)Find the instantaneous accelerations at 2s, 10s, and 18s. [aavg=0, +1.6m/s2, +.8m/s2, M=0, +1.6m/s2, 0] Problem 2.21(Plotting velocity vs. time) The engine of a model rocket accelerates the rocket vertically upward for 2s as follows: at t=0, its speed is zero; at t=1s, its speed is 5m/s; at t=2s, its speed is 16m/s. Plot a velocity-time graph for this motion, and from it determine a)the average acceleration during the 2s interval and b)the instantaneous acceleration at t=1.5s. [aavg=+8m/s2, M=+11m/s2] Homework assignment #10 Problem 9.21 “Ferry boat” A small ferry boat is 4m wide and 6m long. When a loaded truck pulls onto it, the boat sinks an additional 4cm into the water. What is the weight of the truck? [Answer: W=9.41x103] Problem 9.68 “Floating soap” A 2cm thick bar of soap is floating on a water surface so that 1.5cm of the bar is underwater. Bath oil of specific gravity .6 is poured into the water and floats on top of the water. What is the depth of the oil layer when the top of the soap is just level with the uper surface of the oil? [Answer: x=1.3cm] Problems 9.23 and 9.65 “Helium balloon” An empty rubber balloon has a mass of .012kg. The balloon is filled with helium at a density of .181kg/m3. At this density the balloon is spherical with a radius of .5m. If the filled balloon is fastened to a vertical line, what is the tension in the line? A helium-filled balloon at atmospheric pressure is tied to a 2m-long, 0.05kg string. The balloon is spherical with a radius of .4m. When released, it lifts a length (h) of the string and then remains in equilibrium, as in Figure P9.65. Determine the value of h. When deflated, the balloon has a mass of .25kg. (Hint: Only that part of the string above the floor contributes to the weight of the system in equilibrium.) [Answers: T=5.57N ,h=1.9m] Problem 9.30 “Two scales” A 1kg beaker containing 2kg of oil (density=916kg/m3) rests on a scale. A 2kg block of iron is suspended from a spring scale and completely submerged in the oil (Figure P9.30). Find the equilibrium readings of both scales. [Answer: T1=17.3N, T2=31.7N] Homework assignment #9 (Due Monday) Problem 8.2 “Tooth” A steel band exerts a horizontal force of 80N on a tooth at point B in Figure P8.2. What is the torque on the root of the tooth about point A? [Answer: J=.642Nm] Problem 8.55 “Slipping ladder” A uniform ladder of length L and weight W is leaning against a vertical wall. The coefficient of static friction between the ladder and the floor is the same as that between the ladder and the wall. If this coefficient of static friction is µs=.5, determine the smallest angle the ladder can make with the floor without slipping. [Answer: 2=36.9o] Problems 8.53, 8.23, 8.18, 8.11 (Choose two or more) “Muscles: Back, Leg, Upper Arm, Forearm” A person bends and lifts a 200N weight, Figure P8.53a, with his back horizontal. The muscle that attaches 2/3 up the spine maintains the position of the back; the angle between the spine and this muscle is 12o. Using the mechanical model in Figure P8.53b and taking the weight of the upper body to be 350N, find the tension in the back muscle and the compression force in the spine. [Answer: T=2700N, R=2650T] The large quadriceps muscle in the upper leg terminates at its lower end in a tendon attached to the upper end of the tibia (Figure P8.23a). The forces on the lower leg when the leg is extended are modeled as in Figure P8.23b, where T is the tension in the tendon, C is the force of gravity acting on the lower leg, and F is the force of gravity acting on the foot. Find T when the tendon is at an angle of 25o with the tibia, assuming that C=30N, F=12.5N, and the leg is extended at an angle of 40o with the vertical. Assume that the center of gravity of the lower leg is at its center, and that the tendon attaches to the lower leg at a point 1/5 down the leg. [Answer: T=209N] The arm in Figure P8.18 weighs 41.5N. The force of gravity acting on the arm acts through point A. Determine the magnitudes of tension force Ft in the deltoid muscle and the force Fs of the shoulder on the humerus (upper-arm bone) to hold the arm in the position shown. [Answer: Ft=724N, Fs=716N, 2=8.75o] A cook holds a 2kg carton of milk at arm’s length (Figure P8.11). What force FB must be exerted by the biceps muscle, with no forearm weight? [F=312N] Problem 8.65 “Window washers” Two window washers Bob and Joe are on a 3m- long 345N scaffold supported by two cables attached to its ends. Bob, who weighs 750N, stands 1m from the left end, as shown in Figure P8.65. Two meters from the