Derive the equation for the vertical acceleration of a

Problem 1Q A bicycle odometer (which counts revolutions and is calibrated to report distance traveled) is attached near the wheel axle and is calibrated for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Problem 1PE (a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of 7.50 m/s . (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of 600 m/s . (c) What is the momentum of the 90.0-kg hunter running at 7.40 m/s after missing the elephant?

Problem 2CQ An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum?

Problem 2PE (a) What is the mass of a large ship that has a momentum of 1.60×109 kg · m/s , when the ship is moving at a speed of 48.0 km/h? (b) Compare the ship’s momentum to the momentum of a 1100-kg artillery shell fired at a speed of 1200 m/s .

Problem 2Q Suppose a disk rotates at constant angular velocity, (a) Does a point on the rim have radial and or tangential acceleration? (b) If the disk's angular velocity increases uniformly, does the point have radial and or tangential acceleration? (c) For which cases would the magnitude of either component of linear acceleration change?

(1) A laser beam is directed at the Moon, \(380,000 \mathrm{~km}\) from Earth. The beam diverges at an angle \(\theta\) (Fig. 8-37) of \(1.4 \times 10^{-5}\) rad. What diameter spot will it make on the Moon?

Problem 3CQ Professional Application Football coaches advise players to block, hit, and tackle with their feet on the ground rather than by leaping through the air. Using the concepts of momentum, work, and energy, explain how a football player can be more effective with his feet on the ground.

Problem 3PE (a) At what speed would a 2.00×104 -kg airplane have to fly to have a momentum of 1.60×109 kg · m/s (the same as the ship’s momentum in the problem above)? (b) What is the plane’s momentum when it is taking off at a speed of 60.0 m/s ? (c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship.

Problem 4Q Can a small force ever exert a greater torque than a larger force? Explain.

Problem 4PE Find the current when 2.00 nC jumps between your comb and hair over a 0.500 - µs time interval.

Problem 5CQ Professional Application Explain in terms of impulse how padding reduces forces in a collision. State this in terms of a real example, such as the advantages of a carpeted vs. tile floor for a day care center.

Problem 5PE A runaway train car that has a mass of 15,000 kg travels at a speed of 5.4 m/s down a track. Compute the time required for a force of 1500 N to bring the car to rest.

Problem 6CQ While jumping on a trampoline, sometimes you land on your back and other times on your feet. In which case can you reach a greater height and why?

Problem 6PE The mass of Earth is 5.972×1024 kg and its orbital radius is an average of 1.496×1011 m . Calculate its linear momentum.

Problem 6Q Why is it more difficult to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer this.

Problem 7PE A bullet is accelerated down the barrel of a gun by hot gases produced in the combustion of gun powder. What is the average force exerted on a 0.0300-kg bullet to accelerate it to a speed of 600 m/s in a time of 2.00 ms (milliseconds)?

Problem 8CQ Professional Application If you dive into water, you reach greater depths than if you do a belly flop. Explain this difference in depth using the concept of conservation of energy. Explain this difference in depth using what you have learned in this chapter.

(II) A rotating merry-go-round makes one complete revolution in (Fig. ). (a) What is the linear speed of a child seated from the center? What is her acceleration (give components)? FIGURE 8-38 Problem 8

Problem 12PE Professional Application One hazard of space travel is debris left by previous missions. There are several thousand objects orbiting Earth that are large enough to be detected by radar, but there are far greater numbers of very small objects, such as flakes of paint. Calculate the force exerted by a 0.100-mg chip of paint that strikes a spacecraft window at a relative speed of 4.00×103 m/s , given the collision lasts 6.00×10 – 8 s .

(II) A turntable of radius R1 is turned by a circular rubber roller of radius R2 in contact with it at their outer edges. What is the ratio of their angular velocities, w1 fw2

Problem 13PE Professional Application A 75.0-kg person is riding in a car moving at 20.0 m/s when the car runs into a bridge abutment. (a) Calculate the average force on the person if he is stopped by a padded dashboard that compresses an average of 1.00 cm. (b) Calculate the average force on the person if he is stopped by an air bag that compresses an average of 15.0 cm.

A sphere and a cylinder have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

Problem 14CQ Must the total energy of a system be conserved whenever its momentum is conserved? Explain why or why not.

Problem 14PE Professional Application Military rifles have a mechanism for reducing the recoil forces of the gun on the person firing it. An internal part recoils over a relatively large distance and is stopped by damping mechanisms in the gun. The larger distance reduces the average force needed to stop the internal part. (a) Calculate the recoil velocity of a 1.00-kg plunger that directly interacts with a 0.0200-kg bullet fired at 600 m/s from the gun. (b) If this part is stopped over a distance of 20.0 cm, what average force is exerted upon it by the gun? (c) Compare this to the force exerted on the gun if the bullet is accelerated to its velocity in 10.0 ms (milliseconds).

Problem 14Q Problem V\fe claim that momentum and angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explain.

Problem 15CQ What is an elastic collision?

Problem 15PE A cruise ship with a mass of 1.00×107 kg strikes a pier at a speed of 0.750 m/s. It comes to rest 6.00 m later, damaging the ship, the pier, and the tugboat captain’s finances. Calculate the average force exerted on the pier using the concept of impulse. (Hint: First calculate the time it took to bring the ship to rest.)

Problem 16CQ What is an inelastic collision? What is a perfectly inelastic collision?

Problem 16PE Calculate the final speed of a 110-kg rugby player who is initially running at 8.00 m/s but collides head-on with a padded goalpost and experiences a backward force of 1.76×104 N for 5.50×10–2 s .

Problem 16Q Can the diver of Fig. 8-28 do a somersault without having any initial rotation when she leaves the board? Explain.

