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Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett

Chapter 6

Circular Motion and Other Applications of Newton’s Laws - all with Video Answers

Educators

Chapter Questions

02:13

Problem 1

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

Anand Jangid
Anand Jangid
Numerade Educator
02:17

Problem 2

A curve in a road forms part of a horizontal circle. As a car goes around it at constant speed $14.0 \mathrm{~m} / \mathrm{s}$, the total force on the driver has magnitude 130 N . What is the total vector force on the driver if the speed is $18.0 \mathrm{~m} / \mathrm{s}$ instead?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:57

Problem 3

In the Bohr model of the hydrogen atom, 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.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
06:36

Problem 4

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 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
Katie Mcalpine
Numerade Educator
01:18

Problem 5

A coin placed 30.0 cm from the center of a rotating horizontal turntable slips when its speed is $50.0 \mathrm{~cm} / \mathrm{s}$. (a) What force causes the centripetal acceleration when the coin is stationary relative to the turntable? (b) What is the coefficient of static friction between the coin and turntable?

Luis Mendoza
Luis Mendoza
Numerade Educator
02:16

Problem 6

In a cyclotron (one type of particle accelerator), a deuteron (of mass 2.00 u ) reaches a final speed of $10.0 \%$ of the speed of light while moving in a circular path of radius 0.480 m . The deuteron is maintained in the circular path by a magnetic force. What magnitude of force is required?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:41

Problem 7

A space station, in the form of a wheel 120 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 rotation of the wheel (in revolutions per minute) that will produce this effect.

Luis Mendoza
Luis Mendoza
Numerade Educator
05:05

Problem 8

Consider a conical pendulum (Fig. 6.3) with an $80.0-\mathrm{kg}$ bob on a $10.0-\mathrm{m}$ wire making 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
Vishal Gupta
Numerade Educator
01:46

Problem 9

A crate of eggs is located in the middle of the flatbed of a pickup truck as the truck negotiates an unbanked curve in the road. The curve may be regarded as an arc of a circle of radius 35.0 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?

Narayan Hari
Narayan Hari
Numerade Educator
08:43

Problem 10

A car initially traveling eastward turns north by traveling in a circular path at uniform speed as shown in Figure P6.10. The length of the arc $A B C$ is 235 m , and the car completes the turn in 36.0 s . (a) What is the acceleration when the car is at $B$ located at an angle of $35.0^{\circ}$ ? Express your answer in terms of the unit vectors $\hat{\mathbf{i}}$ and $\hat{\mathbf{j}}$. Determine (b) the car's average speed and (c) its average acceleration during the $36.0-\mathrm{s}$ interval.

Vishal Gupta
Vishal Gupta
Numerade Educator
06:57

Problem 11

A $4.00-\mathrm{kg}$ object is attached to a vertical rod by two strings as shown in Figure P6.11. The object rotates in a horizontal circle at constant speed $6.00 \mathrm{~m} / \mathrm{s}$. Find the tension in (a) the upper string and (b) the lower string.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:11

Problem 12

A hawk flies in a horizontal arc of radius 12.0 m at a constant speed of $4.00 \mathrm{~m} / \mathrm{s}$. (a) Find its centripetal acceleration. (b) It continues to fly along the same horizontal arc but increases its speed at the rate of $1.20 \mathrm{~m} / \mathrm{s}^2$. Find the acceleration (magnitude and direction) under these conditions.

Vishal Gupta
Vishal Gupta
Numerade Educator
03:51

Problem 13

A $40.0-\mathrm{kg}$ child swings in a swing supported by two chains, each 3.00 m long. The tension in each chain at the lowest point is 350 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
Vishal Gupta
Numerade Educator
01:43

Problem 14

A roller-coaster car (Fig. P6.14) has a mass of 500 kg when fully loaded with passengers. (a) If the vehicle has a speed of $20.0 \mathrm{~m} / \mathrm{s}$ at point (A), what is the force exerted by the track on the car at this point? (b) What is the maximum speed the vehicle can have at point (B) and still remain on the track?

Narayan Hari
Narayan Hari
Numerade Educator
02:19

Problem 15

$\Delta$ Tarzan $(m=85.0 \mathrm{~kg})$ tries to cross a river by swinging on a vine. The vine is 10.0 m long, and his speed at the bottom of the swing (as he just clears the water) will be $8.00 \mathrm{~m} / \mathrm{s}$. Tarzan doesn't know that the vine has a breaking strength of 1000 N . Does he make it across the river safely?

