A 10.0-kg block rests on a 5.0-kg bracket, as shown in

Various objects lie on the bed of a truck that is moving along a straight horizontal road. If the truck gradually speeds up, what force acts on the objects to cause them to speed up too? Explain why some of the objects might stay stationary on the floor while others might slip backward on the floor.

Blocks made of the same material but differing in size lie on the bed of a truck that is moving along a straight horizontal road. All of the blocks will slide if the trucks acceleration is sufficiently great. How does the minimum acceleration at which a small block slips compare with the minimum acceleration at which a much heavier block slips?

A block of mass m rests on a plane that is inclined at an angle with the horizontal. It follows that the coefficient of static friction between the block and plane is (a) (b) (c) (d) ms ms _ tanu. _ tan u, ms ms _ tanu, _ g, u SSM

A block of mass m is at rest on a plane that is inclined at an angle of with the horizontal, as shown in Figure 5-56. Which of the following statements about the magnitude of the static frictional force is necessarily true? (a) (b) (c) (d) (e) None of these statements is true.

On an icy winter day, the coefficient of friction between the tires of a car and a roadway is reduced to one-quarter of its value on a dry day. As a result, the maximum speed at which the car can safely negotiate a curve of radius R is reduced. The new value for this speed is (a) (b) (c) (d) (e) reduced by an unknown amount depending on the cars mass.

If it is started properly on the frictionless inside surface of a cone (Figure 5-57), a block is capable of maintaining uniform circular motion. Draw the free-body diagram of the block and identify clearly which force (or forces, or force components) is responsible for the centripetal acceleration of the block.

Here is an interesting experiment that you can perform at home: take a wooden block and rest it on the floor or some other flat surface. Attach a rubber band to the block and pull gently on the rubber band in the horizontal direction. Keep your hand moving at constant speed. At some point, the block will start moving, but it will not move smoothly. Instead, it will start moving, stop again, start moving again, stop again, and so on. Explain why the block moves this way. (The start-stop motion is sometimes called stick-slip motion.)

Viewed from an inertial reference frame, an object is seen to be moving in a circle. Which, if any, of the following statements must be true. (a) Anonzero net force acts on the object. (b) The object cannot have a radially outward force acting on it. (c) At least one of the forces acting on the object must point directly toward the center of the circle.

Aparticle is traveling in a vertical circle at constant speed. One can conclude that the magnitude of its _____ is (are) constant. (a) velocity, (b) acceleration, (c) net force, (d) apparent weight.

You place a lightweight piece of iron on a table and hold a small kitchen magnet above the iron at a distance of 1.00 cm. You find that the magnet cannot lift the iron, even though there is obviously a force between the iron and the magnet. Next, holding the magnet in one hand and the piece of iron in the other, with the magnet 1.00 cm above the iron, you simultaneously drop them from rest. As they fall, the magnet and the piece of iron bang into each other before hitting the floor. (a) Draw free-body diagrams illustrating all of the forces on the magnet and the iron for each demonstration. (b) Explain why the magnet and iron move closer together while they are falling, even though the magnet cannot lift the piece of iron when it is sitting on the table.

The following question is an excellent braintwister, invented by Boris Korsunsky:* Two identical blocks are attached by a massless string running over a pulley, as shown in Figure 5-58. The rope initially runs over the pulley at the ropes midpoint, and the surface that block 1 rests on is frictionless. Blocks 1 and 2 are initially at rest when block 2 is released with the string taut and horizontal. Will block 1 hit the pulley before or after block 2 hits the wall? (Assume that the initial distance from block 1 to the pulley is the same as the initial distance from block 2 to the wall.) There is a very simple solution.

In class, most professors do the following experiment while discussing the conditions under which air drag can be neglected while analyzing free-fall. First, a flat piece of paper and a small lead weight are dropped next to each other, and clearly the papers acceleration is less than that of the lead weight. Then, the paper is crumpled into a small wad and the experiment repeated. Over the distance of a meter or two, it is clear the acceleration of the paper is now very close to that of the lead weight. To your dismay, the professor calls on you to explain why the papers acceleration changed so dramatically. Repeat your explanation here!

CONTEXT-RICH Jim decides to attempt to set a record for terminal speed in skydiving. Using the knowledge he has gained from a physics course, he makes the following plans. He will be dropped from as high an altitude as possible (equipping himself with oxygen), on a warm day and go into a knife position, in which his body is pointed vertically down and his hands are pointed ahead. He will outfit himself with a special sleek helmet and rounded protective clothing. Explain how each of these factors helps Jim attain the record.

CONTEXT-RICH You are sitting in the passenger seat in a car driving around a circular, horizontal, flat racetrack at a high speed. As you sit there, you feel a force pushing you toward the outside of the track. What is the true direction of the force acting on you, and where does it come from? (Assume that you do not slide across the seat.) Explain the sensation of an outward force on you in terms of the Newtonian perspective.

The mass of the moon is only about 1% of that of Earth. Therefore, the force that keeps the moon in its orbit around Earth (a) is much smaller than the gravitational force exerted on the moon by Earth, (b) is much greater than the gravitational force exerted on the moon by Earth, (c) actually is the gravitational force exerted on the moon by Earth, (d) cannot be answered yet, because we have not yet studied Newtons law of gravity.

Ablock is sliding on a frictionless surface along a loop-the-loop, as in Figure 5-59a. The block is moving fast enough so that it never loses contact with the track. Match the points along the track to the appropriate free-body diagrams in Figure 5.59b.

(a) Arock and a feather held at the same height above the ground are simultaneously dropped. During the first few milliseconds following release, the drag force on the rock is smaller than the drag force on the feather, but later on during the fall the opposite is true. Explain. (b) In light of this result, explain how the rocks acceleration can be so obviously larger than that of the feather. Hint: Draw a free-body diagram of each object.