Problem 17CQ Mixed-pair ice skaters performing in a show are standing motionless at arms length just before starting a routine. They reach out, clasp hands, and pull themselves together by only using their arms. Assuming there is no friction between the blades of their skates and the ice, what is their velocity after their bodies meet?

Problem 17PE Water from a fire hose is directed horizontally against a wall at a rate of 50.0 kg/s and a speed of 42.0 m/s. Calculate the magnitude of the force exerted on the wall, assuming the water’s horizontal momentum is reduced to zero.

Problem 18CQ A small pickup truck that has a camper shell slowly coasts toward a red light with negligible friction. Two dogs in the back of the truck are moving and making various inelastic collisions with each other and the walls. What is the effect of the dogs on the motion of the center of mass of the system (truck plus entire load)? What is their effect on the motion of the truck?

Problem 18PE A 0.450-kg hammer is moving horizontally at 7.00 m/s when it strikes a nail and comes to rest after driving the nail 1.00 cm into a board. (a) Calculate the duration of the impact. (b) What was the average force exerted on the nail?

Problem 19CQ Figure 8.16 shows a cube at rest and a small object heading toward it. (a) Describe the directions (angle ?1 ) at which the small object can emerge after colliding elastically with the cube. How does ?1 depend on b , the so-called impact parameter? Ignore any effects that might be due to rotation after the collision, and assume that the cube is much more massive than the small object. (b) Answer the same questions if the small object instead collides with a massive sphere.

Problem 19PE Starting with the definitions of momentum and kinetic energy, derive an equation for the kinetic energy of a particle expressed as a function of its momentum.

Problem 19Q Two spheres look identical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which.

Problem 20CQ Professional Application Suppose a fireworks shell explodes, breaking into three large pieces for which air resistance is negligible. How is the motion of the center of mass affected by the explosion? How would it be affected if the pieces experienced significantly more air resistance than the intact shell?

Problem 20PE A ball with an initial velocity of 10 m/s moves at an angle 60º above t?he + ?x -direction. The ball hits a vertical wall and bounces off so that it is moving 60º above? the ? ?x -direction with the same speed. What is the impulse delivered by the wall?

Problem 21CQ Professional Application During a visit to the International Space Station, an astronaut was positioned motionless in the center of the station, out of reach of any solid object on which he could exert a force. Suggest a method by which he could move himself away from this position, and explain the physics involved.

Problem 20Q In what direction is the Earth's angular velocity vector as it rotates daily about its axis, north or south?

Problem 21PE When serving a tennis ball, a player hits the ball when its velocity is zero (at the highest point of a vertical toss). The racquet exerts a force of 540 N on the ball for 5.00 ms, giving it a final velocity of 45.0 m/s. Using these data, find the mass of the ball.

Problem 21Q The angular velocity of a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Problem 22CQ Professional Application It is possible for the velocity of a rocket to be greater than the exhaust velocity of the gases it ejects. When that is the case, the gas velocity and gas momentum are in the same direction as that of the rocket. How is the rocket still able to obtain thrust by ejecting the gases?

Problem 22PE A punter drops a ball from rest vertically 1 meter down onto his foot. The ball leaves the foot with a speed of 18 m/s at an angle 55º above the horizontal. What is the impulse delivered by the foot (magnitude and direction)?

Problem 23PE Professional Application Train cars are coupled together by being bumped into one another. Suppose two loaded train cars are moving toward one another, the first having a mass of 150,000 kg and a velocity of 0.300 m/s, and the second having a mass of 110,000 kg and a velocity of ?0.120 m/s . (The minus indicates direction of motion.) What is their final velocity?

A shortstop may leap into the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. ). Explain. FIGURE 8-36 Question 23. A shortshop in the air, throwing the ball.

Problem 24PE Suppose a clay model of a koala bear has a mass of 0.200 kg and slides on ice at a speed of 0.750 m/s. It runs into another clay model, which is initially motionless and has a mass of 0.350 kg. Both being soft clay, they naturally stick together. What is their final velocity?

Problem 24Q On the basis of the law of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate in order to keep the helicopter stable.

Problem 25PE Professional Application Consider the following question: A car moving at 10 m/s crashes into a tree and stops in 0.26 s. Calculate the force the seat belt exerts on a passenger in the car to bring him to a halt. The mass of the passenger is 70 kg. Would the answer to this question be different if the car with the 70-kg passenger had collided with a car that has a mass equal to and is traveling in the opposite direction and at the same speed? Explain your answer.

Problem 26PE What is the velocity of a 900-kg car initially moving at 30.0 m/s, just after it hits a 150-kg deer initially running at 12.0 m/s in the same direction? Assume the deer remains on the car.

Problem 27PE A 1.80-kg falcon catches a 0.650-kg dove from behind in midair. What is their velocity after impact if the falcon’s velocity is initially 28.0 m/s and the dove’s velocity is 7.00 m/s in the same direction?

Problem 28PE Two identical objects (such as billiard balls) have a one dimensional collision in which one is initially motionless. After the collision, the moving object is stationary and the other moves with the same speed as the other originally had. Show that both momentum and kinetic energy are conserved.

Problem 29PE Professional Application Two manned satellites approach one another at a relative speed of 0.250 m/s, intending to dock. The first has a mass of 4.00×103 kg , and the second a mass of 7.50×103 kg . If the two satellites collide elastically rather than dock, what is their final relative velocity?

Problem 30PE A 70.0-kg ice hockey goalie, originally at rest, catches a 0.150-kg hockey puck slapped at him at a velocity of 35.0 m/ s. Suppose the goalie and the ice puck have an elastic collision and the puck is reflected back in the direction from which it came. What would their final velocities be in this case?

