Solved: In a loop-the-loop roller coaster, show that a car moving too slowly would leave

Compare the net force on a heavy trunk when its (a) at rest on the floor; (b) being slid across the floor at constant speed; (c) being pulled upward in an elevator whose cable tension equals the combined weight of the elevator and trunk; and (d) sliding down a frictionless ramp.

The force of static friction acts only between surfaces at rest. Yet that force is essential in walking and in accelerating or braking a car. Explain.

A jet plane flies at constant speed in a vertical circular loop. At what point in the loop does the seat exert the greatest force on the pilot? The least force?

In cross-country skiing, skis should easily glide forward but should remain at rest when the skier pushes back against the snow. What frictional properties should the ski wax have to achieve this goal?

Why do airplanes bank when turning?

Why is it easier for a child to stand nearer the inside of a rotating merry-go-round?

Gravity pulls a satellite toward Earths center. So why doesnt the satellite actually fall to Earth?

Explain why a car with ABS brakes can have a shorter stopping distance.

A fishing line has a 20-lb breaking strength. Is it possible to break the line while reeling in a 15-lb fish? Explain.

. Two blocks rest on slopes of unequal angles, connected by a rope passing over a pulley (Fig. 5.30). If the blocks have equal masses, will they remain at rest? Why? Neglect friction. Figure 5.30 For Thought and Discussion 10; Exercises 21 and 22

Youre on a plane undergoing a banked turn, so steep that out the window you see the ground below. Yet your pretzels stay put on the seatback tray, rather than sliding downward. Why?

A backcountry skier weighing 700 N skis down a steep slope, unknowingly crossing a snow bridge that spans a deep, hidden crevasse. If the bridge can support 580 Nmeaning thats the maximum normal force it can sustain without collapsingis there any chance the mountaineer can cross safely? Explain.

Two forces, both in the x@y plane, act on a 3.25-kg mass that accelerates at 5.48 m/s2 in a direction 38.0 counterclockwise from the x-axis. One force has magnitude 8.63 N and points in the +xdirection. Find the other force.

Two forces act on a 3.1-kg mass that undergoes acceleration a u = 0.91ni - 0.27nj m/s2 . If one force is -1.2ni - 2.5nj N, whats the other?

At what angle should you tilt an air table to simulate free fall at the surface of Mars, where g = 3.71 m/s2 ?

A skier starts from rest at the top of a 24 slope 1.3 km long. Neglecting friction, how long does it take to reach the bottom?

A tow truck is connected to a 1400-kg car by a cable that makes a 25 angle to the horizontal. If the truck accelerates at 0.57 m/s2 , whats the magnitude of the cable tension? Neglect friction and the cables mass.

Studies of gymnasts show that their high rate of injuries to the Achilles tendon is due to tensions in the tendon that typically reach 10 times body weight. That force is provided by a pair of muscles, each exerting a force at 25 to the vertical, with their horizontal components opposite. For a 55-kg gymnast, find the force in each of these muscles. 25 Achilles tendon 25 F2 S F1 S Figure 5.31

Find the minimum slope angle for which the skier in Question 12 can safely traverse the snow bridge.

Your 12-kg baby sister pulls on the bottom of the tablecloth with all her weight. On the table, 60 cm from the edge, is a 6.8-kg roast turkey. (a) Whats the turkeys acceleration? (b) From the time your sister starts pulling, how long do you have to intervene before the turkey goes over the edge? Neglect friction.

If the left-hand slope in Fig. 5.30 makes a 60 angle with the horizontal, and the right-hand slope makes a 20 angle, how should the masses compare if the objects are not to slide along the frictionless slopes?

Suppose the angles shown in Fig. 5.30 are 60 and 20. If the left-hand mass is 2.1 kg, what should the right-hand mass be so that it accelerates (a) downslope at 0.64 m/s2 and (b) upslope at 0.76 m/s2 ?

Two unfortunate climbers, roped together, are sliding freely down an icy mountainside. The upper climber (mass 75 kg) is on a slope at 12 to the horizontal, but the lower climber (mass 63 kg) has gone over the edge to a steeper slope at 38. (a) Assuming frictionless ice and a massless rope, whats the acceleration of the pair? (b) The upper climber manages to stop the slide with an ice ax. After the climbers have come to a complete stop, what force must the ax exert against the ice?

Suppose the Moon were held in its orbit not by gravity but by tension in a massless cable. Estimate the magnitude of the cable tension. (Hint: See Appendix E.)

Show that the force needed to keep a mass m in a circular path of radius r with period T is 4p2 mr/T2 .