Two pucks of masses and are lying on a frictionless table and are connected by a massless spring of force constant k. A horizontal force directed away from is then exerted on What is the magnitude of the resulting acceleration of the center of mass of the two-puck system? (a) (b) (c) where x is the amount the spring is stretched, (d)

The two pucks in Problem 18 lie unconnected on a frictionless table. A horizontal force directed away from is then exerted on How does the magnitude of the resulting acceleration of the center of mass of the two-puck system compare to the magnitude acceleration of ? Explain your reasoning.

If only external forces can cause the center of mass of a system of particles to accelerate, how can a car on level ground ever accelerate? We normally think of the cars engine as supplying the force needed to accelerate the car, but is this true? Where does the external force that accelerates the car come from?

When you push on the brake pedal to slow down a car, a brake pad is pressed against the rotor so that the friction of the pad slows the wheels rotation. However, the friction of the pad against the rotor cannot be the force that slows the car down, because it is an internal force (both the rotor and the wheel are parts of the car, so any forces between them are purely internal to the system). What is the external force that slows down the car? Give a detailed explanation of how this force operates.

Give an example of each of the following: (a) a threedimensional object that has no matter at its center of mass, (b) a solid object whose center of mass is outside of it, (c) a solid sphere whose center of mass does not lie at its geometrical center, (d) a long wooden stick whose center of mass does not lie at its middle.

BIOLOGICAL APPLICATION When you are standing upright, your center of mass is located within the volume of your body. However, as you bend over (say to pick up a package), its location changes. Approximately where is it when you are bent over at right angles and what change in your body caused the center of mass location to change? Explain.

ENGINEERING APPLICATION Early on their three-day (one-way) trip to the moon, the Apollo team (late 1960s to early 1970s) would explosively separate the lunar ship from the thirdstage booster (that provided the final boost) while still fairly close to Earth. During the explosion, how did the velocity of each of the two pieces of the system change? How did the velocity of the center of mass of the system change?

You throw a boomerang and for a while it flies horizontally in a straight line at a constant speed, while spinning rapidly. Draw a series of pictures, as viewed vertically down from overhead, of the boomerang in different rotational positions as it moves parallel to the surface of Earth. On each picture, indicate the location of the boomerangs center of mass and connect the dots to trace the trajectory of its center of mass. What is the center of masss acceleration during this part of the flight?

ENGINEERING APPLICATION To determine the aerodynamic drag on a car, automotive engineers often use the coastdown method. The car is driven on a long, flat road at some convenient speed (60 mi/h is typical), shifted into neutral, and allowed to coast to a stop. The time that it takes for the speed to drop by successive 5-mi/h intervals is measured and used to compute the net force slowing the car down. (a) One day, a group measured that a Toyota Tercel with a mass of 1020 kg coasted down from 60.0 mi/h to 55.0 mi/h in 3.92 s. Estimate the average net force slowing the car down in this speed range. (b) If the coefficient of rolling friction for this car is known to be 0.020, what is the force of rolling friction that is acting to slow it down? Assuming that the only two forces acting on the car are rolling friction and aerodynamic drag, what is the average drag force acting on the car? (c) The drag force has the form , where A is the cross-sectional area of the car facing into the air, v is the cars speed, is the density of air, and C is a dimensionless constant of order-of-magnitude 1. If the cross-sectional area of the car is determine C from the data given. (The density of air is ; use 57.5 mi/h for the speed of the car in this computation.)

Using dimensional analysis, determine the units and dimensions of the constant b in the retarding force if (a) and (b) (c) Newton showed that the air resistance of a falling object with a circular cross section should be approximately where the density of air. Show that this is consistent with your dimensional analysis for part (b). (d) Find the terminal speed for a 56.0-kg skydiver; approximate his cross-sectional area as a disk of radius 0.30 m. The density of air near the surface of Earth is (e) The density of the atmosphere decreases with height above the surface of Earth; at a height of 8.0 km, the density is only What is the terminal velocity at this height?

Estimate the terminal velocity of an average sized raindrop and a golf-ball-sized hailstone. Hint: See Problems 26 and 27.

Estimate the minimum coefficient of static friction needed between a cars tires and the pavement in order to complete a left turn at a city street intersection at the posted straight-ahead speed limit of 25 mph and on narrow inner-city streets. Comment on the wisdom of attempting such a turn at that speed.

Estimate the widest stance you can take when standing on a dry, icy surface. That is, how wide can you safely place your feet and not slip into an undesired split? Let the coefficient of static friction of rubber on ice be roughly 0.25.

A block of mass m slides at constant speed down a plane inclined at an angle of with the horizontal. It follows that (a) (b) (c) (d)

A block of wood is pulled at constant velocity by a horizontal string across a horizontal surface with a force of 20 N. The coefficient of kinetic friction between the surfaces is 0.3. The force of friction is (a) impossible to determine without knowing the mass of the block, (b) impossible to determine without knowing the speed of the block, (c) 0.30 N, (d) 6.0 N, (e) 20 N.

A block weighing 20-N rests on a horizontal surface. The coefficients of static and kinetic friction between the surface and the block are and A horizontal string is then attached to the block and a constant tension T is maintained in the string. What is the magnitude of the force of friction acting on the block if (a) (b)

A block of mass m is pulled at a constant velocity across a horizontal surface by a string as shown in Figure 5-60. The magnitude of the frictional force is (a) (b) (c) (d) or (e) mk .

A 100-kg crate rests on a thick-pile carpet. A weary worker then pushes on the crate with a horizontal force of 500 N. The coefficients of static and kinetic friction between the crate and the carpet are 0.600 and 0.400, respectively. Find the magnitude of the frictional force exerted by the carpet on the crate.

A box weighing 600 N is pushed along a horizontal floor at constant velocity with a force of 250 N parallel to the floor. What is the coefficient of kinetic friction between the box and the floor?

The coefficient of static friction between the tires of a car and a horizontal road is 0.60. Neglecting air resistance and rolling friction, (a) what is the magnitude of the maximum acceleration of the car when it is braked? (b) What is the shortest distance in which the car can stop if it is initially traveling at \(30 \mathrm{~m} / \mathrm{s}\)?