Problem 31PE A 0.240-kg billiard ball that is moving at 3.00 m/s strikes the bumper of a pool table and bounces straight back at 2.40 m/s (80% of its original speed). The collision lasts 0.0150 s. (a) Calculate the average force exerted on the ball by the bumper. (b) How much kinetic energy in joules is lost during the collision? (c) What percent of the original energy is left?

Problem 32PE Problem During an ice show, a 60.0-kg skater leaps into the air and is caught by an initially stationary 75.0-kg skater. (a) What is their final velocity assuming negligible friction and that the 60.0-kg skater original horizontal velocity is 4.00 m/s? (b) How much kinetic energy is lost?

Problem 33PE Problem Professional Application Using mass and speed data from Example 8.1 and assuming that the football player catches the ball with his feet off the ground with both of them moving horizontally, calculate: (a) the final velocity if the ball and player are going in the same direction and (b) the loss of kinetic energy in this case. (c) Repeat parts (a) and (b) for the situation in which the ball and the player are going in opposite directions. Might the loss of kinetic energy be related to how much it hurts to catch the pass? Example 8.1 Calculating Momentum: A Football Player and a Football (a) Calculate the momentum of a 110-kg football player running at 8.00 m/s. (b) Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s.

Problem 34PE Problem A battleship that is 6.00×107 kg and is originally at rest fires a 1100-kg artillery shell horizontally with a velocity of 575 m/s. (a) If the shell is fired straight aft (toward the rear of the ship), there will be negligible friction opposing the ship’s recoil. Calculate its recoil velocity. (b) Calculate the increase in internal kinetic energy (that is, for the ship and the shell). This energy is less than the energy released by the gunpowder—significant heat transfer occurs.

Problem 35PE Problem Professional Application Two manned satellites approaching one another, at a relative speed of 0.250 m/s, intending to dock. The first has a mass of 4.00×103 kg , and the second a mass of 7.50×103 kg . (a) Calculate the final velocity (after docking) by using the frame of reference in which the first satellite was originally at rest. (b) What is the loss of kinetic energy in this inelastic collision? (c) Repeat both parts by using the frame of reference in which the second satellite was originally at rest. Explain why the change in velocity is different in the two frames, whereas the change in kinetic energy is the same in both.

Problem 36PE Problem Professional Application A 30,000-kg freight car is coasting at 0.850 m/s with negligible friction under a hopper that dumps 110,000 kg of scrap metal into it. (a) What is the final velocity of the loaded freight car? (b) How much kinetic energy is lost?

Problem 37PE Problem Professional Application Space probes may be separated from their launchers by exploding bolts. (They bolt away from one another.) Suppose a 4800-kg satellite uses this method to separate from the 1500-kg remains of its launcher, and that 5000 J of kinetic energy is supplied to the two parts. What are their subsequent velocities using the frame of reference in which they were at rest before separation?

(II) The forearm in Fig. accelerates a ball at \(7.0 \mathrm{~m} / \mathrm{s}^{2}\) by means of the triceps muscle, as shown. Calcu. late (a) the torque needed, and ( ) the force that must be exerted by the triceps muscle. Ignore the mass of the arm. FIGURE 8-45 Problem 38 Equation Transcription: Text Transcription: 7.0 \mathrm{~m} / \mathrm{s}^{2}

Problem 38PE Problem A 0.0250-kg bullet is accelerated from rest to a speed of 550 m/s in a 3.00-kg rifle. The pain of the rifle’s kick is much worse if you hold the gun loosely a few centimeters from your shoulder rather than holding it tightly against your shoulder. (a) Calculate the recoil velocity of the rifle if it is held loosely away from the shoulder. (b) How much kinetic energy does the rifle gain? (c) What is the recoil velocity if the rifle is held tightly against the shoulder, making the effective mass 28.0 kg? (d) How much kinetic energy is transferred to the rifle shoulder combination? The pain is related to the amount of kinetic energy, which is significantly less in this latter situation. (e) Calculate the momentum of a 110-kg football player running at 8.00 m/s. Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s. Discuss its relationship to this problem.

Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8-45. The ball is accelerated uniformly from rest to in 10.0 m/s in 0.350 s, at which point it is released. Calculate (a) the angular acceleration of the arm, and (b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.

Problem 39PE Problem Professional Application One of the waste products of a nuclear reactor is plutonium-239 (239 Pu) . This nucleus is radioactive and decays by splitting into a helium-4 nucleus and a uranium-235 nucleus (4 He + 235 U), the latter of which is also radioactive and will itself decay some time later. The energy emitted in the plutonium decay is 8.40×10– 13 J and is entirely converted to kinetic energy of the helium and uranium nuclei. The mass of the helium nucleus is 6.68×10–27 kg , while that of the uranium is 3.92×10–25 kg (note that the ratio of the masses is 4 to 235). (a) Calculate the velocities of the two nuclei, assuming the plutonium nucleus is originally at rest. (b) How much kinetic energy does each nucleus carry away? Note that the data given here are accurate to three digits only.

Problem 40PE Problem Professional Application The Moon’s craters are remnants of meteorite collisions. Suppose a fairly large asteroid that has a mass of 5.00× kg (about a kilometer across) strikes the Moon at a speed of 15.0 km/s. (a) At what speed does the Moon recoil after the perfectly inelastic collision (the mass of the Moon is 7.36×1022 kg ) ? (b) How much kinetic energy is lost in the collision? Such an event may have been observed by medieval English monks who reported observing a red glow and subsequent haze about the Moon. (c) In October 2009, NASA crashed a rocket into the Moon, and analyzed the plume produced by the impact. (Significant amounts of water were detected.) Answer part (a) and (b) for this real-life experiment. The mass of the rocket was 2000 kg and its speed upon impact was 9000 km/h. How does the plume produced alter these results?

