Principles of Physics a Calculus Based Text

Raymond A. Serway, John W. Jewett, Jr.

Chapter 5

More Applications of Newton's Laws - all with Video Answers

Educators

Chapter Questions

Problem 1

To determine the coefficients of friction between rubber and various surfaces, a student uses a rubber eraser and an incline. In one experiment, the eraser begins to slip down the incline when the angle of inclination is $36.0^{\circ}$ and then moves down the incline with constant speed when the angle is reduced to $30.0^{\circ} .$ From these data, determine the coefficients of static and kinetic friction for this experiment.

Vishal Gupta

Vishal Gupta

Numerade Educator

Problem 2

Before $1960,$ people believed that the maximum attainable coefficient of static friction for an automobile tire on a roadway was $\mu_{s}=1 .$ Around $1962,$ three companies independently developed racing tires with coefficients of 1.6 This problem shows that tires have improved further since then. The shortest time interval in which a pistonengine car initially at rest has covered a distance of one-quarter mile is about $4.43 \mathrm{s}$. (a) Assume the car's rear wheels lift the front wheels off the pavement as shown in Figure P5.2. What minimum value of $\mu_{3}$ is necessary to achieve the record time? (b) Suppose the driver were able to increase his or her engine power, keeping other things equal. How would this change affect the elapsed time?

Dominador Tan

Numerade Educator

Problem 3

A 25.0 -kg block is initially at rest on a horizontal surface. A horizontal force of $75.0 \mathrm{N}$ is required to set the block in motion, after which a horizontal force of $60.0 \mathrm{N}$ is required to keep the block moving with constant speed. Find (a) the coefficient of static friction and (b) the coefficient of kinetic friction between the block and the surface.

Vishal Gupta

Numerade Educator

Problem 4

Review. A car is traveling at $50.0 \mathrm{mi} / \mathrm{h}$ on a horizontal highway. (a) If the coefficient of static friction between road and tires on a rainy day is 0.100 , what is the minimum distance in which the car will stop? (b) What is the stopping distance when the surface is dry and $\mu_{s}=0.600 ?$

Michael Cao

Numerade Educator

Problem 5

To meet a U.S. Postal Service requirement, employees' footwear must have a coefficient of static friction of 0.5 or more on a specified tile surface. A typical athletic shoe has a coefficient of static friction of $0.800 .$ In an emergency, what is the minimum time interval in which a person starting from rest can move $3.00 \mathrm{m}$ on the tile surface if she is wearing (a) footwear meeting the Postal Service minimum and (b) a typical athletic shoe?

Vishal Gupta

Numerade Educator

Problem 6

The person in Figure P5.6 weighs 170 lb. As seen from the front, each light crutch makes an angle of $22.0^{\circ}$ with the vertical. Half of the person's weight is supported by the crutches. The other half is supported by the vertical forces of the ground on the person's feet. Assuming that the person is moving with constant velocity and the force exerted by the ground on the crutches acts along the crutches, determine (a) the smallest possible coefficient of friction between crutches and ground and (b) the magnitude of the compression force in each crutch.

Vishal Gupta

Numerade Educator

Problem 7

A 9.00 -kg hanging object is connected by a light, inextensible cord over a light, frictionless pulley to a $5.00-\mathrm{kg}$ block that is sliding on a flat table (Fig. P5.7). Taking the coefficient of kinetic friction as $0.200,$ find the tension in the string.

Vishal Gupta

Numerade Educator

Problem 8

Consider a large truck carrying a heavy load, such as steel beams. A significant hazard for the driver is that the load may slide forward, crushing the cab, if the truck stops suddenly in an accident or even in braking. Assume, for example, that a $10000-\mathrm{kg}$ load sits on the flatbed of a $20000-\mathrm{kg}$ truck moving at $12.0 \mathrm{m} / \mathrm{s}$. Assume that the load is not tied down to the truck, but has a coefficient of friction of 0.500 with the flatbed of the truck. (a) Calculate the minimum stopping distance for which the load will not slide forward relative to the truck. (b) Is any piece of data unnecessary for the solution?

Vishal Gupta

Numerade Educator

Problem 9

A $3.00-\mathrm{kg}$ block starts from rest at the top of a $30.0^{\circ}$ incline and slides a distance of $2.00 \mathrm{m}$ down the incline in 1.50 s. Find (a) the magnitude of the acceleration of the block, (b) the coefficient of kinetic friction between block and plane, (c) the friction force acting on the block, and (d) the speed of the block after it has slid $2.00 \mathrm{m}$.