Vishal Gupta
Vishal Gupta
Numerade Educator
06:27

Problem 16

One end of a cord is fixed and a small $0.500-\mathrm{kg}$ object is attached to the other end, where it swings in a section of a vertical circle of radius 2.00 m as shown in Figure 6.9 . When $\theta=20.0^{\circ}$, the speed of the object is $8.00 \mathrm{~m} / \mathrm{s}$. At this instant, find (a) the tension in the string, (b) the tangential and radial components of acceleration, and (c) the total acceleration. (d) Is your answer changed if the object is swinging up instead of swinging down? Explain.

Vishal Gupta
Vishal Gupta
Numerade Educator
01:32

Problem 17

A pail of water is rotated in a vertical circle of radius 1.00 m . What is the pail's minimum speed at the top of the circle if no water is to spill out?

Vishal Gupta
Vishal Gupta
Numerade Educator
07:05

Problem 18

A roller coaster at 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. P6.18). The cars ride on the inside of the loop at the top, and the speeds are fast enough to ensure that the cars remain on the track. The biggest loop is 40.0 m high, with a maximum speed of $31.0 \mathrm{~m} / \mathrm{s}$ (nearly $70 \mathrm{mi} / \mathrm{h}$ ) at the bottom. Suppose the speed at the top is $13.0 \mathrm{~m} / \mathrm{s}$ and the corresponding centripetal acceleration 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 m . If the cars have the same speed, $18.0 \mathrm{~m} / \mathrm{s}$ at the top, what is the centripetal acceleration at the top? Comment on the normal force at the top in this situation.

Vishal Gupta
Vishal Gupta
Numerade Educator
03:19

Problem 19

An object of mass 5.00 kg , attached to a spring scale, rests on a frictionless, horizontal surface as shown in Figure P6.19. The spring scale, attached to the front end of a boxcar, has a constant reading of 18.0 N when the car is in motion. (a) The spring scale reads zero when the car is at rest. Determine the acceleration of the car. (b) What constant reading will the spring scale show if the car moves with constant velocity? (c) Describe the forces on the object as observed by someone in the car and by someone at rest outside the car.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:23

Problem 20

A small container of water is placed on a carousel inside a microwave oven at a radius of 12.0 cm from the center. The turntable rotates steadily, turning one revolution each 7.25 s . What angle does the water surface make with the horizontal?

Bret Rosen
Bret Rosen
Numerade Educator
04:04

Problem 21

A $0.500-\mathrm{kg}$ object is suspended from the ceiling of an accelerating boxcar as shown in Figure 6.12. Taking $a=3.00 \mathrm{~m} / \mathrm{s}^2$, find (a) the angle that the string makes with the vertical and (b) the tension in the string.

Vishal Gupta
Vishal Gupta
Numerade Educator
04:46

Problem 22

A student stands in an elevator that is continuously accelerating upward with acceleration $a$. Her backpack is sitting on the floor next to the wall. The width of the elevator car is $L$. The student gives her backpack a quick kick at $t=0$, imparting to it speed $v$ and making it slide across the elevator floor. At time $t$, the backpack hits the opposite wall. Find the coefficient of kinetic friction $\mu_k$ between the backpack and the elevator floor.

Vishal Gupta
Vishal Gupta
Numerade Educator
04:35

Problem 23

A person stands on a scale in an elevator. As the elevator starts, the scale has a constant reading of 591 N . Later, as the elevator stops, the scale reading is 391 N . Assume the magnitude of the acceleration is the same during starting and stopping. Determine (a) the weight of the person, (b) the person's mass, and (c) the acceleration of the elevator.

Vishal Gupta
Vishal Gupta
Numerade Educator
03:03

Problem 24

A child on vacation wakes up. She is lying on her back. The tension in the muscles on both sides of her neck is 55.0 N as she raises her head to look past her toes and out the motel window. Finally it is not raining! Ten minutes later she is screaming feet first down a water slide at terminal speed $5.70 \mathrm{~m} / \mathrm{s}$, riding high on the outside wall of a horizontal curve of radius 2.40 m (Fig. P6.24). She raises her head to look forward past her toes. Find the tension in the muscles on both sides of her neck.