A 940-g rock is whirled in a horizontal circle at the end of a 1.30-m-long string. (a) If the breaking strength of the string is 120 N, whats the minimum angle the string can make with the horizontal? (b) At this minimum angle, whats the rocks speed?

Youre investigating a subway accident in which a train derailed while rounding an unbanked curve of radius 150 m, and youre asked to estimate whether the train exceeded the 35-km/h speed limit for this curve. You interview a passenger who had been standing and holding onto a strap; she noticed that an unused strap was hanging at about a 15 angle to the vertical just before the accident. What do you conclude?

A tetherball on a 1.55-m rope is struck so that it goes into circular motion in a horizontal plane, with the rope making a 12.0 angle to the horizontal. Whats the balls speed?

An airplane goes into a turn 3.6 km in radius. If the banking angle required is 28 from the horizontal, whats the planes speed?

Movers slide a 73-kg file cabinet along a floor where the coefficient of kinetic friction is 0.81. Whats the frictional force on the cabinet?

A hockey puck is given an initial speed of 14 m/s. If it comes to rest in 56 m, whats the coefficient of kinetic friction?

Starting from rest, a skier slides 100 m down a 28 slope. How much longer does the run take if the coefficient of kinetic friction is 0.17 instead of 0?

A car moving at 40 km/h negotiates a 130-m-radius banked turn designed for 60 km/h. What coefficient of friction is needed to keep the car on the road?

Repeat Example 5.1, this time using a horizontal/vertical coordinate system.

A block is launched with initial speed 2.2 m/s up a 35 frictionless ramp. How far up the ramp does it slide?

In the process of mitosis (cell division), two motor proteins pull on a spindle pole, each with a 7.3-pN force. The two force vectors make a 65 angle. Whats the magnitude of the force the two motor proteins exert on the spindle pole?

A 14.6-kg monkey hangs from the middle of a massless rope, each half of which makes an 11.0 angle with the horizontal. Whats the rope tension? Compare with the monkeys weight.

A camper hangs a 26-kg pack between two trees using separate ropes of different lengths, as shown in Fig. 5.32. Find the tension in each rope. 71 28 Figure 5.32 Problem 38

A mass m1 undergoes circular motion of radius R on a horizontal frictionless table, connected by a massless string through a hole in the table to a second mass m2 (Fig. 5.33). If m2 is stationary, find expressions for (a) the string tension and (b) the period of the circular motion.

Patients with severe leg breaks are often placed in traction, with an external force countering muscles that would pull too hard on the broken bones. In the arrangement shown in Fig. 5.34, the mass m is 4.8 kg, and the pulleys can be considered massless and frictionless. Find the horizontal traction force applied to the leg. u1 = 70 u2 = 20 m Figure 5.34 Problem 40

Riders on the Great American Revolution loop-the-loop roller coaster of Example 5.7 wear seatbelts as the roller coaster negotiates its 6.3-m-radius loop at 9.7 m/s. At the top of the loop, what are the magnitude and direction of the force exerted on a 60-kg rider (a) by the roller-coaster seat and (b) by the seatbelt? (c) What would happen if the rider unbuckled at this point?

A 45-kg skater rounds a 5.0-m-radius turn at 6.3 m/s. (a) What are the horizontal and vertical components of the force the ice exerts on her skate blades? (b) At what angle can she lean without falling over?

When a plane turns, it banks as shown in Fig. 5.35 to give the wings lifting force F S w a horizontal component that turns the plane. If a plane is flying level at 950 km/h and the banking angle u is not to exceed 40, whats the minimum curvature radius for the turn? Fw S Fg S u Figure 5.35 Problem 43

You whirl a bucket of water in a vertical circle of radius 85 cm. Whats the minimum speed that will keep the water from falling out?

A child sleds down an 8.5 slope at constant speed. Whats the frictional coefficient between slope and sled?

The handle of a 22-kg lawnmower makes a 35 angle with the horizontal. If the coefficient of friction between lawnmower and ground is 0.68, what magnitude of force, applied in the direction of the handle, is required to push the mower at constant velocity? Compare with the mowers weight.

Repeat Example 5.4, now assuming that the coefficient of kinetic friction between rock and ice is 0.057.

A bat crashes into the vertical front of an accelerating subway train. If the frictional coefficient between bat and train is 0.86, whats the minimum acceleration of the train that will allow the bat to remain in place?

The coefficient of static friction between steel train wheels and steel rails is 0.58. The engineer of a train moving at 140 km/h spots a stalled car on the tracks 150 m ahead. If he applies the brakes so the wheels dont slip, will the train stop in time?