Vishal Gupta

Numerade Educator

Problem 10

A woman at an airport is towing her 20.0 -kg suitcase at constant speed by pulling on a strap at an angle $\theta$ above the horizontal (Fig. P5.10). She pulls on the strap with a $35.0-\mathrm{N}$ force, and the friction force on the suitcase is $20.0 \mathrm{N}$. (a) Draw a free-body diagram of the suitcase. (b) What angle does the strap make with the horizontal? (c) What is the magnitude of the normal force that the ground exerts on the suitcase?

Michael Cao

Numerade Educator

Problem 11

Review. One side of the roof of a house slopes up at $37.0^{\circ} .$ A roofer kicks a round, flat rock that has been thrown onto the roof by a neighborhood child. The rock slides straight up the incline with an initial speed of $15.0 \mathrm{m} / \mathrm{s}$. The coefficient of kinetic friction between the rock and the roof is 0.400 . The rock slides $10.0 \mathrm{m}$ up the roof to its peak. It crosses the ridge and goes into free fall, following a parabolic trajectory above the far side of the roof, with negligible air resistance. Determine the maximum height the rock reaches above the point where it was kicked.

Vishal Gupta

Numerade Educator

Problem 12

A block of mass $3.00 \mathrm{kg}$ is pushed un against a wall by a force $\mathbf{P}$ that makes an angle of $\theta=50.0^{\circ}$ with the horizontal as shown in Figure P5.12. The coefficient of static friction between the block and the wall is 0.250 . (a) Determine the possible values for the magnitude of $\overrightarrow{\mathbf{P}}$ that allow the block to remain stationary. (b) Describe what happens if $|\overrightarrow{\mathbf{P}}|$ has a larger value and what happens if it is smaller. (c) Repeat parts (a) and (b), assuming the force makes an angle of $\theta=13.0^{\circ}$ with the horizontal.

Vishal Gupta

Numerade Educator

Problem 13

Two blocks connected by a rope of negligible mass are being dragged by a horizontal force (Fig. P5.13). Suppose $F=68.0 \mathrm{N}, m_{1}=12.0 \mathrm{kg}$ $m_{2}=18.0 \mathrm{kg},$ and the coef ficient of kinetic friction between each block and the surface is 0.100 . (a) Draw a free-body diagram for each block. Determine (b) the acceleration of the system and (c) the tension $T$ in the rope.

Vishal Gupta

Numerade Educator

Problem 14

Three objects are connected on a table as shown in Figure P5.14. The coefficient of kinetic friction between the block of mass $m_{2}$ and the table is 0.350 . The objects have masses of $m_{1}=4.00 \mathrm{kg}, m_{2}=1.00 \mathrm{kg},$ and $m_{3}=2.00 \mathrm{kg},$ and
the pulleys are frictionless. (a) Draw a free-body diagram of each object. (b) Determine the acceleration of each object, including its direction. (c) Determine the tensions in the two cords. What If? (d) If the tabletop were smooth, would the tensions increase, decrease, or remain the same? Explain.

Vishal Gupta

Numerade Educator

Problem 15

Why is the following situation impossible? Your 3.80 -kg physics book is placed next to you on the horizontal seat of your car. The coefficient of static friction between the book and the seat is 0.650 , and the coefficient of kinetic friction is 0.550 . You are traveling forward at $72.0 \mathrm{km} / \mathrm{h}$ and brake to a stop with constant acceleration over a distance of $30.0 \mathrm{m} .$ Your physics book remains on the seat rather than sliding forward onto the floor.

Vishal Gupta

Numerade Educator

Problem 16

In the Bohr model of the hydrogen atom, an electron moves in a circular path around a proton. The speed of the electron is approximately $2.20 \times 10^{6} \mathrm{m} / \mathrm{s}$. Find (a) the force acting on the electron as it revolves in a circular orbit of radius $0.530 \times$ $10^{-10} \mathrm{m}$ and (b) the centripetal acceleration of the electron.

Luis Mendoza

Numerade Educator

Problem 17

A light string can support a stationary hanging load of 25.0 kg before breaking. An object of mass $m=3.00$ kg attached to the string rotates on a frictionless, horizontal table in a circle of radius $r=0.800 \mathrm{m},$ and the other end of the string is held fixed as in Figure $\mathrm{P} 5.17$ What range of speeds can the object have before the string breaks?