Vishal Gupta
Vishal Gupta
Numerade Educator
01:10

Problem 25

A plumb bob does not hang exactly along a line directed to the center of the Earth's rotation. How much does the plumb bob deviate from a radial line at $35.0^{\circ}$ north latitude? Assume the Earth is spherical.

Mayukh Banik
Mayukh Banik
Numerade Educator
02:54

Problem 26

A skydiver of mass 80.0 kg jumps from a slow-moving aircraft and reaches a terminal speed of $50.0 \mathrm{~m} / \mathrm{s}$. (a) What is the acceleration of the skydiver when her speed is $30.0 \mathrm{~m} / \mathrm{s}$ ? What is the drag force on the skydiver when her speed is (b) $50.0 \mathrm{~m} / \mathrm{s}$ ? (c) When it is $30.0 \mathrm{~m} / \mathrm{s}$ ?

Ajay Singhal
Ajay Singhal
Numerade Educator
05:31

Problem 27

A small piece of Styrofoam packing material is dropped from a height of 2.00 m above the ground. Until it reaches terminal speed, the magnitude of its acceleration is given by $a=g-b u$. After falling 0.500 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 $t=0$ ? (c) What is the acceleration when the speed is $0.150 \mathrm{~m} / \mathrm{s}$ ?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
05:24

Problem 28

(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 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?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:44

Problem 29

Calculate the force required to pull a copper ball of radius 2.00 cm upward through a fluid at the constant speed $9.00 \mathrm{~cm} / \mathrm{s}$. Take the drag force to be proportional to the speed, with proportionality constant $0.950 \mathrm{~kg} / \mathrm{s}$. Ignore the buoyant force.

Luis Mendoza
Luis Mendoza
Numerade Educator
01:38

Problem 30

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
Luis Mendoza
Numerade Educator
05:44

Problem 31

A small, spherical bead of mass 3.00 g is released from rest at $t=0$ in a bottle of liquid shampoo. The terminal speed is observed to be $\pi_T=2.00 \mathrm{~cm} / \mathrm{s}$. Find (a) the value of the constant $b$ in Equation 6.2, (b) the time $t$ at which the bead reaches $0.632 v_p$, and (c) the value of the resistive force when the bead reaches terminal speed.

Vishal Gupta
Vishal Gupta
Numerade Educator
06:39

Problem 32

Review problem. An undercover police agent pulls a rubber squeegee down a very tall vertical window. The squeegee has mass 160 g and is mounted on the end of a light rod. The coefficient of kinetic friction between the squeegee and the dry glass is 0.900 . The agent presses it against the window with a force having a horizontal component of 4.00 N . (a) If she pulls the squeegee down the window at constant velocity, what vertical force component must she exert? (b) The agent increases the downward force component by $25.0 \%$, but all other forces remain the same. Find the acceleration of the squeegee in this situation. (c) The squeegee then moves into a wet portion of the window, where its motion is now resisted by a fluid drag force proportional to its velocity according to $R=-(20.0 \mathrm{~N} \cdot \mathrm{~s} / \mathrm{m}) v$. Find the terminal velocity that the squeegee approaches, assuming the agent exerts the same force described in part (b).

Keshav Singh
Keshav Singh
Numerade Educator
20:14

Problem 33

A $9.00-\mathrm{kg}$ object starting from rest falls through a viscous medium and experiences a resistive force $\overrightarrow{\mathbf{R}}=-b \overrightarrow{\mathbf{v}}$, where $\overrightarrow{\mathbf{v}}$ is the velocity of the object. The object reaches one-half of its terminal speed in 5.54 s . (a) Determine the terminal speed. (b) At what time is the speed of the object three-fourths of the terminal speed? (c) How far has the object traveled in the first 5.54 s of motion?

Donald Albin
Donald Albin
Numerade Educator
01:05

Problem 34

Consider an object on which the net force is a resistive force proportional to the square of its speed. For example, assume the resistive force acting on a speed skater is $f=-k m v^2$, 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 $\nu_0$ 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 $v(t)=v_0 /\left(1+k t v_0\right)$. This problem also provides the background for the next two problems.

Mayukh Banik
Mayukh Banik
Numerade Educator
00:51

Problem 35

(a) Use the result of Problem 34 to find the position $x$ as a function of time for an object of mass $m$, located at $x=0$ and moving with velocity $v_0 \hat{\mathbf{i}}$ at time $t=0$, and thereafter experiencing a net force $-k m v^2 \hat{\mathbf{i}}$. (b) Find the object's velocity as a function of position.