Problem 41PE Professional Application Two football players collide head-on in midair while trying to catch a thrown football. The first player is 95.0 kg and has an initial velocity of 6.00 m/s, while the second player is 115 kg and has an initial velocity of –3.50 m/s. What is their velocity just after impact if they cling together?

Problem 42PE What is the speed of a garbage truck that is 1.20×104 kg and is initially moving at 25.0 m/s just after it hits and adheres to a trash can that is 80.0 kg and is initially at rest?

Problem 43PE During a circus act, an elderly performer thrills the crowd by catching a cannon ball shot at him. The cannonball has a mass of 10.0 kg and the horizontal component of its velocity is 8.00 m/s when the 65.0-kg performer catches it. If the performer is on nearly frictionless roller skates, what is his recoil velocity?

(a) During an ice skating performance, an initially motionless 80.0-kg clown throws a fake barbell away. The clown’s ice skates allow her to recoil frictionlessly. If the clown recoils with a velocity of 0.500 m/s and the barbell is thrown with a velocity of 10.0 m/s, what is the mass of the barbell? (b) How much kinetic energy is gained by this maneuver? (c) Where does the kinetic energy come from?

Problem 46P (II) Estimate the kinetic energy of the Earth with respect to the Sun as the sum of two terms, (a) that due to its daily rotation about its axis, and (b) that due to its yearly revolution about the Sun. [Assume the Earth is a uniform sphere with mass 6.0 x 1024 kg, radius = 6.4 x 106 m, and is 1.5 x 108 km from the Sun.)

Problem 47PE A 3000-kg cannon is mounted so that it can recoil only in the horizontal direction. (a) Calculate its recoil velocity when it fires a 15.0-kg shell at 480 m/s at an angle of 20.0º above the horizontal. (b) What is the kinetic energy of the cannon? This energy is dissipated as heat transfer in shock absorbers that stop its recoil. (c) What happens to the vertical component of momentum that is imparted to the cannon when it is fired?

Problem 45PE Two identical pucks collide on an air hockey table. One puck was originally at rest. (a) If the incoming puck has a speed of 6.00 m/s and scatters to an angle of 30.0º ,what is the velocity (magnitude and direction) of the second puck? (You may use the result that ?1 ? ?2 = 90º for elastic collisions of objects that have identical masses.) (b) Confirm that the collision is elastic.

Problem 46PE Confirm that the results of the example Example 8.7 to conserve momentum in bo?th the ?x? - a? -directions. Example 8.7 Determining the Final Velocity of an Unseen Object from the Scattering of Another Object Suppose the following experiment is performed. A 0.250- ? kg object (?m1) is slid on a frictionless surface into a dark room, where it strikes an initially stationary object with mass? of 0.400 kg (?m2) . The 0.250-kg object emerges from the room at an angle of 45.0° with its incoming direction. The speed of the 0.250-kg object is originally 2.00 m/s and is 1.50 m/s after the collision. Calculate the magnitude and directio?n of the? velocity ? ?2 and ??2) of the 0.400-kg object after the collision.

Professional Application A 5.50-kg bowling ball moving at 9.00 m/s collides with a 0.850-kg bowling pin, which is scattered at an angle of \(85.0^\circ\) to the initial direction of the bowling ball and with a speed of 15.0 m/s. (a) Calculate the final velocity (magnitude and direction) of the bowling ball. (b) Is the collision elastic? (c) Linear kinetic energy is greater after the collision. Discuss how spin on the ball might be converted to linear kinetic energy in the collision.

Problem 49PE Professional Application Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei (4He) from gold-197 nuclei (197 Au). The energy of the incoming helium nucleus was 8.00×10?13 J , and the masses of the helium and gold nuclei were 6.68×10?27 kg and 3.29×10?25 kg , respectively (note that their mass ratio is 4 to 197). (a) If a helium nucleus scatters to an angle of 120º during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and direction) of the gold nucleus. (b) What is the final kinetic energy of the helium nucleus?

Professional Application Two cars collide at an icy intersection and stick together afterward. The first car has a mass of 1200 kg and is approaching at 8.00 m/s due south. The second car has a mass of 850 kg and is approaching at 17.0 m/s due west. (a) Calculate the final velocity (magnitude and direction) of the cars. (b) How much kinetic energy is lost in the collision? (This energy goes into deformation of the cars.) Note that because both cars have an initial velocity, you cannot use the equations for conservation of momentum along the x-axis and y-axis; instead, you must look for other simplifying aspects.

Problem 51PE Starting with equations for conservation of momentum in the x - and y -directions and assuming that one object is originally stationary, prove that for an elastic collision of two objects of equal masses, as discussed in the text.

Problem 53PE Professional Application Antiballistic missiles (ABMs) are designed to have very large accelerations so that they may intercept fast-moving incoming missiles in the short time available. What is the takeoff acceleration of a 10,000-kg ABM that expels 196 kg of gas per second at an exhaust velocity of 2.50×103 m/s?

Problem 52PE Integrated Concepts A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless, how far does the player recoil in the time it takes the puck to reach the goal 15.0 m away?

(II) A diver (such as the one shown in Fig. 8-29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in \(1.5 \mathrm{~s}\) when in the tuck position, what is her angular speed (rev/s) when in the straight position?

Problem 54PE Professional Application What is the acceleration of a 5000-kg rocket taking off from the Moon, where the acceleration due to gravity is only 1.6 m/s2 , if the rocket expels 8.00 kg of gas per second at an exhaust velocity of 2.20×10 3 m/s?

