Solved: In an amusement park ride called The Roundup,

Are Newtons laws different in two dimensions?

How do we analyze projectile-like motion?

How do we analyze circular motion?

Does this analysis apply to orbits?

Why doesnt the water fall out of the bucket?

Why is planar motion important?

A small rocket for gathering weather data has a mass of 30 kg and generates 1500 N of thrust. On a windy day, the wind exerts a 20 N horizontal force on the rocket. If the rocket is launched straight up, what is the shape of its trajectory, and by how much has it been deflected sideways when it reaches a height of 1.0 km? Because the rocket goes much higher than this, assume theres no significant mass loss during the first 1.0 km of flight

This force will cause the particle to a. Speed up and curve upward. b. Speed up and curve downward. c. Slow down and curve upward. d. Slow down and curve downward. e. Move to the right and down. f. Reverse direction.

An energetic father places his 20 kg child on a 5.0 kg cart to which a 2.0-m-long rope is attached. He then holds the end of the rope and spins the cart and child around in a circle, keeping the rope parallel to the ground. If the tension in the rope is 100 N, how many revolutions per minute (rpm) does the cart make? Rolling friction between the carts wheels and the ground is negligible.

A block on a string spins in a horizontal circle on a frictionless table. Rank in order, from largest to smallest, the tensions Ta to Te acting on blocks a to e. (a) (b) (c) (d) (e) r r = 100 cm v = 50 rpm r = 25 cm v = 200 rpm r r r r r = 100 cm v = 100 rpm r = 50 cm v = 100 rpm r = 50 cm v = 200 rpm

What is the maximum speed with which a 1500 kg car can make a left turn around a curve of radius 50 m on a level (unbanked) road without sliding?

An out-of-gas car is rolling over the top of a hill at speed v. At this instant, a. n 7 FG b. n 6 FG c. n = FG d. We cant tell about n without knowing v.

A highway curve of radius 70 m is banked at a 15 angle. At what speed v0 can a car take this curve without assistance from friction?

A ball on a string is swung in a vertical circle. The string happens to break when it is parallel to the ground and the ball is moving up. Which trajectory does the ball follow?

A Stone Age hunter places a 1.0 kg rock in a sling and swings it in a horizontal circle around his head on a 1.0-m-long vine. If the vine breaks at a tension of 200 N, what is the maximum angular speed, in rpm, with which he can swing the rock?

A 1500 kg car drives around a flat, 50-m-diameter track, starting from rest. The drive wheels supply a small but steady 525 N force in the forward direction. The coefficient of static friction between the car tires and the road is 0.90. How many revolutions of the track have been made when the car slides out of the curve?

The 250 g ball shown in FIGURE 8.18 revolves in a horizontal plane as the vertical shaft spins. What is the critical angular speed, in rpm, that the shaft must exceed to keep both strings taut?

In uniform circular motion, which of the following are constant: speed, velocity, angular velocity, centripetal acceleration, magnitude of the net force?

A car runs out of gas while driving down a hill. It rolls through the valley and starts up the other side. At the very bottom of the valley, which of the free-body diagrams in FIGURE Q8.2 is correct? The car is moving to the right, and drag and rolling friction are negligible.

FIGURE Q8.3 is a birds-eye view of particles on strings moving in horizontal circles on a tabletop. All are moving at the same speed. Rank in order, from largest to smallest, the tensions Ta to Td. Give your answer in the form a 7 b = c 7 d and explain your ranking.

Tarzan swings through the jungle on a massless vine. At the lowest point of his swing, is the tension in the vine greater than, less than, or equal to the gravitational force on Tarzan? Explain.

FIGURE Q8.5 shows two balls of equal mass moving in vertical circles. Is the tension in string A greater than, less than, or equal to the tension in string B if the balls travel over the top of the circle (a) with equal speed and (b) with equal angular velocity?

Ramon and Sally are observing a toy car speed up as it goes around a circular track. Ramon says, The cars speeding up, so there must be a net force parallel to the track. I dont think so, replies Sally. Its moving in a circle, and that requires centripetal acceleration. The net force has to point to the center of the circle. Do you agree with Ramon, Sally, or neither? Explain.