The force that accelerates a car along a flat road is the frictional force exerted by the road on the cars tires. (a) Explain why the acceleration can be greater when the wheels do not slip. (b) If a car is to accelerate from 0 to in what is the minimum coefficient of friction needed between the road and tires? Assume that the drive wheels support exactly half the weight of the car.

A 5.00-kg block is held at rest against a vertical wall by a horizontal force of 100 N. (a) What is the frictional force exerted by the wall on the block? (b) What is the minimum horizontal force needed to prevent the block from falling if the static coefficient of friction between the wall and the block is 0.400?

Atired and overloaded student is attempting to hold a large physics textbook wedged under his arm, as shown in Figure 5-61. The textbook has a mass of 3.2 kg, while the coefficient of static friction of the textbook against the students underarm is 0.320 and the coefficient of static friction of the book against the students shirt is 0.160. (a) What is the minimum horizontal force that the student must apply to the textbook to prevent it from falling? (b) If the student can only exert a force of 61 N, what is the acceleration of the textbook as it slides from under his arm? The coefficient of kinetic friction of arm against textbook is 0.200, while that of shirt against textbook is 0.090.

ENGINEERING APPLICATION You are racing in a rally on a snowy day when the temperature is near the freezing point. The coefficient of static friction between a cars tires and an icy road is 0.080. Your crew boss is concerned about some of the hills on the course and wants you to think about switching to studded tires. To address the issue, he wants to compare the actual hill angles on the course to see which of them your car can negotiate. (a) What is the angle of the steepest incline that a vehicle with four-wheel drive can climb at constant speed? (b) Given that the hills are icy, what is the steepest possible hill angle for the same four-wheel drive car to descend at constant speed?

A 50-kg box that is resting on a level floor must be moved. The coefficient of static friction between the box and the floor is 0.60. One way to move the box is to push down on the box at an angle below the horizontal. Another method is to pull up on the box at an angle above the horizontal. (a) Explain why one method requires less force than the other. (b) Calculate the minimum force needed to move the box by each method if and compare the answer with the results when

A block of mass is at rest on a plane that makes an angle of with the horizontal. The coefficient of kinetic friction between the block and the plane is 0.100. The block is attached to a second block of mass that hangs freely by a string that passes over a frictionless, massless pulley (Figure 5-62). When the second block has fallen 30.0 cm, what will be its speed? SSM

In Figure 5-62, \(m_1=4.0 \ \mathrm {kg}\) and the coefficient of static friction between the block and the incline is 0.40. (a) Find the range of possible values for \(m_2\) for which the system will be in static equilibrium. (b) Find the frictional force on the 4.0-kg block if \(m_2= 1.0 \ \mathrm {kg}\).

In Figure 5-62, and the coefficient of kinetic friction between the inclined plane and the 4.0-kg block is Find the magnitude of the acceleration of the masses and the tension in the cord.

A 12-kg turtle rests on the bed of a zookeepers truck, which is traveling down a country road at 55 mi/h. The zookeeper spots a deer in the road, and slows to a stop in 12 s. Assuming constant acceleration, what is the minimum coefficient of static friction between the turtle and the truck bed surface that is needed to prevent the turtle from sliding?

A 150-g block is projected up a ramp with an initial speed of The coefficient of kinetic friction between the ramp and the block is 0.23. (a) If the ramp is inclined with the horizontal, how far along the surface of the ramp does the block slide before coming to a stop? (b) The block then slides back down the ramp. What is the minimum coefficient of static friction between the block and the ramp if the block is not to slide back down the ramp?

An automobile is going up a grade at a speed of The coefficient of static friction between the tires and the road is 0.70. (a) What minimum distance does it take to stop the car? (b) What minimum distance would it take to stop if the car were going down the grade?

ENGINEERING APPLICATION A rear-wheel-drive car supports 40 percent of its weight on its two drive wheels and has a coefficient of static friction of 0.70 with a horizontal straight road. (a) Find the vehicles maximum acceleration. (b) What is the shortest possible time in which this car can achieve a speed of (Assume the engine can provide unlimited power.)

You and your best pal make a friendly bet that you can place a 2.0-kg box against the side of a cart, as in Figure 5-63, and that the box will not fall to the ground, even though you guarantee to use no hooks, ropes, fasteners, magnets, glue, or adhesives of any kind. When your friend accepts the bet, you begin pushing the cart in the direction shown in the figure. The coefficient of static friction between the box and the cart is 0.60. (a) Find the minimum acceleration for which you will win the bet. (b) What is the magnitude of the frictional force in this case? (c) Find the force of friction on the box if the acceleration is twice the minimum needed for the box not to fall. (d) Show that, for a box of any mass, the box will not fall if the magnitude of the forward acceleration is where is the coefficient of static friction.

Two blocks attached by a string (Figure 5-64) slide down a incline. Block 1 has mass and block 2 has mass In addition, the kinetic coefficients of friction between the blocks and the incline are 0.30 for block 1 and 0.20 for block 2. Find (a) the magnitude of the acceleration of the blocks, and (b) the tension in the string.

Two blocks of masses and are sliding down an incline as shown in Figure 5-64. They are connected by a massless rod. The coefficients of kinetic friction between the block and the surface are m1 for block 1 and m2 for block 2. (a) Determine the acceleration of the two blocks. (b) Determine the force that the rod exerts on each of the two blocks. Show that these forces are both 0 when and give a simple, nonmathematical argument why this is true.