Luis Mendoza

Numerade Educator

Problem 18

Why is the following situation impossible? The object of mass $m=4.00 \mathrm{kg}$ in Figure $\mathrm{P} 5.18$ is at tached to a vertical rod by two strings of length $\ell=2.00 \mathrm{m} .$ The strings are attached to the rod at points a distance $d=3.00 \mathrm{m}$ apart. The object rotates in a horizontal circle at a constant speed of $v=3.00 \mathrm{m} / \mathrm{s}$ and the strings remain taut. The rod rotates along with the object so that the strings do not wrap onto the rod. What If? Could this situation be possible on another planet?

Vishal Gupta

Numerade Educator

Problem 19

A crate of eggs is located in the middle of the flatbed of a pickup truck as the truck negotiates a curve in the flat road. The curve may be regarded as an arc of a circle of radius $35.0 \mathrm{m}$. If the coefficient of static friction between crate and truck is 0.600 , how fast can the truck be moving without the crate sliding?

Luis Mendoza

Numerade Educator

Problem 20

Whenever two Apollo astronauts were on the surface of the Moon, a third astronaut orbited the Moon. Assume the orbit to be circular and $100 \mathrm{km}$ above the surface of the Moon, where the acceleration due to gravity is $1.52 \mathrm{m} / \mathrm{s}^{2}$. The radius of the Moon is $1.70 \times 10^{6} \mathrm{m} .$ Determine (a) the astronaut's orbital speed and (b) the period of the orbit.

Katie Mcalpine

Numerade Educator

Problem 21

Consider a conical pendulum (Fig. P5.21) with a bob of mass $m=80.0 \mathrm{kg}$ on a string of length $L=10.0 \mathrm{m}$ that makes an angle of $\theta=5.00^{\circ}$ with the vertical. Determine (a) the horizontal and vertical components of the force exerted by the wire on the pendulum and (b) the radial acceleration of the bob.

Vishal Gupta

Numerade Educator

Problem 22

A roller coaster at the Six Flags Great America amusement park in Gurnee, Illinois, incorporates some clever design technology and some basic physics. Each vertical loop, instead of being circular, is shaped like a teardrop (Fig. P5.22). The cars ride on the inside of the loop at the top, and the speeds are fast enough to ensure the cars remain on the track. The biggest loop is $40.0 \mathrm{m}$ high. Suppose the speed at the top of the loop is $13.0 \mathrm{m} / \mathrm{s}$ and the corresponding centripetal acceleration of the riders is $2 g$. (a) What is the radius of the arc of the teardrop at the top? (b) If the total mass of a car plus the riders is $M$, what force does the rail exert on the car at the top? (c) Suppose the roller coaster had a circular loop of radius $20.0 \mathrm{m}$. If the cars have the same speed, $13.0 \mathrm{m} / \mathrm{s}$ at the top, what is the centripetal acceleration of the riders at the top? (d) Comment on the normal force at the top in the situation described in part (c) and on the advantages of having teardrop-shaped loops.

Vishal Gupta

Numerade Educator

Problem 23

Disturbed by speeding cars outside his workplace, Nobel laureate Arthur Holly Compton designed a speed bump (called the "Holly hump") and had it installed. Suppose a $1800-\mathrm{kg}$ car passes over a $1800-\mathrm{kg}$ car passes over a hump in a roadway that follows the arc of a circle of radius $20.4 \mathrm{m}$ as shown in Figure P5.23. (a) If the car travels at $30.0 \mathrm{km} / \mathrm{h},$ what force does the road exert on the car as the car passes the highest point of the hump? (b) What If? What is the maximum speed the car can have without losing contact with the road as it passes this highest point?

Supratim Pal

Numerade Educator

Problem 24

A car of mass $m$ passes over a hump in a road that follows the arc of a circle of radius $R$ as shown in Figure $\mathrm{P} 5.23$.
(a) If the car travels at a speed $v$, what force does the road exert on the car as the car passes the highest point of the hump?
(b) What If? What is the maximum speed the car can have without losing contact with the road as it passes this highest point?