A jet plane is flying on a level course at constant speed. The engines are at full throttle. a. What is the net force on the plane? Explain. b. Draw a free-body diagram of the plane as seen from the side with the plane flying to the right. Name (dont just label) any and all forces shown on your diagram. c. Airplanes bank when they turn. Draw a free-body diagram of the plane as seen from behind as it makes a right turn. d. Why do planes bank as they turn? Explain

A small projectile is launched parallel to the ground at height h = 1 m with sufficient speed to orbit a completely smooth, airless planet. A bug rides inside a small hole inside the projectile. Is the bug weightless? Explai

You can swing a ball on a string in a vertical circle if you swing it fast enough. But if you swing too slowly, the string goes slack as the ball nears the top. Explain why theres a minimum speed to keep the ball moving in a circle

A golfer starts with the club over her head and swings it to reach maximum speed as it contacts the ball. Halfway through her swing, when the golf club is parallel to the ground, does the acceleration vector of the club head point (a) straight down, (b) parallel to the ground, approximately toward the golfers shoulders, (c) approximately toward the golfers feet, or (d) toward a point above the golfers head? Explain.

As a science fair project, you want to launch an 800 g model rocket straight up and hit a horizontally moving target as it passes 30 m above the launch point. The rocket engine provides a constant thrust of 15.0 N. The target is approaching at a speed of 15 m/s. At what horizontal distance between the target and the rocket should you launch?

A 500 g model rocket is on a cart that is rolling to the right at a speed of 3.0 m/s. The rocket engine, when it is fired, exerts an 8.0 N vertical thrust on the rocket. Your goal is to have the rocket pass through a small horizontal hoop that is 20 m above the ground. At what horizontal distance left of the hoop should you launch?

A 4.0 * 1010 kg asteroid is heading directly toward the center of the earth at a steady 20 km/s. To save the planet, astronauts strap a giant rocket to the asteroid perpendicular to its direction of travel. The rocket generates 5.0 * 109 N of thrust. The rocket is fired when the asteroid is 4.0 * 106 km away from earth. You can ignore the earths gravitational force on the asteroid and their rotation about the sun.a. If the mission fails, how many hours is it until the asteroid impacts the earth? b. The radius of the earth is 6400 km. By what minimum angle must the asteroid be deflected to just miss the earth? c. What is the actual angle of deflection if the rocket fires at full thrust for 300 s before running out of fuel?

A 55 kg astronaut who weighs 180 N on a distant planet is pondering whether she can leap over a 3.5-m-wide chasm without falling in. If she leaps at a 15 angle, what initial speed does she need to clear the chasm?

A 1500 kg car drives around a flat 200-m-diameter circular track at 25 m/s. What are the magnitude and direction of the net force on the car? What causes this force?

A 1500 kg car takes a 50-m-radius unbanked curve at 15 m/s. What is the size of the friction force on the car?

A 200 g block on a 50-cm-long string swings in a circle on a horizontal, frictionless table at 75 rpm. a. What is the speed of the block? b. What is the tension in the string?

In the Bohr model of the hydrogen atom, an electron 1mass m = 9.1 * 10-31 kg2 orbits a proton at a distance of 5.3 * 10-11 m. The proton pulls on the electron with an electric force of 8.2 * 10-8 N. How many revolutions per second does the electron make?

Suppose the moon were held in its orbit not by gravity but by a massless cable attached to the center of the earth. What would be the tension in the cable? Use the table of astronomical data inside the back cover of the book

A highway curve of radius 500 m is designed for traffic moving at a speed of 90 km/h. What is the correct banking angle of the road?

It is proposed that future space stations create an artificial gravity by rotating. Suppose a space station is constructed as a 1000-m-diameter cylinder that rotates about its axis. The inside surface is the deck of the space station. What rotation period will provide normal gravity?

A 5.0 g coin is placed 15 cm from the center of a turntable. The coin has static and kinetic coefficients of friction with the turntable surface of ms = 0.80 and mk = 0.50. The turntable very slowly speeds up to 60 rpm. Does the coin slide off?

Mass m1 on the frictionless table of FIGURE EX8.13 is connected by a string through a hole in the table to a hanging mass m2. With what speed must m1 rotate in a circle of radius r if m2 is to remain hanging at rest?

A satellite orbiting the moon very near the surface has a period of 110 min. What is free-fall acceleration on the surface of the moon?

What is free-fall acceleration toward the sun at the distance of the earths orbit? Astronomical data are inside the back cover of the book.

A 9.4 * 1021 kg moon orbits a distant planet in a circular orbit of radius 1.5 * 108 m. It experiences a 1.1 * 1019 N gravitational pull from the planet. What is the moons orbital period in earth days?