A block of mass m rests on a horizontal table (Figure 5-65). The block is pulled by a massless rope with a force at an angle The coefficient of static friction is 0.60. The minimum value of the force needed to move the block depends on the angle (a) Discuss qualitatively how you would expect the magnitude of this force to depend on (b) Compute the force for the angles and and make a plot of F versus for From your plot, at what angle is it most efficient to apply the force to move the block? SSM

Consider the block in Figure 5-65. Show that, in general, the following results hold for a block of mass m resting on a horizontal surface whose coefficient of static friction is (a) If you want to apply the minimum possible force to move the block, you should apply it with the force pulling upward at an angle (b) The minimum force necessary to start the block moving is (c) Once the block starts moving, if you want to apply the least possible force to keep it moving, should you keep the angle at which you are pulling the same, increase it, or decrease it?

Answer the questions in Problem 54, but for a force that pushes down on the block at an angle below the horizontal.

A 100-kg mass is pulled along a frictionless surface by a horizontal force \(\vec{F}\) such that its acceleration is \(a_{1}=6.00 \mathrm{\ m} / \mathrm{s}^{2}\) (Figure 5-66). A 20.0-kg mass slides along the top of the 100-kg mass and has an acceleration of \(a_{2}=4.00 \mathrm{\ m} / \mathrm{s}^{2}\). (It thus slides backward relative to the 100-kg mass.) (a) What is the frictional force exerted by the 100-kg mass on the 20.0-kg mass? (b) What is the net force acting on the 100-kg mass? What is the force F? (c) After the 20.0-kg mass falls off the 100-kg mass, what is the acceleration of the 100-kg mass? (Assume that the force F does not change.)

A 60-kg block slides along the top of a 100-kg block. The 60-kg block has an acceleration of while a horizontal force of 320 N is applied to it, as shown in Figure 5-67. There is no friction between the 100-kg block and a horizontal frictionless surface, but there is friction between the two blocks. (a) Find the coefficient of kinetic friction between the blocks. (b) Find the acceleration of the 100-kg block during the time that the 60-kg block remains in contact.

The coefficient of static friction between a rubber tire and the road surface is 0.85. What is the maximum acceleration of a 1000-kg four-wheel-drive truck if the road makes an angle of with the horizontal and the truck is (a) climbing, and (b) descending?

A 2.0-kg block sits on a 4.0-kg block that is on a frictionless table (Figure 5-68). The coefficients of friction between the blocks are and (a) What is the maximum horizontal force F that can be applied to the 4.0-kg block if the 2.0-kg block is not to slip? (b) If F has half this value, find the acceleration of each block and the force of friction acting on each block. (c) If F is twice the value found in (a), find the acceleration of each block.

A 10.0-kg block rests on a 5.0-kg bracket, as shown in Figure 5-69. The 5.0-kg bracket sits on a frictionless surface. The coefficients of friction between the 10.0-kg block and the bracket on which it rests are and (a) What is the maximum force F that can be applied if the 10.0-kg block is not to slide on the bracket? (b) What is the corresponding acceleration of the 5.0-kg bracket?

You and your friends push a 75.0-kg greased pig up an aluminum slide at the county fair, starting from the low end of the slide. The coefficient of kinetic friction between the pig and the slide is 0.070. (a) All of you pushing together (parallel to the incline) manage to accelerate the pig from rest at the constant rate of over a distance of 1.5 m, at which point you release the pig. The pig continues up the slide, reaching a maximum vertical height above its release point of 45 cm. What is the angle of inclination of the slide? (b) At the maximum height the pig turns around and begins to slip down the slide, how fast is it moving when it arrives at the low end of the slide?

A 100-kg block on an inclined plane is attached to another block of mass m via a string, as in Figure 5-70. The coefficients of static and kinetic friction for the block and the incline are and and the plane is inclined with horizontal. (a) Determine the range of values for m, the mass of the hanging block, for which the 100-kg block will not move unless disturbed, but if nudged, will slide down the incline. (b) Determine a range of values for m for which the 100-kg block will not move unless nudged, but if nudged will slide up the incline.

A block of mass 0.50 kg rests on the inclined surface of a wedge of mass 2.0 kg, as in Figure 5-71. The wedge is acted on by a horizontal applied force and slides on a frictionless surface. (a) If the coefficient of static friction between the wedge and the block is and the wedge is inclined with the horizontal, find the maximum and minimum values of the applied force for which the block does not slip. (b) Repeat part (a) with ms _ 0.40.

SPREADSHEET In your physics lab, you and your lab partners push a block of wood with a mass of 10.0 kg (starting from rest), with a constant horizontal force of 70 N across a wooden floor. In the previous weeks laboratory meeting, your group determined that the coefficient of kinetic friction was not exactly constant, but instead was found to vary with the objects speed according to Write a spreadsheet program using Eulers method to calculate and graph both the speed and the position of the block as a function of time from 0 to 10 s. Compare this result to the result you would get if you assumed the coefficient of kinetic friction had a constant value of 0.11.

MULTISTEP In order to determine the coefficient of kinetic friction of a block of wood on a horizontal table surface, you are given the following assignment: Take the block of wood and give it an initial velocity across the surface of the table. Using a stopwatch, measure the time it takes for the block to come to a stop and the total displacement the block slides following the push. (a) Using Newtons laws and a free-body diagram of the block, show that the expression for the coefficient of kinetic friction is given by (b) If the block slides a distance of 1.37 m in 0.97 s, calculate (c) What was the initial speed of the block?

SPREADSHEET (a) A block is sliding down an inclined plane. The coefficient of kinetic friction between the block and the plane is . Show that a graph of versus tan (where is the acceleration down the incline and is the angle the plane is inclined with the horizontal) would be a straight line with slope g and intercept (b) The following data show the acceleration of a block sliding down an inclined plane as a function of the angle that the plane is inclined with the horizontal:* Using a spreadsheet program, graph these data and fit a straight line to them to determine and g. What is the percentage difference between the obtained value of g and the commonly specified value of ?

A Ping-Pong ball has a mass of 2.3 g and a terminal speed of The drag force is of the form What is the value of b?

A small pollution particle settles toward Earth in still air. The terminal speed of the particle is the mass of the particle is and the drag force of the particle is of the form bv. What is the value of b?