Vishal Gupta

Numerade Educator

Problem 25

An adventurous archeologist $(m=85.0 \mathrm{kg})$ tries to cross a river by swinging from a vine. The vine is $10.0 \mathrm{m}$ long, and his speed at the bottom of the swing is $8.00 \mathrm{m} / \mathrm{s}$. The archeologist doesn't know that the vine has a breaking strength of $1000 \mathrm{N}$. Does he make it across the river without falling in?

Luis Mendoza

Numerade Educator

Problem 26

A pail of water is rotated in a vertical circle of radius $1.00 \mathrm{m} .$ (a) What two external forces act on the water in the pail? (b) Which of the two forces is most important in causing the water to move in a circle? (c) What is the pail's minimum speed at the top of the circle if no water is to spill out? (d) Assume the pail with the speed found in part (c) were to suddenly disappear at the top of the circle. Describe the subsequent motion of the water. Would it differ from the motion of a projectile?

Vishal Gupta

Numerade Educator

Problem 27

A 40.0 -kg child swings in a swing supported by two chains, each $3.00 \mathrm{m}$ long. The tension in each chain at the lowest point is $350 \mathrm{N}$. Find (a) the child's speed at the lowest point and (b) the force exerted by the seat on the child at the lowest point. (Ignore the mass of the seat.)

Vishal Gupta

Numerade Educator

Problem 28

A child of mass $m$ swings in a swing supported by two chains, each of length $R$. If the tension in each chain at the lowest point is $T,$ find (a) the child's speed at the lowest point and (b) the force exerted by the seat on the child at the lowest point. (Ignore the mass of the seat.)

Vishal Gupta

Numerade Educator

Problem 29

A small, spherical bead of mass $3.00 \mathrm{g}$ is released from rest at $t=0$ from a point under the surface of a viscous liquid. The terminal speed is observed to be $v_{T}=2.00$ $\mathrm{cm} / \mathrm{s} .$ Find (a) the value of the constant $b$ that appears in Equation $5.4,$ (b) the time $t$ at which the bead reaches $0.632 v_{T}$ and $(c)$ the value of the resistive force when the bead reaches terminal speed.

Dominador Tan

Numerade Educator

Problem 30

Review. (a) Estimate the terminal speed of a wooden sphere (density $0.830 \mathrm{g} / \mathrm{cm}^{3}$ ) falling through air, taking its radius as $8.00 \mathrm{cm}$ and its drag coefficient as $0.500 .$ (b) From what height would a freely falling object reach this speed in the absence of air resistance?

Katie Mcalpine

Numerade Educator

Problem 31

A small piece of Styrofoam packing material is dropped from a height of $2.00 \mathrm{m}$ above the ground. Until it reaches terminal speed, the magnitude of its acceleration is given by $a=g-B v$. After falling $0.500 \mathrm{m},$ the Styrofoam effectively reaches terminal speed and then takes 5.00 s more to reach the ground. (a) What is the value of the constant $B$ ? (b) What is the acceleration at $t=0 ?(\mathrm{c})$ What is the acceleration when the speed is $0.150 \mathrm{m} / \mathrm{s}$ ?

Katie Mcalpine

Numerade Educator

Problem 32

Assume the resistive force acting on a speed skater is proportional to the square of the skater's speed $v$ and is given by $f=-k m v^{2},$ where $k$ is a constant and $m$ is the skater's mass. The skater crosses the finish line of a straight-line race with speed $v_{i}$ and then slows down by coasting on his skates. Show that the skater's speed at any time $t$ after crossing the finish line is $v(t)=v_{i} /\left(1+k t v_{i}\right)$.

Luis Mendoza

Numerade Educator

Problem 33

A motorboat cuts its engine when its speed is $10.0 \mathrm{m} / \mathrm{s}$ and then coasts to rest. The equation describing the motion of the motorboat during this period is $v=v_{i} e^{-d},$ where $v$ is the speed at time $t, v_{i}$ is the initial speed at $t=0,$ and $c$ is a constant. At $t=20.0 \mathrm{s}$, the speed is $5.00 \mathrm{m} / \mathrm{s}$. (a) Find the constant $c .(\mathrm{b})$ What is the speed at $t=40.0 \mathrm{s}$ ? (c) Differentiate the expression for $v(t)$ and thus show that the acceleration of the boat is proportional to the speed at any time.

Dominador Tan

Numerade Educator

Problem 34

A 9.00 -kg object starting from rest falls through a viscous medium and experiences a resistive force given by Equation 5.4 The object reaches one-half its terminal speed in 5.54 s. (a) Determine the terminal speed. (b) At what time is the speed of the object three-fourths the terminal speed? (c) How far has the object traveled in the first 5.54 s of motion?