Communications satellites are placed in circular orbits where they stay directly over a fixed point on the equator as the earth rotates. These are called geosynchronous orbits. The altitude of a geosynchronous orbit is 3.58 * 107 m (22,000 miles). a. What is the period of a satellite in a geosynchronous orbit? b. Find the value of g at this altitude. c. What is the weight of a 2000 kg satellite in a geosynchronous orbit?

A car drives over the top of a hill that has a radius of 50 m. What maximum speed can the car have at the top without flying off the road?

The weight of passengers on a roller coaster increases by 50% as the car goes through a dip with a 30 m radius of curvature. What is the cars speed at the bottom of the dip?

A roller coaster car crosses the top of a circular loop-the-loop at twice the critical speed. What is the ratio of the normal force to the gravitational force?

The normal force equals the magnitude of the gravitational force as a roller coaster car crosses the top of a 40-m-diameter loop-the-loop. What is the cars speed at the top?

A student has 65-cm-long arms. What is the minimum angular velocity (in rpm) for swinging a bucket of water in a vertical circle without spilling any? The distance from the handle to the bottom of the bucket is 35 cm.

While at the county fair, you decide to ride the Ferris wheel. Having eaten too many candy apples and elephant ears, you find the motion somewhat unpleasant. To take your mind off your stomach, you wonder about the motion of the ride. You estimate the radius of the big wheel to be 15 m, and you use your watch to find that each loop around takes 25 s. a. What are your speed and the magnitude of your acceleration? b. What is the ratio of your weight at the top of the ride to your weight while standing on the ground? c. What is the ratio of your weight at the bottom of the ride to your weight while standing on the ground?

A 500 g ball swings in a vertical circle at the end of a 1.5-m-long string. When the ball is at the bottom of the circle, the tension in the string is 15 N. What is the speed of the ball at that point?

A 500 g ball moves in a vertical circle on a 102-cm-long string. If the speed at the top is 4.0 m/s, then the speed at the bottom will be 7.5 m/s. (Youll learn how to show this in Chapter 10.) a. What is the gravitational force acting on the ball? b. What is the tension in the string when the ball is at the top? c. What is the tension in the string when the ball is at the bottom?

A heavy ball with a weight of 100 N 1m = 10.2 kg2 is hung from the ceiling of a lecture hall on a 4.5-m-long rope. The ball is pulled to one side and released to swing as a pendulum, reaching a speed of 5.5 m/s as it passes through the lowest point. What is the tension in the rope at that point?

A toy train rolls around a horizontal 1.0-m-diameter track. The coefficient of rolling friction is 0.10. How long does it take the train to stop if its released with an angular speed of 30 rpm?

A new car is tested on a 200-m-diameter track. If the car speeds up at a steady 1.5 m/s2 , how long after starting is the magnitude of its centripetal acceleration equal to the tangential acceleration?

An 85,000 kg stunt plane performs a loop-the-loop, flying in a 260-m-diameter vertical circle. At the point where the plane is flying straight down, its speed is 55 m/s and it is speeding up at a rate of 12 m/s per second. a. What is the magnitude of the net force on the plane? You can neglect air resistance. b. What angle does the net force make with the horizontal? Let an angle above horizontal be positive and an angle below horizontal be negative

Three cars are driving at 25 m/s along the road shown in FIGURE EX8.30. Car B is at the bottom of a hill and car C is at the top. Both hills have a 200 m radius of curvature. Suppose each car suddenly brakes hard and starts to skid. What is the tangential acceleration (i.e., the acceleration parallel to the road) of each car? Assume mk = 1.0.

Derive Equations 8.3 for the acceleration of a projectile subject to drag

A 100 g bead slides along a frictionless wire with the parabolic shape y = 12 m-1 2x2 . a. Find an expression for ay, the vertical component of acceleration, in terms of x, vx, and ax. Hint: Use the basic definitions of velocity and acceleration. b. Suppose the bead is released at some negative value of x and has a speed of 2.3 m/s as it passes through the lowest point of the parabola. What is the net force on the bead at this instant? Write your answer in component form

Space scientists have a large test chamber from which all the air can be evacuated and in which they can create a horizontal uniform electric field. The electric field exerts a constant horizontal force on a charged object. A 15 g charged projectile is launched with a speed of 6.0 m/s at an angle 35 above the horizontal. It lands 2.9 m in front of the launcher. What is the magnitude of the electric force on the projectile?

| A 5000 kg interceptor rocket is launched at an angle of 44.7. The thrust of the rocket motor is 140,700 N. a. Find an equation y(x) that describes the rockets trajectory. b. What is the shape of the trajectory? c. At what elevation does the rocket reach the speed of sound, 330 m/s?