A common classroom demonstration involves dropping basket-shaped coffee filters and measuring the time required for them to fall a given distance. A professor drops a single basketshaped coffee filter from a height h above the floor, and records the time for the fall as How far will a stacked set of n identical filters fall during the same time interval ? Consider the filters to be so light that they instantaneously reach their terminal velocities. Assume a drag force that varies as the square of the speed and assume the filters are released oriented right-side up.

A skydiver of mass 60.0 kg can slow herself to a constant speed of by orienting her body horizontally, looking straight down with arms and legs extended. In this position, she presents the maximum cross-sectional area and thus maximizes the air-drag force on her. (a) What is the magnitude of the drag force on the skydiver? (b) If the drag force is given by what is the value of b? (c) At some instant she quickly flips into a knife position, orienting her body vertically with her arms straight down. Suppose this reduces the value of b to 55 percent of the value in Parts (a) and (b). What is her acceleration at the instant she achieves the knife position?

ENGINEERING APPLICATION, CONTEXT-RICH Your team of test engineers is to release the parking brake so an 800-kg car will roll down a very long 6.0 percent grade in preparation for a crash test at the bottom of the incline. (On a 6.0 percent grade the change in altitude is 6.0 percent of the horizontal distance traveled.) The total resistive force (air drag plus rolling friction) for this car has been previously established to be where v is the speed of the car. What is the terminal speed for the car rolling down this grade?

APPROXIMATION Small, slowly moving spherical particles experience a drag force given by Stokes law: where r is the radius of the particle, v is its speed, and is the coefficient of viscosity of the fluid medium. (a) Estimate the terminal speed of a spherical pollution particle of radius and density of (b) Assuming that the air is still and that is estimate the time it takes for such a particle to fall from a height of 100 m.

ENGINEERING APPLICATION, CONTEXT-RICH You have an environmental chemistry internship, and are in charge of a sample of air that contains pollution particles of the size and density given in Problem 72. You capture the sample in an 8.0-cm-long test tube. You then place the test tube in a centrifuge with the midpoint of the test tube 12 cm from the rotation axis of the centrifuge. You set the centrifuge to spin at 800 revolutions per minute. (a) Estimate the time you have to wait so that nearly all of the pollution particles settle to the end of the test tube. (b) Compare this to the time required for a pollution particle to fall 8.0 cm under the action of gravity and subject to the drag force given in Problem 72.

A rigid rod with a 0.050-kg ball at one end rotates about the other end so the ball travels at constant speed in a vertical circle with a radius of 0.20 m. What is the maximum speed of the ball so that the force of the rod on the ball does not exceed 10 N?

A 95-g stone is whirled in a horizontal circle on the end of an 85-cm-long string. The stone takes 1.2 s to make each complete revolution. Determine the angle that the string makes with the horizontal.

A0.20-kg stone is whirled in a horizontal circle on the end of an 0.80-m-long string. The string makes an angle of with the horizontal. Determine the speed of the stone.

A 0.75-kg stone attached to a string is whirled in a horizontal circle of radius 35 cm as in the tetherball in Example 5-11. The string makes an angle of with the vertical. (a) Find the speed of the stone. (b) Find the tension in the string.

BIOLOGICAL APPLICATION A pilot with a mass of 50 kg comes out of a vertical dive in a circular arc such that at the bottom of the arc her upward acceleration is 3.5g. (a) How does the magnitude of the force exerted by the airplane seat on the pilot at the bottom of the arc compare to her weight? (b) Use Newtons laws of motion to explain why the pilot might be subject to a blackout. This means that an above normal volume of blood pools in her lower limbs. How would an inertial reference frame observer describe the cause of the blood pooling?

A 80.0-kg airplane pilot pulls out of a dive by following, at a constant speed of the arc of a circle whose radius is 300 m. (a) At the bottom of the circle, what are the direction and magnitude of his acceleration? (b) What is the net force acting on him at the bottom of the circle? (c) What is the force exerted on the pilot by the airplane seat?

An small object of mass moves in a circular path of radius r on a frictionless horizontal tabletop (Figure 5-72). It is attached to a string that passes through a small frictionless hole in the center of the table. Asecond object with a mass of is attached to the other end of the string. Derive an expression for r in terms of m and the time T for one revolution.

A block of mass is attached to a cord of length which is fixed at one end. The block moves in a horizontal circle on a frictionless tabletop. A second block of mass is attached to the first by a cord of length and also moves in a circle on the same frictionless tabletop, as shown in Figure 5-73. If the period of the motion is T, find the tension in each cord in terms of the given symbols. SSM L2 m2 L1 m , 1

MULTISTEP A particle moves with constant speed in a circle of radius 4.0 cm. It takes 8.0 s to complete each revolution. (a) Draw the path of the particle to scale, and indicate the particles position at 1.0-s intervals. (b) Sketch the displacement vectors for each interval. These vectors also indicate the directions for the average- velocity vectors for each interval. (c) Graphically find the magnitude of the change in the average velocity for two consecutive 1-s intervals. Compare measured in this way, with the magnitude of the instantaneous acceleration computed from

You are swinging your younger sister in a circle of radius 0.75 m, as shown in Figure 5-74. If her mass is 25 kg and you arrange it so she makes one revolution every 1.5 s, (a) what is the magnitude and direction of the force that must be exerted by you on her? (Model her as a point particle.) (b) What is the magnitude and direction of the force she exerts on you?

The string of a conical pendulum is 50.0 cm long and the mass of the bob is 0.25 kg. (a) Find the angle between the string and the horizontal when the tension in the string is six times the weight of the bob. (b) Under those conditions, what is the period of the pendulum?

A100-g coin sits on a horizontally rotating turntable. The turntable makes exactly 1.00 revolution each second. The coin is located 10 cm from the axis of rotation of the turntable. (a) What is the frictional force acting on the coin? (b) If the coin slides off the turntable when it is located more than 16.0 cm from the axis of rotation, what is the coefficient of static friction between the coin and the turntable?