Problem 55PE Professional Application Calculate the increase in velocity of a 4000-kg space probe that expels 3500 kg of its mass at an exhaust velocity of 2.00×103 m/s . You may assume the gravitational force is negligible at the probe’s location.

Problem 56PE Professional Application Ion-propulsion rockets have been proposed for use in space. They employ atomic ionization techniques and nuclear energy sources to produce extremely high exhaust velocities, perhaps as great as 8.00×106 m/s . These techniques allow a much more favorable payload-to-fuel ratio. To illustrate this fact: (a) Calculate the increase in velocity of a 20,000-kg space probe that expels only 40.0-kg of its mass at the given exhaust velocity. (b) These engines are usually designed to produce a very small thrust for a very long time—the type of engine that might be useful on a trip to the outer planets, for example. Calculate the acceleration of such an engine if it expels 4.50×10?6 kg/s at the given velocity, assuming the acceleration due to gravity is negligible.

Problem 57PE Derive the equation for the vertical acceleration of a rocket.

Determine the angular momentum of the Earth (a) about its rotation axis (assume the Earth is a uniform sphere), and (6) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun).

Problem 58PE Professional Application (a) Calculate the maximum rate at which a rocket can expel gases if its acceleration cannot exceed seven times that of gravity. The mass of the rocket just as it runs out of fuel is 75,000-kg, and its exhaust velocity is 2.40×103 m/s . Assume that the acceleration of gravity is the same as on Earth’s surface (9.80 m/s2). (b) Why might it be necessary to limit the acceleration of a rocket?

(II) A nonrotating cylindrical disk of moment of inertia I is dropped onto an identical disk rotating at angular speed co. Assuming no external torques, what is the final common angular speed of the two disks?

Problem 59PE Given the following data for a fire extinguisher-toy wagon rocket experiment, calculate the average exhaust velocity of the gases expelled from the extinguisher. Starting from rest, the final velocity is 10.0 m/s. The total mass is initially 75.0 kg and is 70.0 kg after the extinguisher is fired.

Problem 60PE How much of a single-stage rocket that is 100,000 kg can be anything but fuel if the rocket is to have a final speed of 8.00 km/s , given that it expels gases at an exhaust velocity of 2.20×103 m/s?

Problem 61PE Professional Application (a) A 5.00-kg squid initially at rest ejects 0.250-kg of fluid with a velocity of 10.0 m/s. What is the recoil velocity of the squid if the ejection is done in 0.100 s and there is a 5.00-N frictional force opposing the squid’s movement. (b) How much energy is lost to work done against friction?

Problem 62PE Unreasonable Results Squids have been reported to jump from the ocean and travel 30.0 m (measured horizontally) before re-entering the water. (a) Calculate the initial speed of the squid if it leaves the water at an angle of 20.0º , assuming negligible lift from the air and negligible air resistance. (b) The squid propels itself by squirting water. What fraction of its mass would it have to eject in order to achieve the speed found in the previous part? The water is ejected at 12.0 m/s ; gravitational force and friction are neglected. (c) What is unreasonable about the results? (d) Which premise is unreasonable, or which premises are inconsistent?

(II) Suppose our Sun eventually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, (a) what would the Sun's new rotation rate be? Take the Sun's current period to be about 30 days, (b) What would be its final kinetic energy in terms of its initial kinetic energy of today?

Problem 63PE Construct Your Own Problem Consider an astronaut in deep space cut free from her space ship and needing to get back to it. The astronaut has a few packages that she can throw away to move herself toward the ship. Construct a problem in which you calculate the time it takes her to get back by throwing all the packages at one time compared to throwing them one at a time. Among the things to be considered are the masses involved, the force she can exert on the packages through some distance, and the distance to the ship.

(II) A person stands on a platform, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform is \(I_p\). The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia \(I_w\) and angular velocity \(\omega_{\mathrm{W}}\), What will be the angular velocity \(\omega_{\mathrm{p}}\) of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a \(60^{\circ}\) angle to the vertical, (c) vertically downward? (d) What will \(\omega_{\mathrm{p}}\) be if the person reaches up and stops the wheel in part (a)?

Problem 64PE Construct Your Own Problem Consider an artillery projectile striking armor plating. Construct a problem in which you find the force exerted by the projectile on the plate. Among the things to be considered are the mass and speed of the projectile and the distance over which its speed is reduced. Your instructor may also wish for you to consider the relative merits of depleted uranium versus lead projectiles based on the greater density of uranium.

A large spool of rope rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8-50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool's center of mass move? FIGURE 8-50 Problem 68.

Problem 69GP The Moon orbits the Earth such that the same side always faces the Earth. Determine the ratio of the Moon's spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)

Figure illustrates an \(\mathrm{H}_{2} \mathrm{O} \) molecule. The \(\mathrm{O}-\mathrm{H}\) bond length is and the \(\mathrm{H}-\mathrm{O}-\mathrm{H}\) bonds make an angle of \(104^{\circ}\). Calculate the moment of inertia for the \(\mathrm{H}_{2} \mathrm{O} \) molecule about an axis passing through the center of the oxygen atom (a) perpendicular to the plane of the molecule, and in the plane of the molecule, bisecting the \(\mathrm{H}-\mathrm{O}-\mathrm{H}\) bonds. FIGURE 8-53 Problem 76. Equation Transcription: Text Transcription: H2O O-H H-O-H 104° H2O H-O-H

(a) For a bicycle, how is the angular speed of the rear wheel \(\left(\omega_{R}\right)\) related to that of the pedals and front sprocket \(\left(\omega_{F}\right)\), Fig. That is, derive a formula for \(\omega_{R} / \omega_{F}\). Let \(N_{F} \text { and } N_{R}\) be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly. (b) Evaluate the ratio \(\omega_{R} / \omega_{F}\) when the front and rear sprockets have 52 and 13 teeth, respectively, and when they have 42 and 28 teeth. FIGURE 8-52 Problem 73. Equation Transcription: Text Transcription: (\omega_R) (\omega_F) \omega_R / \omega_F NF and NR \omega_R / \omega_F

A uniform rod of mass and length can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. . The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8-21g.] FIGURE 8-54 Problem 78.