left end is the 500N washing equipment. Joe is .5m from the right end and weighs 1000N. Given that the scaffold is in rotational and translational equilibrium, what are the forces on each cable? [Answer: TR=1589N, TL=1006N] Problem 8.60 “Massive pulley” A 4kg mass is connected by a light cord to a 3kg mass on a smooth surface (Figure P8.60). The pulley rotates about a frictionless axle and has moment of inertia .5kg m2 and radius .3m. If the cord does not slip on the pulley, find (a) acceleration of the two masses and (b) tensions T1 and T2. [Answer: a=3.12m/s2, T1=26.7N, T2=9.36N] Homework Assignment #8 Problem 4.29 “Dockworker & crate” [coefficient of friction, horizontal motion, equilibrium with additional force] A dockworker loading crates on a ship finds that a 20kg crate, initially at rest on a horizontal surface, requires a 75N horizontal force to set it in motion. However, after the crate is in motion, a horizontal force of 60N is required to keep it moving with a constant speed. Find the coefficients of static and kinetic friction between crate and floor. [Answers: µS=.38, µk=.31] Problem 4.33 “Crate on an incline” [coefficient of friction, inclined plane, equilibrium with additional force] The coefficient of static friction between the 3kg crate and the 35o incline of Figure P4.33 is .3 . What minimum force F must be applied to the crate perpendicular to the incline to prevent the crate from sliding down the incline? [Answer: Fmin=32.1N] Problem 4.48b,c “Falling mass and horizontal motion” [coefficient of friction, horizontal and vertical motion] (b) What coefficient of static friction between the 100N block and the table ensures equilibrium? (C) If the coefficient of kinetic friction between the 100N block and the table is .25, what hanging weight should replace the 50N weight to allow the system to move at a constant speed once it is set in motion. [Answer: µk=.5, W=25N]] Problem 4.65 “C-clamp” [coefficient of friction, vertical plane] The board sandwiched between two other boards in Figure P4.65 weighs 95.5N. If the coefficient of friction between the boards is .663, what must be the magnitude of the compression forces (assume horizontal) acting on both sides of the center board to keep it from slipping? [Answer: N=72N] Problems 4.54 & 4.60 “Penguin on a sled” [coefficient of friction, equilibrium, applied forces] A 5kg penguin sits on a 10kg sled, as in Figure P4.54. A horizontal force of 45N is applied to the sled, but the penguin attempts to impede the motion by holding onto a cord attached to a tree. The coefficient of kinetic friction beween the sled and snow as well as that between the sled and the penguin is .2 . (a) Draw a freebody diagram for the penguin and one for the sled, and identify the reaction force for each force you include. Determine (b) the tension in the cord and, (c) the acceleration of the sled, and (d) A sled weighing 60N is pulled horizontally across snow and the coefficient of kinetic friction between the sled runners and the snow is .1 . A penguin weighing 70N rides on the sled. (See Figure P4.60.) If the coefficient of static friction between penguin and sled is .7, find the maximum horizontal force that can be exerted on the sled before the penguin begins to slide off. [Answers: T=9.8N, a=.58m/s2, Fmax=104N] Homework assignment #7 Problem 9.10 “Tires” [pressure, weight, & area] The four tires of an automobile are inflated to a gauge pressure of 2x105Pa. Each tire has an area of .024m2 in contact with the ground. Determine the weight of the automobile. [Answer: W=1.9x104N ] Problem 9.12 “Concrete pillar” [pressure, height, & compression strength] If 1m3 of concrete weighs 5x104N, what is the height of the tallest cylindrical concrete pillar that will not collapse under its own weight? The compression strength of concrete is 1.7x107Pa. [Answer: Hmax=3.4x102m] Problem 9.14 “Glucose” [gauge pressure & specific gravity] A collapsible plastic bag (Figure P9.14) contains a glucose solution. If the average gauge pressure in the artery is 1.33x104Pa, what must be the minimum height h of the bag in order to infuse glucose into the artery? Assume that the specific gravity of the solution is 1.02. [Answer: h=1.33m] Problem 9.15 “Air bubble” [absolute pressure in bubble] Air trapped above liquid ethanol in a rigid container, as shown in Figure P9.15. If the air pressure above the liquid is 1.10atm, determine the pressure inside a bubble 4m below the surface of the ethanol. [Answer: P=1.4atm] Problem 9.17 “Oil on water” [absolute pressure under 2 layers] A