A 0.25-kg tether ball is attached to a vertical pole by a 1.2-m cord. Assume the radius of the ball is negligible. If the ball moves in a horizontal circle with the cord making an angle of with the vertical, (a) what is the tension in the cord? (b) What is the speed of the ball?

A small bead with a mass of \(100 \mathrm{~g}\) (Figure 5-75) slides without friction along a semicircular wire with a radius of \(10 \mathrm{~cm}\) that rotates about a vertical axis at a rate of 2.0 revolutions per second. Find the value of \(\theta\) for which the bead will remain stationary relative to the rotating wire.

A car speeds along the curved exit ramp of a freeway. The radius of the curve is 80.0 m. A 70.0-kg passenger holds the armrest of the car door with a 220-N force in order to keep from sliding across the front seat of the car. (Assume the exit ramp is not banked and ignore friction with the car seat.) What is the cars speed?

The radius of curvature of the track at the top of a loop-the-loop on a roller-coaster ride is 12.0 m. At the top of the loop, the force that the seat exerts on a passenger of mass m is 0.40mg. How fast is the roller-coaster car moving as it moves through the highest point of the loop.

ENGINEERING APPLICATION On a runway of a decommissioned airport, a 2000-kg car travels at a constant speed of At 100-km/h the air drag on the car is 500 N. Assume that rolling friction is negligible. (a) What is the force of static friction exerted on the car by the runway surface, and what is the minimum coefficient of static friction necessary for the car to sustain this speed? (b) The car continues to travel at but now along a path with radius of curvature r. For what value of r will the angle between the static friction force vector and the velocity vector equal and for what value of r will it equal ? What is the minimum coefficient of static friction necessary for the car to hold this last radius of curvature without skidding?

Suppose you ride a bicycle in a 20-m-radius circle on a horizontal surface. The resultant force exerted by the surface on the bicycle (normal force plus frictional force) makes an angle of with the vertical. (a) What is your speed? (b) If the frictional force on the bicycle is half its maximum possible value, what is the coefficient of static friction?

An airplane is flying in a horizontal circle at a speed of The plane is banked for this turn, its wings tilted at an angle of from the horizontal (Figure 5-76). Assume that a lift force acting perpendicular to the wings acts on the aircraft as it moves through the air. What is the radius of the circle in which the plane is flying?

An automobile club plans to race a 750-kg car at the local racetrack. The car needs to be able to travel around several 160-mradius curves at What should the banking angle of the curves be so that the force of the pavement on the tires of the car is in the normal direction? Hint: What does this requirement tell you about the frictional force?

A curve of radius 150 m is banked at an angle of An 800-kg car negotiates the curve at without skidding. Neglect the effects of air drag and rolling friction. Find (a) the normal force exerted by the pavement on the tires, (b) the frictional force exerted by the pavement on the tires, (c) the minimum coefficient of static friction between the pavement and the tires.

On another occasion, the car in Problem 94 negotiates the curve at Neglect the effects of air drag and rolling friction. Find (a) the normal force exerted on the tires by the pavement, and (b) the frictional force exerted on the tires by the pavement.

ENGINEERING APPLICATION As a civil engineering intern during one of your summers in college, you are asked to design a curved section of roadway that meets the following conditions: When ice is on the road, and the coefficient of static friction between the road and rubber is 0.080, a car at rest must not slide into the ditch and a car traveling less than must not skid to the outside of the curve. Neglect the effects of air drag and rolling friction. What is the minimum radius of curvature of the curve and at what angle should the road be banked?

ENGINEERING APPLICATION A curve of radius 30 m is banked so that a 950-kg car traveling at can round it even if the road is so icy that the coefficient of static friction is approximately zero. You are commissioned to tell the local police the range of speeds at which a car can travel around this curve without skidding. Neglect the effects of air drag and rolling friction. If the coefficient of static friction between the road and the tires is 0.300, what is the range of speeds you tell them?

SPREADSHEET, APPROXIMATION You are riding in a hovering hot air balloon when you throw a baseball straight down with an initial speed of The baseball falls with a terminal speed of Assuming air drag is proportional to the speed squared, use Eulers method (spreadsheet) to estimate the speed of the ball after 10.0 s. What is the uncertainty in this estimate? You drop a second baseball, this one is released from rest. How long does it take for it to reach 99 percent of its terminal speed? How far does it fall during this time?

SPREADSHEET, APPROXIMATION You throw a baseball straight up with an initial speed of The balls terminal speed when falling is also (a) Use Eulers method (spreadsheet) to estimate its height 3.50 s after release. (b) What is the maximum height it reaches? (c) How long after release does it reach its maximum height? (d) How much later does it return to the ground? (e) Is the time the ball spends on the way up less than, the same as, or greater than the time it spends on the way down?

SPREADSHEET, APPROXIMATION A 0.80-kg block on a horizontal frictionless surface is held against a massless spring, compressing it 30 cm. The force constant of the spring is The block is released and the spring pushes it 30 cm. Use Eulers method (spreadsheet) with s to estimate the time it takes for the spring to push the block the 30 cm. How fast is the block moving at this time? What is the uncertainty in this speed?

Three point masses of 2.0 kg each are located on the x axis. One is at the origin, another at and another at Find the center of mass of the system.

On a weekend archeological dig, you discover an old club-ax that consists of a symmetrical 8.0-kg stone attached to the end of a uniform 2.5-kg stick. You measure the dimensions of the club-ax as shown in Figure 5-77. How far is the center of mass of the club-ax from the handle end of the club-ax?

Three balls A, B, and C, with masses of 3.0 kg, 1.0 kg, and 1.0 kg, respectively, are connected by massless rods, as shown in Figure 5-78. What are the coordinates of the center of mass of this system?