A wheel of mass has radius . It is standing vertically on the floor, and we want to exert a horizontal force \(F^{-}\) at its axle so that it will climb a step against which it rests (Fig. ). The step has height , where \(h<R\). What minimum force is needed? FIGURE 8-56 Problem 79. Equation Transcription: Text Transcription: F- h<R

Problem 4CQ How can a small force impart the same momentum to an object as a large force?

Problem 13CQ Steel nails are rigid and unbending. Steel wool is soft and squishy. How would you account for this difference?

Problem 7CQ You must lean quite far forward as you rise from a chair (try it!). Explain why.

Problem 6P A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do the wheels make?

Problem 1P Express the following angles in radians: (a) 30°, (b) 57°, (c) 90°, (d) 360°, and (e) 420°. Give as numerical values and as fractions of ?.

Problem 1CQ A 64 kg student stands on a very light, rigid board that rests on a bathroom scale at each end, as shown in Figure P8.1 . What is the reading on each of the scales?

Problem 21P The tires of a car make 65 revolutions as the car reduces its speed uniformly from 95 km/h to 45 km/h. The tires have a diameter of 0.80 m. (a) What was the angular acceleration of the tires? (b) If the car continues to decelerate at this rate, how much more time is required for it to stop?

Problem 23P A person exerts a force of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exerted (a) perpendicular to the door, and (b) at a 45° angle to the face of the door?

Problem 24P Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0.40 m · N opposes the motion.

(II) Two blocks, each of mass m, are attached to the ends of a massless rod which pivots as shown in Fig. 8-40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

Problem 26P The bolts on the cylinder head of an engine require tightening to a torque of 88 m·N. If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end? If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).

Problem 27P (I) Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center.

(I) Calculate the moment of inertia of a bicycle wheel \(66.7 \mathrm{~cm}\) in diameter. The rim and tire have a combined mass of \(1.25 \mathrm{~kg}\). The mass of the hub can be ignored (why?).

Problem 29P A small 650-gram ball on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculate (a) the moment of inertia of the ball about the center of the circle, and (b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.

(II) Calculate the moment of inertia of the array of point objects shown in Fig. about the vertical axis, and the horizontal axis. Assume . , and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis. ( ) About which axis would it be harder to accelerate this array?

(II) To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. . If the satellite has a mass of and a radius of , what is the required steady force of each rocket if the satellite is to reach in ?

Problem 32P (II) An oxygen molecule consists of two oxygen atoms whose total mass is 5.3 X 10-26 kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is 1.9 X10-49 kg.m2 From these data, estimate the effective distance between the atoms.

Problem 34P A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculate (a) its moment of inertia about its center, and (b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s.

(II) Calculate the angular velocity of the Earth (a) in its orbit around the Sun, and (6) about its axis.

Mammals that depend on being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. ). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Problem 9CQ Under what circumstances is momentum conserved?

Problem 8PE Professional Application A car moving at 10 m/s crashes into a tree and stops in 0.26s. Calculate the force the seat belt exerts on a passenger in the car to bring him to a halt. The mass of the passenger is 70 kg.

Problem 9PE A person slaps her leg with her hand, bringing her hand to rest in 2.50 milliseconds from an initial speed of 4.00 m/s. (a) What is the average force exerted on the leg, taking the effective mass of the hand and forearm to be 1.50 kg? (b) Would the force be any different if the woman clapped her hands together at the same speed and brought them to rest in the same time? Explain why or why not.

Problem 10PE Professional Application A professional boxer hits his opponent with a 1000-N horizontal blow that lasts for 0.150 s. (a) Calculate the impulse imparted by this blow. (b) What is the opponent’s final velocity, if his mass is 105 kg and he is motionless in midair when struck near his center of mass? (c) Calculate the recoil velocity of the opponent’s 10.0-kg head if hit in this manner, assuming the head does not initially transfer significant momentum to the boxer’s body. (d) Discuss the implications of your answers for parts (b) and (c).

Problem 10Q 5. If the net force on 3 system is zero, is the net torque also zero? If the net torque on 3 system is zero, is the net force zero? Explain and give examples.

Problem 11PE Professional Application Suppose a child drives a bumper car head on into the side rail, which exerts a force of 4000 N on the car for 0.200 s. (a) What impulse is imparted by this force? (b) Find the final velocity of the bumper car if its initial velocity was 2.80 m/s and the car plus driver have a mass of 200 kg. You may neglect friction between the car and floor.

Problem 11Q Two inclines have the same height but make different angles with the horizontal. The same steel ball rolls without slipping down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Problem 12CQ Professional Application Explain in terms of momentum and Newton’s laws how a car’s air resistance is due in part to the fact that it pushes air in its direction of motion.

(II) What is the linear speed of a point (a) on the equator, (b) on the Arctic Circle (latitude \(66.5^{\circ} \mathrm{N}\) ), and (c) at a latitude of \(45.0^{\circ} \mathrm{N}\), due to the Earth's rotation?

Problem 11P (II) How fast (in rpm) must a centrifuge rotate if a particle 8.0 cm from the axis of rotation is to experience an acceleration of 100,000 gs?