Mayukh Banik
Mayukh Banik
Numerade Educator
00:38

Problem 36

At major league baseball games it is commonplace to flash on the scoreboard a speed for each pitch. This speed is determined with a radar gun aimed by an operator positioned behind home plate. The gun uses the Doppler shift of microwaves reflected from the baseball, as we will study in Chapter 39. The gun determines the speed at some particular point on the baseball's path, depending on when the operator pulls the trigger. Because the ball is subject to a drag force due to air, it slows as it travels 18.3 m toward the plate. Use the result of Problem 35 (b) to find how much its speed decreases. Suppose the ball leaves the pitcher's hand at $90.0 \mathrm{mi} / \mathrm{h}=$ $40.2 \mathrm{~m} / \mathrm{s}$. Ignore its vertical motion. Use data on baseballs from Example 6.11 to determine the speed of the pitch when it crosses the plate.

Mayukh Banik
Mayukh Banik
Numerade Educator
05:44

Problem 37

A The driver of a motorboat cuts its engine when its speed is $10.0 \mathrm{~m} / \mathrm{s}$ and coasts to rest. The equation describing the motion of the motorboat during this period is $v=v_i e^{-a t}$, where $v$ is the speed at time $t, v_i$ is the initial speed, and $c$ is a constant. At $t=20.0 \mathrm{~s}$, the speed is $5.00 \mathrm{~m} / \mathrm{s}$. (a) Find the constant $c$ (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.

Donald Albin
Donald Albin
Numerade Educator
02:50

Problem 38

You can feel a force of air drag on your hand if you stretch your arm out of an open window of a rapidly moving car. Note Do not endanger yourself. What is the order of magnitude of this force? In your solution, state the quantities you measure or estimate and their values.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
View

Problem 39

An object of mass $m$ is projected forward along the $x$ axis with initial speed $v_0$. The only force on it is a resistive force proportional to its velocity, given by $\overrightarrow{\mathbf{R}}=-b \overrightarrow{\mathbf{v}}$. For concreteness, you could visualize an airplane with pontoons landing on a lake. Newton's second law applied to the object is $-b v \hat{\mathbf{i}}=m(d v / d t) \hat{\mathbf{i}}$. From the fundamental theorem of calculus, this differential equation implies that the speed changes according to

$$
\int_{\text {start }}^{a \text { later point }} \frac{d v}{v}=-\frac{b}{m} \int_0^t d t
$$

Carry out the integrations to determine the speed of the object as a function of time. Sketch a graph of the speed as a function of time. Does the object come to a complete stop after a finite interval of time? Does the object travel a finite distance in stopping?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
02:04

Problem 40

A $0.400-\mathrm{kg}$ object is swung in a vertical circular path on a string 0.500 m long. If its speed is $4.00 \mathrm{~m} / \mathrm{s}$ at the top of the circle, what is the tension in the string there?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:52

Problem 41

(a) A luggage carousel at an airport has the form of a section of a large cone, steadily rotating about its vertical axis. Its metallic surface slopes downward toward the outside, making an angle of $20.0^{\circ}$ with the horizontal. A piece of luggage having mass 30.0 kg is placed on the carousel, 7.46 m from the axis of rotation. The travel bag goes around once in 38.0 s . Calculate the force of static friction exerted by the carousel on the bag. (b) The drive motor is shifted to turn the carousel at a higher constant rate of rotation, and the piece of luggage is bumped to another position, 7.94 m from the axis of rotation. Now going around once in every 34.0 s , the bag is on the verge of slipping. Calculate the coefficient of static friction between the bag and the carousel.