A motorcycle daredevil plans to ride up a 2.0-m-high, 20 ramp, sail across a 10-m-wide pool filled with hungry crocodiles, and land at ground level on the other side. He has done this stunt many times and approaches it with confidence. Unfortunately, the motorcycle engine dies just as he starts up the ramp. He is going 11 m/s at that instant, and the rolling friction of his rubber tires (coefficient 0.02) is not negligible. Does he survive, or does he become crocodile food? Justify your answer by calculating the distance he travels through the air after leaving the end of the ramp

A rocket-powered hockey puck has a thrust of 2.0 N and a total mass of 1.0 kg. It is released from rest on a frictionless table, 4.0 m from the edge of a 2.0 m drop. The front of the rocket is pointed directly toward the edge. How far does the puck land from the base of the table?

A 500 g model rocket is resting horizontally at the top edge of a 40-m-high wall when it is accidentally bumped. The bump pushes it off the edge with a horizontal speed of 0.5 m/s and at the same time causes the engine to ignite. When the engine fires, it exerts a constant 20 N horizontal thrust away from the wall. a. How far from the base of the wall does the rocket land? b. Describe the rockets trajectory as it travels to the ground

A 2.0 kg projectile with initial velocity v u = 8.0 ni m/s experiences the variable force F u = -2.0t ni + 4.0t 2 nj N, where t is in s. a. What is the projectiles speed at t = 2.0 s? b. At what instant of time is the projectile moving parallel to the y-axis?

A 75 kg man weighs himself at the north pole and at the equator. Which scale reading is higher? By how much? Assume the earth is spherical

A concrete highway curve of radius 70 m is banked at a 15 angle. What is the maximum speed with which a 1500 kg rubbertired car can take this curve without sliding?

a. An object of mass m swings in a horizontal circle on a string of length L that tilts downward at angle u. Find an expression for the angular velocity v. b. A student ties a 500 g rock to a 1.0-m-long string and swings it around her head in a horizontal circle. At what angular speed, in rpm, does the string tilt down at a 10 angle?

Youve taken your neighbors young child to the carnival to ride the rides. She wants to ride The Rocket. Eight rocketshaped cars hang by chains from the outside edge of a large steel disk. A vertical axle through the center of the ride turns the disk, causing the cars to revolve in a circle. Youve just finished taking physics, so you decide to figure out the speed of the cars while you wait. You estimate that the disk is 5 m in diameter and the chains are 6 m long. The ride takes 10 s to reach full speed, then the cars swing out until the chains are 20 from vertical. What is the cars speed?

A 4.4-cm-diameter, 24 g plastic ball is attached to a 1.2-m-long string and swung in a vertical circle. The balls speed is 6.1 m/s at the point where it is moving straight up. What is the magnitude of the net force on the ball? Air resistance is not negligible

A charged particle of mass m moving with speed v in a plane perpendicular to a magnetic field experiences a force F u = 1qvB, perpendicular to v u 2, where q is the amount of charge and B is the magnetic field strength. Because the force is always perpendicular to the particles velocity, the particle undergoes uniform circular motion. Find an expression for the period of the motion. Gravity can be neglected

Two wires are tied to the 2.0 kg sphere shown in FIGURE P8.45. The sphere revolves in a horizontal circle at constant speed. a. For what speed is the tension the same in both wires? b. What is the tension?

Two wires are tied to the 300 g sphere shown in FIGURE P8.46. The sphere revolves in a horizontal circle at a constant speed of 7.5 m/s. What is the tension in each of the wires?

A conical pendulum is formed by attaching a ball of mass m to a string of length L, then allowing the ball to move in a horizontal circle of radius r. FIGURE P8.47 shows that the string traces out the surface of a cone, hence the name. a. Find an expression for the tension T in the string. b. Find an expression for the balls angular speed v. c. What are the tension and angular speed (in rpm) for a 500 g ball swinging in a 20-cm-radius circle at the end of a 1.0-m-long string?

The 10 mg bead in FIGURE P8.48 is free to slide on a frictionless wire loop. The loop rotates about a vertical axis with angular velocity v. If v is less than some critical value vc, the bead sits at the bottom of the spinning loop. When v 7 vc, the bead moves out to some angle u. a. What is vc in rpm for the loop shown in the figure? b. At what value of v, in rpm, is u = 30?