Problem 14P In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine (a) the angular acceleration, and (b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration.

Problem 15P A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time?

Problem 16P An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate (a) its angular acceleration, assumed constant, and (b) the total number of revolutions the engine makes in this time.

Problem 18P A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time?

Problem 19P A cooling fan is turned off when it is running at 850 rev/min. It turns 1500 revolutions before it comes to a stop. (a) What was the fan’s angular acceleration, assumed constant? (b) How long did it take the fan to come to a complete stop?

Problem 2P Eclipses happen on Earth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.

Problem 3Q Could a nonrigid body be described by a single value of the angular velocity ?? Explain.

The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?

Problem 5P A child rolls a ball on a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diameter?

If a force \(\vec{F}\) acts on an object such that its lever arm is zero, does it have any effect on the object's motion? Explain. Equation Transcription: Text Transcription: \vec F

Problem 7P (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angular velocity in rad/s. (b) What are the linear speed and acceleration of a point on the edge of the grinding wheel?

Problem 7Q A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Why do tightrope walkers (Fig. 8-35) carry a long, narrow beam?

Problem 12P A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determine (a) its angular acceleration, and (b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating.

Two solid spheres simultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

Problem 15Q If there were a great migration of people toward the Earth’s equator, how would this affect the length of the day?

Problem 17P Pilots can be tested for the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed. (a) What was its angular acceleration (assumed constant), and (b) what was its final angular speed in rpm?

The moment of inertia of a rotating solid disk about an axis through its center of mass is \(\frac{1}{2} M R^{2}\) (Fig. ). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller? Equation Transcription: Text Transcription: \frac{1 2 M R^2

Problem 18Q Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

A small rubber wheel is used to drive a large pottery wheel, and they are mounted so that their circular edges touch. The small wheel has a radius of 2.0 cm and accelerates at the rate of \(7.2 \ \mathrm {rad/s}^2\), and it is in contact with the pottery wheel (radius 25.0 cm) without slipping. Calculate (a) the angular acceleration of the pottery wheel, and (b) the time it takes the pottery wheel to reach its required speed of 65 rpm.

Problem 22P A 55-kg person riding a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm. (a) What is the maximum torque she exerts? (b) How could she exert more torque?

Suppose you are standing on the edge of a large freely rotating turntable. What happens if you walk toward the center?

Problem 82GP Model a figure skater’s body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body. Check Sections 8–5 and 8–8.

(II) A potter is shaping a bowl on a potter's wheel rotating at constant angular speed (Fig. ). The friction force between her hands and the clay is total. (a) How large is her torque on the wheel, if the diameter of the bowl is How long would it take for the potter's wheel to stop if the only torque acting on it is due to the potter's hand? The initial angular velocity of the wheel is \(1.6 \mathrm{rev} / \mathrm{s}\), and the moment of inertia of the wheel and the bowl is \(0.11 \mathrm{~kg} \cdot \mathrm{m}^{2}\) Equation Transcription: Text Transcription: 1.6rev/s 0.11 kg \dot m2

A softball player swings a bat, accelerating it from rest to \(3.0 \mathrm{rev} / \mathrm{s}\) in a time of \(0.20 \mathrm{~s}\). Approximate the bat as a \(2.2-\mathrm{kg}\) uniform rod of length \(0.95 \mathrm{~m}\), and compute the torque the player applies to one end of it.

Problem 36P A teenager pushes tangentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. What force is required at the edge?

Problem 37P A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.20 m · N. If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest, and how long will it take?

(II) A helicopter rotor blade can be considered a long thin rod, as shown in Fig. If each of the three rotor helicopter blades is long and has a mass of , calculate the moment of inertia of the three rotor blades about the axis of rotation. How much torque must the motor apply to bring the blades up to a speed of in ?

(III) An Atwood's machine consists of two masses, \(m_{1}\) and \(m_{2}\), which are connected by a massless inelastic cord that passes over a pulley, Fig. 8-47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses \(m_{1}\) and \(m_{2}\), and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions \(F_{T 1}\) and \(F_{T 2}\) are not equal. We discussed this situation in Example 4-13, assuming I = 0 for the pulley.]

(III) A hammer thrower accelerates the hammer (mass \(=7.30 \mathrm{~kg}\) ) from rest within four full turns (revolutions) and releases it at a speed of \(28.0 \mathrm{~m} / \mathrm{s}\). Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius \(1.20 \mathrm{~m}\), calculate (a) the angular acceleration, (b) the (linear) tangential acceleration, (c) the centripetal acceleration just before release, (d) the net force being exerted on the hammer by the athlete just before release, and (e) the angle of this force with respect to the radius of the circular motion.

(I) A centrifuge rotor has a moment of inertia of \(3.75 \times 10^{-2} \mathrm{~kg} \cdot \mathrm{m}^2\). How much energy is required to bring it from rest to \(8250 \mathrm{rpm}\) ?

Problem 44P An automobile engine develops a torque of 280 m · N at 3800 rpm. What is the power in watts and in horsepower?

Problem 45P A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.3 m/s. Calculate its total kinetic energy.

Problem 47P A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.

Problem 48P A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls without slipping down a 30.0° incline that is 10.0 m long. (a) Calculate its translational and rotational speeds when it reaches the bottom. (b) What is the ratio of translational to rotational KEat the bottom? Avoid putting in numbers until the end so you can answer: (c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

(III) Two masses, \(m_{1}=18.0 \mathrm{~kg}\) and \(m_{2}=26.5 \mathrm{~kg}\) , are connected by a rope that hangs over a pulley (as in Fig. 8-47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, \(m_{1}\) is on the ground and \(m_{2}\) rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of \(m_{2}\) just before it strikes the ground. Assume the pulley is frictionless.