container is filled to a depth of 20cm with water. On top of the water floats a 30cm thick layer of oil with specific gravity .7. What is the absolute pressure at the bottom of the container? [Answer: P=1.05x105Pa ] Homework assignment #6 Problem 9.4 “High wire” [elastic limit] If the elastic limit of steel is 5x108Pa, determine the minimum diameter a steel wire can have if it is to support a 70kg circus performer without its elastic limit being exceeded. [Answer: dmin=1.3mm] Problem 9.5 “Bone compression” [Young’s modulus] Bone has a Young’s modulus of about 14.5x109Pa. Under compression, it can withstand a stress of about 160x106Pa before breaking. Assume that a femur is .5m long and calculte the amount of compression this bone can withstand before breaking. [Answer: )L=5.5mm] Problem 9.66 “Orthodontics” [equilibrium & Young’s modulus] A stainless steel orthodontic wire is applied to a tooth, as in Figure P9.66. The wire has an unstretched length of 3.1cm and a diameter of .22mm. If the wire is stretched .1mm, find the magnitude and direction of the force on the tooth. Disregard the width of the tooth and assume that Young’s modulus for stainless steel is 18x1010Pa. [Answer=F=22N, direction=”down” the page in the textbook figure] Problem 9.6 “Plate tectonics” [shear stress] The distortion of the Earth’s crustal plates is an example of shear on a large scale. A particular crustal rock has a shear modulus of 1.5x1010Pa. What shear stress is involved when a 10km layer of this rock is sheared through a distance of 5m? [Answer: shear stress=7.5x106Pa] Problem 9.43 “Plasma” [surface tension] In order to lift a wire ring of radius 1.75cm from the surface of a container of blood plasma, a vertical force of 1.61x10-2N greater than the weight of the ring is required. Calculate the surface tension of blood plasma from this information. [Answer: (=7.32x10-2N/m] Problem 9.52 “Hypodermic needle” [Poiseuille’s Law] A hypodermic needle is 3cm in length and .3mm in diameter. What excess pressure is required along the needle so that the flow rate of water through it will be 1g/s? [Answer: )P=1.5x105Pa] Problem 9.60 “Protein” [terminal velocity] Spherical particles of a protein of density 1.8g/cm3 are shaken up in a solution of 20oC water. The solution is allowed to stand for 1hr. If the depth of water in the tube is 5cm, find the radius of the largest particles that remain in solution at the end of the hour. [Answer: r=2.82x10-6m] Homework assignment #5 Problem 7.21 “Blood in a centrifuge” [centrifugal pseudo-force] A sample of blood is placed in a centrifuge of radius 15cm. The mass of a red corpuscle is 3 x 10- 16 kg, and the magnitude of the force required to make it settle out of the plasma is 4 x 10-11N. At how many revolutions per second should the centrifuge be operated? [Answer: f=150rev/s] Problem 7.26 “Levitating air puck” [orbital motion due to tension in the same plane] An air puck of mass .25kg is tied to a string and allowed to revolve in a circle of radius 1m on a frictionless horizontal table. The other end of the string passes through a hole in the center of the table, and a mass of 1kg is tied to it (Figure P7.26). The suspended mass remains in equilibrium while the puck on the tabletop revolves. (A)What is the tension in the string? (B)What is the force causing the centripetal acceleration of the puck? (C)What is the speed of the puck? [Answer: T=9.8N; Fcentripetal=9.8N; v=6.3m/s] Problem 7.45 “Mass of Io” [orbital motion due to gravitational] Io, a small moon of the giant planet Jupiter, has an orbital period of 1.77 days and an orbital radius equal to 4.22 x 105km. From these data, determine the mass of Jupiter. [Answer: M=1.90x1027kg] Problem 7.48 “Geosynchronous satellites” [geosynchronous orbital motion due to gravitation] Geosynchronous satellites have an angular velocity that matches the rotation of the Earth and follow circular orbits in the equatorial plane of the Earth. (Almost all communications satellites are geosynchronous and appear to be stationary above a point on the Equator.) (A)What must the radius of the orbit of a geosynchronous satellite? (B)How high in miles above the surface of the Earth are geosynchronous satellites located? [Answer: r=4.23x107m; altitude=3.59x107m] Problem 7.55 “Swinging ball” [orbital motion due to tension at a vertical angle] A .5kg ball that is tied to the end of a 1.5m light cord is revolved in a horizontal plane with the cord making a 30deg angle with the vertical (see Figure 7.10). (A)Determine the ball’s speed. (B)If the ball is revolved