Dominador Tan

Numerade Educator

Problem 35

When a falling meteor is at a distance above the Earth's surface of 3.00 times the Earth's radius, what is its free-fall acceleration caused by the gravitational force exerted on it?

Vishal Gupta

Numerade Educator

Problem 36

In a thundercloud, there may be electric charges of $+40.0 \mathrm{C}$ near the top of the cloud and -40.0 C near the bottom of the cloud. These charges are separated by $2.00 \mathrm{km} .$ What is the electric force on the top charge?

Vishal Gupta

Numerade Educator

Problem 37

Two identical isolated particles, each of mass $2.00 \mathrm{kg},$ are separated by a distance of $30.0 \mathrm{cm} .$ What is the magnitude of the gravitational force exerted by one particle on the other?

Vishal Gupta

Numerade Educator

Problem 38

Find the order of magnitude of the gravitational force that you exert on another person $2 \mathrm{m}$ away. In your solution, state the quantities you measure or estimate and their values.

Yaqub Khan

Numerade Educator

Problem 39

The mass of a sports car is 1200 kg. The shape of the body is such that the aerodynamic drag coefficient is 0.250 and the frontal area is $2.20 \mathrm{m}^{2}$. Ignoring all other sources of friction, calculate the initial acceleration the car has if it has been traveling at $100 \mathrm{km} / \mathrm{h}$ and is now shifted into neutral and allowed to coast.

Luis Mendoza

Numerade Educator

Problem 40

Consider a 1300 -kg car presenting front-end area $2.60 \mathrm{m}^{2}$ and having drag coefficient $0.340 .$ It can achieve instantaneous acceleration $3.00 \mathrm{m} / \mathrm{s}^{2}$ when its speed is $10.0 \mathrm{m} / \mathrm{s}$ Ignore any force of rolling resistance. Assume that the only horizontal forces on the car are static friction forward exerted by the road on the drive wheels and resistance exerted by the surrounding air, with density $1.20 \mathrm{kg} / \mathrm{m}^{3}$. (a) Find the friction force exerted by the road. (b) Suppose the car body could be redesigned to have a drag coefficient of $0.200 .$ If nothing else changes, what will be the car's acceleration? (c) Assume that the force exerted by the road remains constant. Then what maximum speed could the car attain with $D=0.340 ?(\mathrm{d}) \mathrm{With} D=0.200 ?$

Dominador Tan

Numerade Educator

Problem 41

In a home laundry dryer, a cylindrical tub containing wet clothes is rotated steadily about a horizontal axis as shown in Figure P5.41. So that the clothes will dry uniformly, they are made to tumble. The rate of rotation of the smoothwalled tub is chosen so that a small piece of cloth will lose contact with the tub when the cloth is at an angle of $\theta=68.0^{\circ}$ above the horizontal. If the radius of the tub is $r=0.330 \mathrm{m},$ what rate of revolution is needed?

Vishal Gupta

Numerade Educator

Problem 42

A crate of weight $F_{g}$ is pushed by a force $\vec{P}$ on a horizontal floor as shown in Figure $\mathrm{P} 5.42 .$ The coefficient of static friction is $\mu_{s}$ and $\overrightarrow{\mathbf{P}}$ is directed at angle $\theta$ below the horizontal.
(a) Show that the minimum value of $P$
that will move the crate is given by
$$P=\frac{\mu_{s} F_{g} \sec \theta}{1-\mu_{s} \tan \theta}$$
(b) Find the condition on $\theta$ in terms of $\mu_{s}$ for which motion of the crate is impossible for any value of $P$.

Vishal Gupta

Numerade Educator

Problem 43

Consider the three connected objects shown in Figure $P 5.43 .$ Assume first that the inclined plane is frictionless and that the system is in equilibrium. In terms of $m$ $g,$ and $\theta,$ find (a) the mass $M$ and $(\mathrm{b})$ the tensions $T_{1}$ and $T_{2} .$ Now assume that the value of $M$ is double the value found in part (a). Find (c) the acceleration of each object and (d) the tensions $T_{1}$ and $T_{2} .$ Next, assume that the coefficient of static friction between $m$ and $2 m$ and the inclined plane is $\mu_{s}$ and that the system is in equilibrium. Find (e) the maximum value of $M$ and (f) the minimum value of $M$. (g) Compare the values of $T_{2}$ when $M$ has its minimum and maximum values.