In an old-fashioned amusement park ride, passengers stand inside a 5.0-m-diameter hollow steel cylinder with their backs against the wall. The cylinder begins to rotate about a vertical axis. Then the floor on which the passengers are standing suddenly drops away! If all goes well, the passengers will stick to the wall and not slide. Clothing has a static coefficient of friction against steel in the range 0.60 to 1.0 and a kinetic coefficient in the range 0.40 to 0.70. A sign next to the entrance says No children under 30 kg allowed. What is the minimum angular speed, in rpm, for which the ride is safe?

The ultracentrifuge is an important tool for separating and analyzing proteins. Because of the enormous centripetal accelerations, the centrifuge must be carefully balanced, with each sample matched by a sample of identical mass on the opposite side. Any difference in the masses of opposing samples creates a net force on the shaft of the rotor, potentially leading to a catastrophic failure of the apparatus. Suppose a scientist makes a slight error in sample preparation and one sample has a mass 10 mg larger than the opposing sample. If the samples are 12 cm from the axis of the rotor and the ultracentrifuge spins at 70,000 rpm, what is the magnitude of the net force on the rotor due to the unbalanced samples?

In an amusement park ride called The Roundup, passengers stand inside a 16-m-diameter rotating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane, as shown in FIGURE P8.51. a. Suppose the ring rotates once every 4.5 s. If a riders mass is 55 kg, with how much force does the ring push on her at the top of the ride? At the bottom? b. What is the longest rotation period of the wheel that will prevent the riders from falling off at the top?

Suppose you swing a ball of mass m in a vertical circle on a string of length L. As you probably know from experience, there is a minimum angular velocity vmin you must maintain if you want the ball to complete the full circle without the string going slack at the top. a. Find an expression for vmin. b. Evaluate vmin in rpm for a 65 g ball tied to a 1.0-m-long string

A 30 g ball rolls around a 40-cm-diameter L-shaped track, shown in FIGURE P8.53, at 60 rpm. What is the magnitude of the net force that the track exerts on the ball? Rolling friction can be neglected. Hint: The track exerts more than one force on the ball.

FIGURE P8.54 shows a small block of mass m sliding around the inside of an L-shaped track of radius r. The bottom of the track is frictionless; the coefficient of kinetic friction between the block and the wall of the track is mk. The blocks speed is v0 at t0 = 0. Find an expression for the blocks speed at a later time t

The physics of circular motion sets an upper limit to the speed of human walking. (If you need to go faster, your gait changes from a walk to a run.) If you take a few steps and watch whats happening, youll see that your body pivots in circular motion over your forward foot as you bring your rear foot forward for the next step. As you do so, the normal force of the ground on your foot decreases and your body tries to lift off from the ground. a. A persons center of mass is very near the hips, at the top of the legs. Model a person as a particle of mass m at the top of a leg of length L. Find an expression for the persons maximum walking speed vmax. b. Evaluate your expression for the maximum walking speed of a 70 kg person with a typical leg length of 70 cm. Give your answer in both m/s and mph, then comment, based on your experience, as to whether this is a reasonable result. A normal walking speed is about 3 mph.

A 100 g ball on a 60-cm-long string is swung in a vertical circle about a point 200 cm above the floor. The tension in the string when the ball is at the very bottom of the circle is 5.0 N. A very sharp knife is suddenly inserted, as shown in FIGURE P8.56, to cut the string directly below the point of support. How far to the right of where the string was cut does the ball hit the floor?

A 60 g ball is tied to the end of a 50-cm-long string and swung in a vertical circle. The center of the circle, as shown in FIGURE P8.57, is 150 cm above the floor. The ball is swung at the minimum speed necessary to make it over the top without the string going slack. If the string is released at the instant the ball is at the top of the loop, how far to the right does the ball hit the ground?

Elm Street has a pronounced dip at the bottom of a steep hill before going back uphill on the other side. Your science teacher has asked everyone in the class to measure the radius of curvature of the dip. Some of your classmates are using surveying equipment, but you decide to base your measurement on what youve learned in physics. To do so, you sit on a spring scale, drive through the dip at different speeds, and for each speed record the scales reading as you pass through the bottom of the dip. Your data are as follows: Speed (m/s) Scale reading (N) 5 599 10 625 15 674 20 756 25 834 Sitting on the scale while the car is parked gives a reading of 588 N. Analyze your data, using a graph, to determine the dips radius of curvature.