Mayukh Banik
Mayukh Banik
Numerade Educator
04:13

Problem 42

In a home laundry dryer, a cylindrical tub containing wet clothes is rotated steadily about a horizontal axis as shown in Figure P6.42. So that the clothes will dry uniformly, they are made to tumble. The rate of rotation of the smooth-walled tub is chosen so that a small piece of cloth will lose contact with the tub when the cloth is at an angle of $68.0^{\circ}$ above the horizontal. If the radius of the tub is 0.380 m , what rate of revolution is needed?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:22

Problem 43

We will study the most important work of Nobel laureate Arthur Compton in Chapter 40. Disturbed by speeding cars outside the physics building at Washington University in St. Louis, Compton designed a speed bump and had it installed. Suppose a $1800-\mathrm{kg}$ car passes over a bump in a roadway that follows the arc of a circle of radius 20.4 m as shown in Figure P6.48. (a) What force does the road exert on the car as the car passes the highest point of the bump if it travels at $30.0 \mathrm{~km} / \mathrm{h}$ ? (b) What If? What is the maximum speed the car can have as it passes this highest point without losing contact with the road?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:28

Problem 44

A car of mass $m$ passes over a bump in a road that follows the arc of a circle of radius $R$ as shown in Figure P6.43. (a) What force does the road exert on the car as the car passes the highest point of the bump if it travels at a speed $v ?$ (b) What If? What is the maximum speed the car can have as it passes this highest point without losing contact with the road?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:14

Problem 45

Interpret the graph in Figure 6.16(b). Proceed as follows. (a) Find the slope of the straight line, including its units. (b) From Equation $6.6, R=\frac{1}{2} D \rho A v^2$, identify the theoretical slope of a graph of resistive force versus squared speed. (c) Set the experimental and theoretical slopes equal to each other and proceed to calculate the drag coefficient of the filters. Use the value for the density of air listed on the book's endpapers. Model the crosssectional area of the filters as that of a circle of radius 10.5 cm . (d) Arbitrarily choose the eighth data point on the graph and find its vertical separation from the line of best fit. Express this scatter as a percentage. (e) In a short paragraph, state what the graph demonstrates and compare what it demonstrates to the theoretical prediction. You will need to make reference to the quantities plotted on the axes, to the shape of the graph line, to the data points, and to the results of parts (c) and (d).

Mayukh Banik
Mayukh Banik
Numerade Educator
06:13

Problem 46

A basin surrounding a drain has the shape of a circular cone opening upward, having everywhere an angle of $35.0^{\circ}$ with the horizontal. A $25.0-\mathrm{g}$ ice cube is set sliding around the cone without friction in a horizontal circle of radius $R$. (a) Find the speed the ice cube must have as it depends on $R$ (b) Is any piece of data unnecessary for the solution? Suppose $R$ is made two times larger. (c) Will the required speed increase, decrease, or stay constant? If it changes, by what factor? (d) Will the time required for each revolution increase, decrease, or stay constant? If it changes, by what factor? (e) Do the answers to parts (c) and (d) seem contradictory? Explain how they are consistent.

Ajay Singhal
Ajay Singhal
Numerade Educator
04:57

Problem 47

Suppose the boxcar of Figure 6.12 is moving with constant acceleration $a$ up a hill that makes an angle $\phi$ with the horizontal. If the pendulum makes a constant angle $\theta$ with the perpendicular to the ceiling, what is $a$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
06:26

Problem 48

The pilot of an airplane executes a constant-speed loop-the-loop maneuver in a vertical circle. The speed of the airplane is $300 \mathrm{mi} / \mathrm{h}$; the radius of the circle is 1200 ft . (a) What is the pilot's apparent weight at the lowest point if his true weight is 160 lb ? (b) What is his apparent weight at the highest point? (c) What If? Describe how the pilot could experience weightlessness if both the radius and the speed can be varied. Note: His apparent weight is equal to the magnitude of the force exerted by the seat on his body.

Vishal Gupta
Vishal Gupta
Numerade Educator
04:01

Problem 49

$\Delta$ Because the Earth rotates about its axis, a point on the equator experiences a centripetal acceleration of $0.0387 \mathrm{~m} / \mathrm{s}^2$, whereas a point at the poles experiences no centripetal acceleration. (a) Show that at the equator the gravitational force on an object must exceed the normal force required to support the object. That is, show that the object's true weight exceeds its apparent weight. (b) What is the apparent weight at the equator and at the poles of a person having a mass of 75.0 kg ? Assume the Earth is a uniform sphere and take $g=9.800 \mathrm{~m} / \mathrm{s}^2$.

Vishal Gupta
Vishal Gupta
Numerade Educator
04:38

Problem 50

An air 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 a load of mass $m_2$ is tied to the string (Fig. P6.50). The suspended load remains in equilibrium while the puck on the tabletop revolves. What are (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 somewhat increased by placing an additional load on it. (e) Qualitatively describe what will happen in the motion of the puck if the value of $m_2$ is instead decreased by removing a part of the hanging load.