By symmetry, locate the center of mass of a uniform sheet in the shape of an equilateral triangle with edges of length a. The triangle has one vertex on the y axis and the others at and

Find the center of mass of the uniform sheet of plywood in Figure 5-79. Consider this as a system of effectively two sheets, letting one have a negative mass to account for the cutout. Thus, one is a square sheet of 3.0-m edge length and mass and the second is a rectangular sheet measuring with a mass of Let the coordinate origin be at the lower left corner of the sheet. SSM

A can in the shape of a symmetrical cylinder with mass M and height H is filled with water. The initial mass of the water is M, the same mass as the can. Asmall hole is punched in the bottom of the can, and the water drains out. (a) If the height of the water in the can is x, what is the height of the center of mass of the can plus the water remaining in the can? (b) What is the minimum height of the center of mass as the water drains out?

Two identical thin uniform rods each of length L are glued together at the ends so that the angle at the joint is Determine the location of the center of mass (in terms of L) of this configuration relative to the origin taken to be at the joint. Hint: You do not need the mass of the rods, but you should start by assuming a mass m and see that it cancels out.

Repeat the analysis of Problem 107 with a general angle at the joint instead of Does your answer agree with the specific -angle answer in Problem 107 if you set equal to ? Does your answer give plausible results for angles of zero and ?

FINDING THE CENTER OF MASS BY INTEGRATION Show that the center of mass of a uniform semicircular disk of radius R is at a point from the center of the circle.

Find the location of the center of mass of a nonuniform rod 0.40 m in length if its density varies linearly from 1.00 g/cm at one end to 5.00 g/cm at the other end. Specify the center-of-mass location relative to the less-massive end of the rod.

You have a thin uniform wire bent into part of a circle that is described by a radius R and angle (see Figure 5-80). Show that the location of its center of mass is on the x axis and located a distance where is expressed in radians. Check your answer by showing that this answer gives the physically expected limit for Verify that your answer gives you the result in the text (in the subsection Finding the Center of Mass by Integration) for the special case of um _ 90.

A long, thin wire of length L has a linear mass density given by where A and B are positive constants and x is the distance from the more massive end. (a) A condition for this problem to be realistic is that Explain why. (b) Determine in terms of L, A, and B. Does your answer makes sense if ? Explain.

Two 3.0-kg particles have velocities and Find the velocity of the center of mass of the system.

A 1500-kg car is moving westward with a speed of and a 3000-kg truck is traveling east with a speed of Find the velocity of the center of mass of the cartruck system.

A force is applied to the 3.0-kg ball in Figure 5-78 in Problem 103. (No forces act on the other two balls.) What is the acceleration of the center of mass of the three-ball system?

A block of mass m is attached to a string and suspended inside an otherwise empty box of mass M. The box rests on a scale that measures the systems weight. (a) If the string breaks, does the reading on the scale change? Explain your reasoning. (b) Assume that the string breaks and the mass m falls with constant acceleration g. Find the magnitude and direction of the acceleration of the center of mass of the boxblock system. (c) Using the result from (b), determine the reading on the scale while m is in free-fall.

The bottom end of a massless, vertical spring of force constant k rests on a scale and the top end is attached to a massless cup, as in Figure 5-81. Place a ball of mass gently into the cup and ease it down into an equilibrium position where it sits at rest in the cup. (a) Draw the separate free-body diagrams for the ball and the spring. (b) Show that in this situation, the spring compression d is given by (c) What is the scale reading under these conditions? SSM

In the Atwoods machine in Figure 5-82 the string passes over a fixed cylinder of mass The cylinder does not rotate. Instead, the string slides on its frictionless surface. (a) Find the acceleration of the center of mass of the two-blockcylinder-string system. (b) Use Newtons second law for systems to find the force F exerted by the support. (c) Find the tension T in the string connecting the blocks and show that F _ mcg _ 2T.

Starting with the equilibrium situation in Problem 117, the whole system (scale, spring, cup, and ball) is now subjected to an upward acceleration of magnitude a (for example, in an elevator). Repeat the free-body diagrams and calculations in Problem 117?

In designing your new house in California, you are prepared for it to withstand a maximum horizontal acceleration of 0.50g. What is the minimum coefficient of static friction between the floor and your prized Tuscan vase so that the vase does not slip on the floor under these conditions?

A 4.5-kg block slides down an inclined plane that makes an angle of with the horizontal. Starting from rest, the block slides a distance of 2.4 m in 5.2 s. Find the coefficient of kinetic friction between the block and plane.

You plan to fly a model airplane of mass 0.400 kg that is attached to a horizontal string. The plane will travel in a horizontal circle of radius 5.70 m. (Assume the weight of the plane is balanced by the upward lift force of the air on the wings of the plane.) The plane will make 1.20 revolutions every 4.00 s. (a) Find the speed at which you must fly the plane. (b) Find the force exerted on your hand as you hold the string (assume the string is massless).

CONTEXT-RICH Your moving company is to load a crate of books on a truck with the help of some planks that slope upward at The mass of the crate is 100 kg, and the coefficient of sliding friction between it and the planks is 0.500. You and your employees push horizontally with a combined net force Once the crate has started to move, how large must F be in order to keep the crate moving at constant speed?

Three forces act on an object in static equilibrium (Figure 5-83). (a) If and represent the magnitudes of the forces acting on the object, show that (b) Show that F21

In a carnival ride, you sit on a seat in a compartment that rotates with constant speed in a vertical circle of radius 5.0 m. The ride is designed so your head always points toward the center of the circle. (a) If the ride completes one full circle in 2.0 s, find the direction and magnitude of your acceleration. (b) Find the slowest rate of rotation (in other words, the longest time to complete one full circle) if the seat belt is to exert no force on you at the top of the ride.

A flat-topped toy cart moves on frictionless wheels, pulled by a rope under tension T. The mass of the cart is Aload of mass rests on top of the cart with the coefficient of static friction between the cart and the load. The cart is pulled up a ramp that is inclined at angle above the horizontal. The rope is parallel to the ramp. What is the maximum tension T that can be applied without causing the load to slip?