so that its speed is 4m/s, what angle does the cord make with the vertical? (C)If the cord can withstand a maximum tension of 9.8N, what is the highest speed at which the ball can move? [Answer: v=2.1m/s; 2=54degrees; v=4.7m/s] Problem 7.25 “Banked exit” [orbital motion due to normal force at a vertical angle] An engineer wishes to design a curved exit ramp for a toll road in such a way that a car will not have to rely on friction to round the curve without skidding. He does so by banking the road in such a way that the necessary force causing the centripetal acceleration will be supplied by the component of the normal force toward the center of the circular path. (A)Show that for a given speed of v and a radius of 4, the curve must be banked at the angle 2 2 such that tan2 = v2/rg. (B)Find the angle at which the curve should be banked if a typical car rounds it at a 50m radius and a speed of 13.4m/s. [Answer: (show); 2=20.1o] Homework assignment #4 Problem 4.15 “Block on inclined plane” [Equilibrium on inclined plane with 3 forces: horizontal, vertical, normal] A block of mass m=2kg is held in equilibrium on an incline of angle 2=60deg by the horizontal force F, as shown in Figure P4.15. (A)Determine the value of F. (B)Determine the normal force exerted by the incline on the block (ignore friction). Is the block moving? Explain. [F=33.9N, N=39N] Problem 4.20 “Accelerating block on inclined plane” [Acceleration on inclined plane with 3 forces: falling mass tension, vertical, normal] Two objects of masses 10kg and 5kg are connected by a light string that passes over a frictionless pulley as in Figure P4.20. The 5kg object lies on a smooth incline of angle 40deg. Find the acceleration of the 5kg object and the tension in the string. [a=4.43m/s2, T=53.7N] Problem 4.51 “Connected blocks on the level and a ramp” [Horizontal acceleration with force due to tension of falling mass on an inclined plane] A 2kg aluminum block and a 6kg copper block are connected by a light string over a frictionless pulley. They are allowed to move on a fixed steel block-wedge (of angle 30deg) as shown in Figure P4.51. Assuming no friction, determine the (a)acceleration of the two blocks and (b)the tension in the string. [a=.232m/s2, T=57.2N & 24.5N] Problem 4.21 “Connected and unconnected blocks on the level” [Horizontal acceleration and tension with 1 force and 3 objects] Assume that the three blocks in Figure P4.21 move on a frictionless surface and that a 42N force acts as shown on the 3kg block. Determins (a) the acceleration given this system, (b)the tension in the cord connecting the 3kg and the 1kg blocks, and (c)the force exerted on the 2kg block by the 1kg block. [a=7m/s2 horiz, T=21N, F=14N horiz to right] Problem 4.27 “Blocks in the elevator” [Vertical acceleration and tensions due to gravity and vertical driving force] Two blocks are fasted to the ceiling of an elevator, as in Figure P4.26. The elevator accelerates upward at 2m/s2. Find the tension in each rope. [F=2150N, T=645N, T=645N, F=10,190N at 2=15.9o left of vertical] Homework assignment #3 Problem 4.25 “Horizontal acceleration driven by falling mass” [Horizontal and vertical acceleration via pulley” A mass, m1=5kg, resting on a frictionless horizontal table is connected to a cable that passes over a pulley and then is fastened to a hanging mass, m2=10kg, as in Figure P4.25. Find the acceleration of each mass and the tension in the cable. [a=6.53m/s2, T=32.7N] Problem 4.48 “Horizontal equilibrium with falling mass” [Horizontal and vertical equilibrium via pulley] (a) What is the minimum force of friction required to hold the system of Figure P4.48 in equilibrium? [f=50N] Problem 4.66 “Accelerating elevator” [Vertical acceleration and equilibrium] A 72kg man stands on a spring scale in an elevator. Starting from rest, the elevator ascends, attaining its maximum speed of 1.2m/s in .8s [which means the acceleration is 1.5m/s2]. It travels with this constant speed for 5s, undergoes a uniform negative acceleration [of -.8m/s2 ] for 1.5s, and comes to rest. What does the spring scale register (a)before the elevator starts to move? (b)during the first .8s? (c)while the elevator is traveling at constant speed? (d)during the negative acceleration? [a=0 & T=710N, a=1.5m/s2 & T=820N, , a=0 & T=710N, a=-.8m/s2 & T=650N] Problem 4.57 “Accelerating boat” [Horizontal acceleration with 2 forces] Two people pull as hard as they can on ropes attached to a 200kg boat. If they pull in the same direction, the boat has an acceleration of 1.52m/s2 to the right. If