Vishal Gupta

Numerade Educator

Problem 44

A car rounds a banked curve as discussed in Example 5.7 and shown in Figure $5.11 .$ The radius of curvature of the road is $R,$ the banking angle is $\theta,$ and the coefficient of static friction is $\mu_{s^{*}}$ (a) Determine the range of speeds the car can have without slipping up or down the road. (b) Find the minimum value for $\mu_{s}$ such that the minimum speed is zero.

Vishal Gupta

Numerade Educator

Problem 45

The system shown in Figure $\mathrm{P} 4.47$ (Chapter 4 ) has an acceleration of magnitude $1.50 \mathrm{m} / \mathrm{s}^{2}$. Assume that the coefficient of kinetic friction between block and incline is the same for both inclines. Find (a) the coefficient of kinetic friction and
(b) the tension in the string.

Vishal Gupta

Numerade Educator

Problem 46

An aluminum block of mass $m_{1}=2.00 \mathrm{kg}$ and a copper block of mass $m_{2}=6.00 \mathrm{kg}$ are connected by a light string over a frictionless pulley. They sit on a steel surface as shown in Figure $\mathrm{P} 5.46,$ where $\theta=30.0^{\circ} .$ (a) When they are released from rest, will they start to move? If they do, determine (b) their acceleration and (c) the tension in the string. If they do not move, determine (d) the sum of the magnitudes of the forces of friction acting on the blocks.

Michael Cao

Numerade Educator

Problem 47

Figure $\mathrm{P} 5.47$ shows a photo of a swing ride at an amusement park. The structure consists of a horizontal, rotating, circular platform of diameter $D$ from which seats of mass $m$ are suspended at the end of massless chains of length $d$. When the system rotates at constant speed, the chains swing outward and make an angle $\theta$ with the vertical. Consider such a ride with the following parameters: $D=8.00 \mathrm{m}, d=2.50 \mathrm{m}$ $m=10.0 \mathrm{kg},$ and $\theta=28.0^{\circ} .$ (a) What is the speed of each seat? (b) Draw a diagram of forces acting on a 40.0 -kg child riding in a seat and (c) find the tension in the chain.

Dominador Tan

Numerade Educator

Problem 48

Why is the following situation impossible? A $1.30-\mathrm{kg}$ toaster is not plugged in. The coefficient of static friction between the toaster and a horizontal countertop is $0.350 .$ To make the toaster start moving, you carelessly pull on its electric cord. Unfortunately, the cord has become frayed from your previous similar actions and will break if the tension in the cord exceeds 4.00 N. By pulling on the cord at a particular angle, you successfully start the toaster moving without breaking the cord.

Michael Cao

Numerade Educator

Problem 49

A space station, in the form of a wheel $120 \mathrm{m}$ in diameter, rotates to provide an "artificial gravity" of $3.00 \mathrm{m} / \mathrm{s}^{2}$ for persons who walk around on the inner wall of the outer rim. Find the rate of the wheel's rotation in revolutions per minute that will produce this effect.

Luis Mendoza

Numerade Educator

Problem 50

A 5.00 -kg block is placed on top of a 10.0 -kg block (Fig. P5.50). A horizontal force of $45.0 \mathrm{N}$ is applied to the 10 -kg block, and the 5 -kg block is tied to the wall. The coefficient of kinetic friction between all moving surfaces is $0.200 .$ (a) Draw a free-body diagram for each block and identify the action-reaction forces between the blocks. (b) Determine the tension in the string and the magnitude of the acceleration of the 10 -kg block.

David Morabito

Numerade Educator

Problem 51

In Example 5.8 , we investigated the forces a child experiences on a Ferris wheel. Assume the data in that example applies to this problem. What force (magnitude and direction ) does the seat exert on a 40.0 -kg child when the child is halfway between top and bottom?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 52

A student builds and calibrates an accelerometer and uses it to determine the speed of her car around a certain unbanked highway curve. The accelerometer is a plumb bob with a protractor that she attaches to the roof of her car. A friend riding in the car with the student observes that the plumb bob hangs at an angle of $15.0^{\circ}$ from the vertical when the car has a speed of $23.0 \mathrm{m} / \mathrm{s}$. (a) What is the centripetal acceleration of the car rounding the curve? (b) What is the radius of the curve? (c) What is speed of the car if the plumb bob deflection is $9.00^{\circ}$ while rounding the same curve?