A 100 g ball on a 60-cm-long string is swung in a vertical circle about a point 200 cm above the floor. The string suddenly breaks when it is parallel to the ground and the ball is moving upward. The ball reaches a height 600 cm above the floor. What was the tension in the string an instant before it broke?

Scientists design a new particle accelerator in which protons 1mass 1.7 * 10-27 kg2 follow a circular trajectory given by r u = c cos1kt 2 2ni + c sin1kt 2 2nj, where c = 5.0 m and k = 8.0 * 104 rad/s2 are constants and t is the elapsed time. a. What is the radius of the circle? b. What is the protons speed at t = 3.0 s? c. What is the force on the proton at t = 3.0 s? Give your answer in component form.

A 1500 kg car starts from rest and drives around a flat 50-m-diameter circular track. The forward force provided by the cars drive wheels is a constant 1000 N. a. What are the magnitude and direction of the cars acceleration at t = 10 s? Give the direction as an angle from the r-axis. b. If the car has rubber tires and the track is concrete, at what time does the car begin to slide out of the circle?

A 500 g steel block rotates on a steel table while attached to a 2.0-m-long massless rod. Compressed air fed through the rod is ejected from a nozzle on the back of the block, exerting a thrust force of 3.5 N. The nozzle is 70 from the radial line, as shown in FIGURE P8.62. The block starts from rest. a. What is the blocks angular velocity after 10 rev? b. What is the tension in the rod after 10 rev?

A 2.0 kg ball swings in a vertical circle on the end of an 80-cm-long string. The tension in the string is 20 N when its angle from the highest point on the circle is u = 30. a. What is the balls speed when u = 30? b. What are the magnitude and direction of the balls acceleration when u = 30?

In Problems 64 and 65 you are given the equation used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation. Be sure that the answer your problem requests is consistent with the equation given. b. Finish the solution of the problem.60 N = 10.30 kg2v2 10.50 m2

In Problems 64 and 65 you are given the equation used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation. Be sure that the answer your problem requests is consistent with the equation given. b. Finish the solution of the problem.11500 kg219.8 m/s2 2 - 11,760 N = 11500 kg2 v 2 /1200 m2

Sam (75 kg) takes off up a 50-m-high, 10 frictionless slope on his jet-powered skis. The skis have a thrust of 200 N. He keeps his skis tilted at 10 after becoming airborne, as shown in FIGURE CP8.66. How far does Sam land from the base of the cliff?

In the absence of air resistance, a projectile that lands at the elevation from which it was launched achieves maximum range when launched at a 45 angle. Suppose a projectile of mass m is launched with speed v0 into a headwind that exerts a constant, horizontal retarding force F u wind = -Fwind ni. a. Find an expression for the angle at which the range is maximum. b. By what percentage is the maximum range of a 0.50 kg ball reduced if Fwind = 0.60 N?

The father of Example 8.2 stands at the summit of a conical hill as he spins his 20 kg child around on a 5.0 kg cart with a 2.0-m-long rope. The sides of the hill are inclined at 20. He again keeps the rope parallel to the ground, and friction is negligible. What rope tension will allow the cart to spin with the same 14 rpm it had in the example?

A small bead slides around a horizontal circle at height y inside the cone shown in FIGURE CP8.69. Find an expression for the beads speed in terms of a, h, y, and g

A 500 g steel block rotates on a steel table while attached to a 1.2-m-long hollow tube as shown in FIGURE CP8.70. Compressed air fed through the tube and ejected from a nozzle on the back of the block exerts a thrust force of 4.0 N perpendicular to the tube.The maximum tension the tube can withstand without breaking is 50 N. If the block starts from rest, how many revolutions does it make before the tube breaks?

If a vertical cylinder of water (or any other liquid) rotates about its axis, as shown in FIGURE CP8.71, the surface forms a smooth curve. Assuming that the water rotates as a unit (i.e., all the water rotates with the same angular velocity), show that the shape of the surface is a parabola described by the equation z = 1v2 /2g2r2 . Hint: Each particle of water on the surface is subject to only two forces: gravity and the normal force due to the water underneath it. The normal force, as always, acts perpendicular to the surface.

What are the overarching findings of Part I?

What laws of physics govern motion?

What are the most common models for applying the laws of physics to moving objects?

What are the most important tools for analyzing the physics of motion?