they pull in opposite directions, the boat has an acceleration of .518m/s2 to the left. What is the force exerted by each person on the boat? (Disregard any other forces on the boat.) [F1=100N, F2=204N] Problem 4.35 “Terminal velocity” [Equilibrium when falling through air] An object falling under the pull of gravity experiences a frictional force of air resistance. The magnitude of this force is approximately proportional to the speed of the object, f=bv. Assume that b=15kg/s and m=50kg. (a)What is the terminal speed that the object reaches while falling? (b)Does your answer to part a depend on the initial speed of the object? Explain. [v=32.7m/s, yes] Homework Assignment #2 Problem 4.27 “Car and trailer” [Horizontal acceleration and acceleration] A 1000kg car is pulling a 300kg trailer. Together the car and trailer have an acceleration of 2.15m/s2 in the forward direction. Neglecting frictional forces on the trailer, determine (a)the net force on the car; (b)the net force on the trailer; (c)the force exerted on the car by the trailer; (d)the resultant force exerted on the road by the car. Problem 4.45 “Traffic light” [Equilibrium with 3 forces: 2 diagonal, 1 vertical] (a)What is the resultant force exerted by the two cables supporting the traffic light in Figure P4.45? (B)What is the weight of the light? Problem 4.52 “Three vertically connected masses” [Vertical acceleration and tension] Three masses are connected by light strings, as shown in Figure P4.52. The string connecting the 4kg mass and the 5kg mass passes over a light frictionless pulley. Determine (a)the acceleration of each mass and (b)the tensions in the two strings. Problem 4.55a “Two horizontally connected blocks” [Horizontal acceleration and tension] Two blocks on a frictionless horizontal surface are connected by a light string as in Figure P4.55, where m1=10kg and m2=20kg. A force of 50N is applied to the 20kg block. (A)Determine the acceleration of each block and the tension in the string. Problem 4.63 “Plane taking off” [Horizontal acceleration, vertical equilibrium] On takeoff, the combined action of the engines and wings of an airplane exerts an 8000N force on the plane, directed upward at an angle of 65 degrees above the horizontal. The plane rises with constant velocity in the vertical direction while continuing to accelerate inthe horizontal direction. (A)What is the weight of the plane? (B)What is its horizontal acceleration? Homework Assignment #1 Problem 4.11 “Cat burglar” [Equilbrium with 3 forces: 1 diagonal, 1 horizontal, 1 vertical] Find the tension in each cable supporting the 600N cat burglar in Figure P4.11. Problem 4.14 “Leg in a cast” [Equilibrium with 3 forces: 2 diagonal, 1 vertical] A leg and cast weigh 220N. The foot is held at a 40 degree angle from the horizontal by a 110N weight connected by a cable attached to the mid-leg. Determine the weight and angle of a rearward placing of tension needed in order that there be no force exerted on the hip joint by the leg plus cast. Problem 4.16 “Constant velocity boating” [Equilbrium with 3 forces: 2 diagonal, 1 horizontal] Two persons are pulling a boat through the water. Each exerts a force of 600N directed at a 30 degree angle relative to the forward motion of the boat. If the boat moves with constant velocity, find the resistive force exerted on the boat by the water. Problem 4.69 “Pat pulling himself up on the swing” [Acceleration with 3 vertical forces] An inventive child Pat wants to reach an apple in a tree without climbing the tree. Sitting in a chair connected to a rope that passes over a frictionless pulley, Pat pulls on the loose end of the rope with such a force that the spring scale reads 259N. Pat’s true weight is 320N, and the chair weighs 160N. Find the force Pat exerts on the chair. Problem 4.70 “Equilibrium with 2 forces: 1 vertical, 1 horizontal” A fire helicopter carries a 620kg bucket of water at the end of a 20.0m long cable. Flying back from a fire at a constant speed of 40.0m/s, the cable makes an angle of 40 degrees with respect to the vertical. Determine the force of air resistance on the bucket. Later: Problem 8.2 “Orthodontics” [Torque of 1 force] A steel band exerts a horizontal force of 80N on a tooth at point B in Figure P8.2. What is the torque on the root of the tooth about point A? Problem 8.11 “Milk carton” [Equilibrium with torque of 2 forces] A cook holds a 2kg carton of milk at arm’s length (Figure P8.11). What force FB must be exerted by the biceps muscle? (Ignore the weight of the forearm.) 8.13 8.18 8.23 8.15