Mayukh Banik

Numerade Educator

Problem 53

Two blocks of masses $m_{1}$ and $m_{2}$ are placed on a table in contact with each other as discussed in Example 4.5 and shown in Active Figure 4.13 a. The coefficient of kinetic friction between the block of mass $m_{1}$ and the table is $\mu_{1},$ and that between the block of mass $m_{2}$ and the table is $\mu_{2}$. A horizontal force of magnitude $F$ is applied to the block of mass $m_{1} .$ We wish to find $P$, the magnitude of the contact force between the blocks. (a) Draw diagrams showing the forces for each block. (b) What is the net force on the system of two blocks? (c) What is the net force acting on $m_{1}$ ? (d) What is the net force acting on $m_{2}$ ? (e) Write Newton's second law in the $x$ direction for each block. (f) Solve the two equations in two unknowns for the acceleration $a$ of the blocks in terms of the masses, the applied force $F$, the coefficients of friction, and $g .(\mathrm{g})$ Find the magnitude $P$ of the contact force between the blocks in terms of the same quantities.

Ren Jie Tuieng

Numerade Educator

Problem 54

Why is the following situation impossible? A mischievous child goes to an amusement park with his family. On one ride, after a severe scolding from his mother, he slips out of his seat and climbs to the top of the ride's structure, which is shaped like a cone with its axis vertical and its sloped sides making an angle of $\theta=20.0^{\circ}$ with the horizontal as shown in Figure $\mathrm{P} 5.54 .$ This part of the structure rotates about the vertical central axis when the ride operates. The child sits on the sloped surface at a point $d=5.32 \mathrm{m}$ down the sloped side from the center of the cone and pouts. The coefficient of static friction between the boy and the cone is 0.700 . The ride operator does not notice that the child has slipped away from his seat and so continues to operate the ride. As a result, the sitting, pouting boy rotates in a circular path at a speed of $3.75 \mathrm{m} / \mathrm{s}$.

Dominador Tan

Numerade Educator

Problem 55

A block of mass $m=2.00$ kg rests on the left edge of a block of mass $M=8.00 \mathrm{kg} .$ The coefficient of kinetic friction between the two blocks is 0.300 , and the surface on which the 8.00 -kg block rests is frictionless. A constant horizontal force of magnitude $F=10.0 \mathrm{N}$ is applied to the 2.00 -kg block, setting it in motion as shown in Figure P5.55a. If the distance $L$ that the leading edge of the smaller block travels on the larger block is $3.00 \mathrm{m},$ (a) in what time interval will the smaller block make it to the right side of the 8.00 -kg block as shown in Figure P5.55b? (Note. Both blocks are set into motion when $\overrightarrow{\mathbf{F}}$ is applied. (b) How far does the 8.00 kg block move in the process?

Dominador Tan

Numerade Educator

Problem 56

A puck of mass $m_{1}$ is tied to a string and allowed to revolve in a circle of radius $R$ on a frictionless, horizontal table. The other end of the string passes through a small hole in the center of the table, and an object of mass $m_{2}$ is tied to it (Fig. P5.56). The suspended object remains in equilibrium while the puck on the tabletop revolves. Find symbolic expressions for
(a) the tension in the string, (b) the radial force acting on the puck, and
(c) the speed of the puck.
(d) Qualitatively describe what will happen in the motion of the puck if the value of $m_{2}$ is increased by placing a small additional load on the hanging load.

Dominador Tan

Numerade Educator

Problem 57

A model airplane of mass $0.750 \mathrm{kg}$ flies with a speed of $35.0 \mathrm{m} / \mathrm{s}$ in a horizontal circle at the end of a 60.0 -m-long control wire as shown in Figure $\mathrm{P} 5.57$ a. The forces exerted on the airplane are shown in Figure $\mathrm{P} 5.57 \mathrm{b}$ : the tension in the control wire, the gravitational force, and aerodynamic lift that acts at $\theta=20.0^{\circ}$ inward from the vertical. Compute the tension in the wire, assuming it makes a constant angle of $\theta=20.0^{\circ}$ with the horizontal.