A bug crawls outward from the center of a CD spinning at 200 revolutions per minute. The coefficient of static friction between the bugs sticky feet and the disc surface is 1.2. How far does the bug get from the center before slipping?

A 310-g paperback book rests on a 1.2-kg textbook. A force is applied to the textbook, and the two books accelerate together from rest to 96 cm/s in 0.42 s. The textbook is then brought to a stop in 0.33 s, during which time the paperback slides off. Within what range does the coefficient of static friction between the two books lie?

Children sled down a 41-m-long hill inclined at 25. At the bottom, the slope levels out. If the coefficient of friction is 0.12, how far do the children slide on the level ground?

In a typical front-wheel-drive car, 70% of the cars weight rides on the front wheels. If the coefficient of friction between tires and road is 0.61, whats the cars maximum acceleration?

A police officer investigating an accident estimates that a moving car hit a stationary car at 25 km/h. Before the collision, the car left 47-m-long skid marks as it braked. The officer determines that the coefficient of kinetic friction was 0.71. What was the initial speed of the moving car?

A slide inclined at 35 takes bathers into a swimming pool. With water sprayed onto the slide to make it essentially frictionless, a bather spends only one-third as much time on the slide as when its dry. Whats the coefficient of friction on the dry slide?

You try to move a heavy trunk, pushing down and forward at an angle of 50 below the horizontal. Show that, no matter how hard you push, its impossible to budge the trunk if the coefficient of static friction exceeds 0.84.

A block is shoved up a 22 slope with an initial speed of 1.4 m/s. The coefficient of kinetic friction is 0.70. (a) How far up the slope will the block get? (b) Once stopped, will it slide back down?

At the end of a factory production line, boxes start from rest and slide down a 30 ramp 5.4 m long. If the slide can take no more than 3.3 s, whats the maximum allowed frictional coefficient?

Youre in traffic court, arguing against a speeding citation. You entered a 210-m-radius banked turn designed for 80 km/h, which was also the posted speed limit. The road was icy, yet you stayed in your lane, so you argue that you must have been going at the design speed. But police measurements show there was a frictional coefficient m = 0.15 between tires and road. Is it possible you were speeding, and if so by how much?

A space station is in the shape of a hollow ring, 450 m in diameter (Fig. 5.36). At how many revolutions per minute should it rotate in order to simulate Earths gravitythat is, so the normal force on an astronaut at the outer edge would equal the astronauts weight on Earth?

In a loop-the-loop roller coaster, show that a car moving too slowly would leave the track at an angle f given by cosf = v2 /rg, where f is the angle made by a vertical line through the center of the circular track and a line from the center to the point where the car leaves the track.

Find an expression for the minimum frictional coefficient needed to keep a car with speed v on a banked turn of radius R designed for speed v0.

An astronaut is training in an earthbound centrifuge that consists of a small chamber whirled horizontally at the end of a 5.1-mlong shaft. The astronaut places a notebook on the vertical wall of the chamber and it stays in place. If the coefficient of static friction is 0.62, whats the minimum rate at which the centrifuge must be revolving?

You stand on a spring scale at the north pole and again at the equator. Which scale reading will be lower, and by what percentage will it be lower than the higher reading? Assume g has the same value at pole and equator.

Driving in thick fog on a horizontal road, you spot a tractortrailer truck jackknifed across the road. To avert a collision, you could brake to a stop or swerve in a circular arc, as suggested in Fig. 5.37. Which option offers the greater margin of safety? Assume that there is the same coefficient of static friction in both cases, and that you maintain constant speed if you swerve. Figure 5.37 Problem 65

A block is projected up an incline at angle u. It returns to its initial position with half its initial speed. Show that the coefficient of kinetic friction is mk = 3 5 tan u.

A 2.1-kg mass is connected to a spring with spring constant k = 150 N/m and unstretched length 18 cm. The two are mounted on a frictionless air table, with the free end of the spring attached to a frictionless pivot. The mass is set into circular motion at 1.4 m/s. Find the radius of its path.

Take mk = 0.75 in Example 5.11, and plot the tension force in units of the trunks weight, as a function of the rope angle u (that is, plot T/mg versus u). Use your plot to determine (a) the minimum tension necessary to move the trunk and (b) the angle at which this minimum tension should be applied.

Repeat the preceding problem for an arbitrary value of mk, by using calculus to find the minimum force needed to move the trunk with constant speed.