(III) A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]

Problem 51P What is the angular momentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10.4 rad/s?

Problem 52P (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? (b) How much torque is required to stop it in 6.0 s?

(II) A person stands, hands at his side, on a platform that is rotating at a rate of . If he raises his arms to a horizontal position, Fig. , the speed of rotation decreases to . Why? By what factor has his moment of inertia changed?

Problem 55P A figure skater can increase her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of 3.0 rev/s. If her initial moment of inertia was 4.6 kg · m2, what is her final moment of inertia? How does she physically accomplish this change?

(II) A potter’s wheel is rotating around a vertical axis through its center at a frequency of 1.5 rev/s. The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?

Problem 57P (a) What is the angular momentum of a figure skater spinning at 3.5 rev/s with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg? (b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?

(II) A uniform disk turns at around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk's diameter, is dropped onto the freely spinning disk, Fig. . They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?

A person of mass \(75 \mathrm{~kg}\) stands at the center of a rotating merry-go-round platform of radius \(3.0 \mathrm{~m}\) and moment of inertia \(920 \mathrm{~kg} \cdot \mathrm{m}^2\). The platform rotates without friction with angular velocity \(2.0 \mathrm{rad} / \mathrm{s}\). The person walks radially to the edge of the platform. (a) Calculate the angular velocity when the person reaches the edge. (b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person's walk.

(II) A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 0.80 rad/s. Its total moment of inertia is \(1760 \ \mathrm{kg}\cdot \mathrm{m}^2\). Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now? What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?

Problem 64P Hurricanes can involve winds in excess of 120 km/h at the outer edge. Make a crude estimate of (a) the energy, and (b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 kg/m3) of radius 100 km and height 4.0 km.

An asteroid of mass \(1.0 \times 10^5 \mathrm{~kg}\), traveling at a speed of \(30 \mathrm{~km} / \mathrm{s}\) relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth's rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.

Problem 67P Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1700 kg-m2. The turntable is at rest initially, but when the person begins running at a speed of 3.8 m/s (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.

A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of 1800 rev/s from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.

A cyclist accelerates from rest at a rate of \(1.00 \mathrm{~m} / \mathrm{s}^{2}\). How fast will a point on the rim of the tire (diameter ) at the top be moving after ? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest-see Fig. Equation Transcription: Text Transcription: 1.00 m/s2

(a) A yo-yo is made of two solid cylindrical disks, each of mass \(0.050 \mathrm{~kg}\) and diameter \(0.075 \mathrm{~m}\), joined by a (concentric) thin solid cylindrical hub of mass \(0.0050 \mathrm{~kg}\) and diameter \(0.010 \mathrm{~m}\). Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string. if it is released from rest. (b) What fraction of its kinetic energy is rotational?

Problem 74GP Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.

Problem 75GP One possibility for a low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.” (a) Make reasonable assumptions (average frictional retarding force = 450 N, twenty acceleration periods from rest to 95 km/h, equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about 1.7 × 108J. (b) What is the angular velocity of the flywheel when it has a full “energy charge”? (c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip?

A hollow cylinder (hoop) is rolling on a horizontal surface at speed v = 3.3 m/s when it reaches a \(15^\circ\) incline. (a) How far up the incline will it go? (b) How long will it be on the incline before it arrives back at the bottom?

A bicyclist traveling with speed \(v=4.2 \ \mathrm{m} / \mathrm{s}\) on a flat road is making a turn with a radius \(=6.4 \ \mathrm{m}\). The forces acting on the cyclist and cycle are the normal force \(\left(\vec{F}_{N}\right)\) and friction force \(\left(\vec{F}_{f r}\right)\) exerted by the road on the tires, and \(m \vec{g}\), the total weight of the cyclist and cycle (see Fig. 8-56 ). (a) Explain carefully why the angle the bicycle makes with the vertical (Fig. 8-56 ) must be given by \(\tan \theta=F_{f r} / F_{N}\) if the cyclist is to maintain balance. (b) Calculate \(\theta\) for the values given. (c) If the coefficient of static friction between tires and road is \(\mu_{s}\) = 0.70, what is the minimum turning radius? Equation Transcription: v=4.2 m/s r=6.4 m () () tan = Text Transcription: v=8.2 m/s r=13 m (vector F_N) (vector F_fr) m vector g theta tan theta = F_fr/F_N mu_s

Problem 81GP Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?

You are designing a clutch assembly which consists of two cylindrical plates, of mass \(M_{\mathrm{A}}=6.0 \mathrm{~kg}\) and \(M_{\mathrm{B}}=9.0 \mathrm{~kg}\), with equal radii \(R=0.60 \mathrm{~m}\). They are initially separated (Fig. 8-57). Plate \(M_{\mathrm{A}}\) is accelerated from rest to an angular velocity \(\omega_1=7.2 \mathrm{rad} / \mathrm{s}\) in time \(\Delta t=2.0 \mathrm{~s}\). Calculate (a) the angular momentum of \(M_{\mathrm{A}}\), and (b) the torque required to have accelerated \(M_{\mathrm{A}}\) from rest to \(\omega_1\). (c) Plate \(M_{\mathrm{B}}\), initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate \(M_{\mathrm{A}}\) (their contact surfaces are high-friction). Before contact, \(M_{\mathrm{A}}\) was rotating at constant \(\omega_1\). After contact, at what constant angular velocity \(\omega_2\) do the two plates rotate?

Problem 86GP The tires of a car make 85 revolutions as the car reduces its speed uniformly from 90.0 km/h to 60.0 km/h. The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? (b) If the car continues to decelerate at this rate, how much more time is required for it to stop?