Dominador Tan

Numerade Educator

Problem 58

Why is the following situation impossible? A book sits on an inclined plane on the surface of the Earth. The angle of the plane with the horizontal is $60.0^{\circ} .$ The coefficient of kinetic friction between the book and the plane of 0.300 . At time $t=0,$ the book is released from rest. The book then slides through a distance of $1.00 \mathrm{m},$ measured along the plane, in a time interval of 0.483 s.

Michael Cao

Numerade Educator

Problem 59

A single bead can slide with negligible friction on a stiff wire that has been bent into a circular loop of radius $15.0 \mathrm{cm}$ as shown in Figure $\mathrm{P} 5.59 .$ The circle is always in a vertical plane and rotates steadily about its vertical diameter with a period of 0.450 s. The position of the bead is described by the angle $\theta$ that the radial line, from the center of the loop to the bead, makes with the vertical. (a) At what angle up from the bottom of the circle can the bead stay motionless relative to the turning circle?
(b) What If? Repeat the problem, this time taking the period of the circle's rotation as 0.850 s. (c) Describe how the solution to part (b) is different from the solution to part (a). (d) For any period or loop size, is there always an angle at which the bead can stand still relative to the loop? (e) Are there ever more than two angles? Arnold Arons suggested the idea for this problem.

Dominador Tan

Numerade Educator

Problem 60

An amusement park ride consists of a large vertical cylinder that spins about its axis fast enough that any person inside is held up against the wall when the floor drops away (Fig. P5.60). The coefficient of staticfriction between person and wall is $\mu_{s}$ and the radius of the cylinder is $R$. (a) Show that the maximum period of revolution necessary to keep the person from falling is $T=\left(4 \pi^{2} R \mu_{s} / g\right)^{1 / 2} .$ (b) If the rate of revolution of the cylinder is made to be somewhat larger, what happens to the magnitude of each one of the forces acting on the person? What happens in the motion of the person? (c) If the rate of revolution of the cylinder is instead made to be somewhat smaller, what happens to the magnitude of each one of the forces acting on the person? What happens in the motion of the person?

Dominador Tan

Numerade Educator

Problem 61

The expression $F=a r v+b r^{2} v^{2}$ gives the magnitude of the resistive force (in newtons) exerted on a sphere of radius $r$ (in meters) by a stream of air moving at speed $v$ (in meters per second $),$ where $a$ and $b$ are constants with appropriate SI units. Their numerical values are $a=3.10 \times 10^{-4}$ and $b=0.870 .$ Using this expression, find the terminal speed for water droplets falling under their own weight in air, taking the following values for the drop radii: (a) $10.0 \mu \mathrm{m}$ (b) $100 \mu \mathrm{m}$ (c) $1.00 \mathrm{mm}$. For parts (a) and (c), you can obtain accurate answers without solving a quadratic equation by considering which of the two contributions to the air resistance is dominant and ignoring the lesser contribution.

Luis Mendoza

Numerade Educator

Problem 62

Members of a skydiving club were given the following data to use in planning their jumps. In the table, $d$ is the distance fallen from rest by a skydiver in a "free-fall stable spread position" versus the time of fall $t$. (a) Convert the distances in feet into meters. (b) Graph $d$ (in meters) versus $t .$ (c) Determine the value of the terminal speed $v_{T}$ by finding the slope of the straight portion of the curve. Use a least-squares fit to determine this slope.

Dominador Tan

Numerade Educator

Problem 63

Because the Earth rotates about its axis, a point on the equator experiences a centripetal acceleration of $0.0337 \mathrm{m} / \mathrm{s}^{2},$ whereasa pointat the poles experiences no centripetal acceleration. If a person at the equator has a mass of $75 \mathrm{kg},$ calculate (a) the gravitational force (true weight) on the person and (b) the normal force (apparent weight) on the person. (c) Which force is greater? Assume the Earth is a uniform sphere and take $g=9.800 \mathrm{m} / \mathrm{s}^{2}$.

Dominador Tan

Numerade Educator

Problem 64

If a single constant force acts on an object that moves on a straight line, the object's velocity is a linear function of time. The equation $v=v_{i}+$ at gives its velocity $v$ as a function of time, where $a$ is its constant acceleration. What if velocity is instead a linear function of position? Assume that as a particular object moves through a resistive medium, its speed decreases as described by the equation $v=v_{i}-k x,$ where $k$ is a constant coefficient and $x$ is the position of the object. Find the law describing the total force acting on this object.

Mayukh Banik

Numerade Educator