A sled weighing 200 N that is held in place by static friction, rests on a incline. The coefficient of static friction between the sled and the incline is 0.50. (a) What is the magnitude of the normal force on the sled? (b) What is the magnitude of the static frictional force on the sled? (c) The sled is now pulled up the incline (Figure 5-84) at constant speed by a child walking up the incline ahead of the sled. The child weighs 500 N and pulls on the rope with a constant force of 100 N. The rope makes an angle of with the incline and has negligible mass. What is the magnitude of the kinetic frictional force on the sled? (d) What is the coefficient of kinetic friction between the sled and the incline? (e) What is the magnitude of the force exerted on the child by the incline?

ENGINEERING APPLICATION In 1976, Gerard ONeill proposed that large space stations be built for human habitation in orbit around Earth and the moon. Because prolonged free-fall has adverse medical effects, he proposed making the stations in the form of long cylinders and spinning them around the cylinder axis to provide the inhabitants with the sensation of gravity. One such ONeill colony is to be built 5.0 miles long, with a diameter of 0.60 mi. A worker on the inside of the colony would experience a sense of gravity, because he would be in an accelerated frame of reference due to the rotation. (a) Show that the acceleration of gravity experienced by the worker in the ONeill colony is equal to his centripetal acceleration. Hint: Consider someone looking in from outside the colony. (b) If we assume that the space station is composed of several decks that are at varying distances (radii) from the axis of rotation, show that the acceleration of gravity becomes weaker the closer the worker gets to the axis. (c) How many revolutions per minute would this space station have to make to give an acceleration of gravity of at the outermost edge of the station?

A child of mass m slides down a slide inclined at in time The coefficient of kinetic friction between her and the slide is She finds that if she sits on a small sled (also of mass m) with frictionless runners, she slides down the same slide in time Find

The position of a particle of mass as a function of time is given by where and (a) Show that the path of this particle is a circle of radius R, with its center at the origin of the xy plane. (b) Compute the velocity vector. Show that (c) Compute the acceleration vector and show that it is directed toward the origin and has the magnitude (d) Find the magnitude and direction of the net force acting on the particle.

MULTISTEP You are on an amusement park ride with your back against the wall of a spinning vertical cylinder. The floor falls away and you are held up by static friction. Assume your mass is 75 kg. (a) Draw a free-body diagram of yourself. (b) Use this diagram with Newtons laws to determine the force of friction on you. (c) If the radius of the cylinder is 4.0 m and the coefficient of static friction between you and the wall is 0.55. What is the minimum number of revolutions per minute necessary to prevent you from sliding down the wall? Does this answer hold only for you? Will other, more massive, patrons fall downward? Explain.

An block of mass is on a horizontal table. The block is attached to a 2.5-kg block (m by a light string that passes over a pulley at the edge of the table. The block of mass dangles 1.5 m above the ground (Figure 5-85). The string that connects them passes over a frictionless, massless pulley. This system is released from rest at and the 2.5-kg block strikes the ground at The system is now placed in its initial configuration and a 1.2-kg block is placed on top of the block of mass Released from rest, the 2.5-kg block now strikes the ground 1.3 s later. Determine the mass and the coefficient of kinetic friction between the block whose mass is m and the table.

Sally claims flying squirrels do not really fly; they jump and use the folds of skin that connect their forelegs and their back legs like a parachute to allow them to glide from tree to tree. Liz decides to test Sallys hypothesis by calculating the terminal speed of a falling outstretched flying squirrel. If the constant b in the drag force is proportional to the area of the object facing the air flow, use the results of Example 5-12 and some assumptions about the size of the squirrel to estimate its terminal (downward) speed. Is Sallys claim supported by Lizs calculation?

BIOLOGICAL APPLICATION After a parachutist jumps from an airplane (but before he pulls the rip cord to open his parachute), a downward speed of up to can be reached. When the parachute is finally opened, the drag force is increased by about a factor of 10, and this can create a large jolt on the jumper. Suppose this jumper falls at before opening his chute. (a) Determine the parachutists acceleration when the chute is just opened, assuming his mass is 60 kg. (b) If rapid accelerations greater than 5.0g can harm the structure of the human body, is this a safe practice?

Find the location of the center of mass of the Earthmoon system relative to the center of Earth. Is it inside or outside the surface of Earth?

A circular plate of radius R has a circular hole of radius R/2 cut out of it (Figure 5-86). Find the center of mass of the plate after the hole has been cut. Hint: The plate can be modeled as two disks superimposed, with the hole modeled as a disk negative mass.

An unbalanced baton consists of a 50-cm-long uniform rod of mass 200 g. At one end there is a 10-cm-diameter uniform solid sphere of mass 500 g, and at the other end there is a 8.0-cmdiameter uniform solid sphere of mass 750 g. (The center-to-center distance between the spheres is 59 cm.) (a) Where, relative to the center of the light sphere, is the center of mass of this baton? (b) If this baton is tossed straight up (but spinning) so that its initial center of mass speed is what is the velocity of the center of mass 1.5 s later? (c) What is the net external force on the baton while in the air? (d) What is the acceleration of the batons center of mass 1.5 s following its release?

You are standing at the very rear of a 6.0-m-long, 120-kg raft that is at rest in a lake with its prow only 0.50 m from the end of the pier (Figure 5-87). Your mass is 60 kg. Neglect frictional forces between the raft and the water. (a) How far from the end of the pier is the center of mass of the youraft system? (b) You walk to the front of the raft and then stop. How far from the end of the pier is the center of mass now? (c) When you are at the front of the raft, how far are you from the end of the pier? 10.0m>s,

An Atwoods machine that has a frictionless massless pulley and massless strings has a 2.00-kg object hanging from one side and 4.00-kg object hanging from the other side. (a) What is the speed of each object 1.50 s after they are simultaneously released from rest? (b) At that time, what is the velocity of the center of mass of the two objects? (c) At that moment, what is the acceleration of the center of mass of the two objects?