Moving through a liquid, an object of mass m experiences a resistive drag force proportional to its velocity, Fdrag = -bv, where b is a constant. (a) Find an expression for the objects speed as a function of time, when it starts from rest and falls vertically through the liquid. (b) Show that it reaches a terminal velocity mg/b.

Suppose the object in Problem 70 had an initial velocity in the horizontal direction equal to the terminal speed, vx0 = mg/b. Show that the horizontal distance it can go is limited to xmax = mvx0/b, and find an expression for its trajectory (y as a function of x).

A block is launched with speed v0 up a slope making an angle u with the horizontal; the coefficient of kinetic friction is mk. (a) Find an expression for the distance d the block travels along the slope. (b) Use calculus to determine the angle that minimizes d.

A florist asks you to make a window display with two hanging pots as shown in Fig. 5.38. The florist is adamant that the strings be as invisible as possible, so you decide to use fishing line but want to use the thinnest line you can. Will fishing line that can withstand 100 N of tension work?

Youre at the state fair. A sideshow barker claims that the star of the show can throw a 7.3-kg Olympic-style hammer faster than a speeding bullet. You recall that bullets travel at several hundreds of meters per second. The burly hammer thrower whirls the hammer in a circle that you estimate to be 2.4 m in diameter. You guess the chain holding the hammer makes an angle of 10 with the horizontal. When the hammer flies off, is it really moving faster than a bullet?

One of the limiting factors in high-performance aircraft is the acceleration to which the pilot can be subjected without blacking out; its measured in gees, or multiples of the gravitational acceleration. The F-22 Raptor fighter can achieve Mach 1.8 (1.8 times the speed of sound, which is about 340 m/s). Suppose a pilot dives in a circle and pulls up. If the pilot cant exceed 6g, whats the tightest circle (smallest radius) in which the plane can turn?

Figure 5.39 shows an apparatus used to verify Newtons second law. A pulling mass m1 hangs vertically from a string of negligible mass that passes over a pulley, also of negligible mass and with nearly frictionless bearings. The other end of the string is attached to a glider of mass m2 riding on an essentially frictionless, horizontal air track. Both m1 and m2 may be varied by placing additional masses on the pulling mass and glider. The experiment consists of starting the glider from rest and letting the pulling mass accelerate it down the track. Three photogates are used to time the glider over two distance intervals, and an experimental value for its acceleration is determined from these data, using constant-acceleration equations from Chapter 2. The table in the next column lists the measured acceleration for a number of mass combinations. (a) Determine a quantity that, when plotted on the horizontal axis of a graph, should result in a straight line of slope g when acceleration is plotted on the vertical axis. (b) Make your plot, fit a line to the plotted data, and report the experimentally determined value of g.

Passage Problems A spiral is an ice-skating position in which the skater glides on one foot with the other foot held above hip level. Its a required element in womens singles figure skating competition and is related to the arabesque performed in ballet. Figure 5.40 shows skater Sarah Hughes executing a spiral during her gold-medal performance at the Winter Olympics in Salt Lake City. From the photo, you can conclude that the skater is a. executing a turn to her left. b. executing a turn to her right. c. moving in a straight line out of the page.

Passage Problems A spiral is an ice-skating position in which the skater glides on one foot with the other foot held above hip level. Its a required element in womens singles figure skating competition and is related to the arabesque performed in ballet. Figure 5.40 shows skater Sarah Hughes executing a spiral during her gold-medal performance at the Winter Olympics in Salt Lake City. The net force on the skater a. points to her left. b. points to her right. c. is zero.

Passage Problems A spiral is an ice-skating position in which the skater glides on one foot with the other foot held above hip level. Its a required element in womens singles figure skating competition and is related to the arabesque performed in ballet. Figure 5.40 shows skater Sarah Hughes executing a spiral during her gold-medal performance at the Winter Olympics in Salt Lake City. If the skater were to execute the same maneuver but at higher speed, the tilt evident in the photo would be a. less. b. greater. c. unchanged.

Passage Problems A spiral is an ice-skating position in which the skater glides on one foot with the other foot held above hip level. Its a required element in womens singles figure skating competition and is related to the arabesque performed in ballet. Figure 5.40 shows skater Sarah Hughes executing a spiral during her gold-medal performance at the Winter Olympics in Salt Lake City. The tilt angle u that the skaters body makes with the vertical is given approximately by u = tan-1 (0.5). From this you can conclude that the skaters centripetal acceleration has approximate magnitude a. 0. b. 0.5 m/s2 . c. 5 m/s2 . d. cant be determined without knowing the skaters speed