Vishal Gupta
Vishal Gupta
Numerade Educator
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Problem 51

While learning to drive, you are in a $1200-\mathrm{kg}$ car moving at $20.0 \mathrm{~m} / \mathrm{s}$ across a large, vacant, level parking lot. Suddenly you realize you are heading straight toward a brick sidewall of a large supermarket and are in danger of running into it. The pavement can exert a maximum horizontal force of 7000 N on the car. (a) Explain why you should expect the force to have a well-defined maximum value. (b) Suppose you apply the brakes and do not turn the steering wheel. Find the minimum distance you must be from the wall to avoid a collision. (c) If you do not brake but instead maintain constant speed and turn the steering wheel, what is the minimum distance you must be from the wall to avoid a collision? (d) Which method, (b) or (c), is better for avoiding a collision? Or, should you use both the brakes and the steering wheel, or neither? Explain. (e) Does the conclusion in part (d) depend on the numerical values given in this problem, or is it true in general? Explain.

Victor Salazar
Victor Salazar
Numerade Educator
07:42

Problem 52

Suppose a Ferris wheel rotates four times each minute. It carries each car around a circle of diameter 18.0 m . (a) What is the centripetal acceleration of a rider? What force does the seat exert on a $40.0-\mathrm{kg}$ rider (b) at the lowest point of the ride and (c) at the highest point of the ride? (d) What force (magnitude and direction) does the seat exert on a rider when the rider is halfway between top and bottom?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:06

Problem 53

An amusement park ride consists of a rotating circular platform 8.00 m in diameter from which $10.0-\mathrm{kg}$ seats are suspended at the end of $2.50-\mathrm{m}$ massless chains (Fig. P6.53). When the system rotates, the chains make an angle $\theta=28.0^{\circ}$ with the vertical. (a) What is the speed of each seat? (b) Draw a free-body diagram of a $40.0-\mathrm{kg}$ child riding in a seat and find the tension in the chain.

Vishal Gupta
Vishal Gupta
Numerade Educator
00:53

Problem 54

A piece of putty is initially located at point $A$ on the rim of a grinding wheel rotating about a horizontal axis. The putty is dislodged from point $A$ when the diameter through $A$ is horizontal. It then rises vertically and returns to $A$ at the instant the wheel completes one revolution. (a) Find the speed of a point on the rim of the wheel in terms of the acceleration due to gravity and the radius $R$ of the wheel. (b) If the mass of the putty is $m$, what is the magnitude of the force that held it to the wheel?

Mayukh Banik
Mayukh Banik
Numerade Educator
06:26

Problem 55

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. P6.55). The coefficient of static friction 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 $T=$ ( $\left.4 \pi^2 R \mu_s / \mathrm{g}\right)^{1 / 2}$. (b) Obtain a numerical value for $T$, taking $R=4.00 \mathrm{~m}$ and $\mu_s=0.400$. How many revolutions per minute does the cylinder make? (c) 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? (d) If instead the cylinder's rate of revolution is 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?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:41

Problem 56

An example of the Coriolis effect. Suppose air resistance is negligible for a golf ball. A golfer tees off from a location precisely at $\phi_i=35.0^{\circ}$ north latitude. He hits the ball due south, with range 285 m . The ball's initial velocity is at $48.0^{\circ}$ above the horizontal. (a) For how long is the ball in flight? The cup is due south of the golfer's location, and he would have a hole in one if the Earth were not rotating. The Earth's rotation makes the tee move in a circle of radius $R_E \cos \phi_i=\left(6.37 \times 10^6 \mathrm{~m}\right) \cos 35.0^{\circ}$ as shown in Figure P6.56. The tee completes one revolution each day. (b) Find the eastward speed of the tee, relative to the stars. The hole is also moving east, but it is 285 m farther south and therefore at a slightly lower latitude $\phi_f$. Because the hole moves in a slightly larger circle, its speed must be greater than that of the tee. (c) By how much does the hole's speed exceed that of the tee? During the time interval the ball is in flight, it moves upward and downward as well as southward with the projectile motion you studied in Chapter 4, but it also moves eastward with the speed you found in part (b). The hole moves to the east at a faster speed, however, pulling ahead of the ball with the relative speed you found in part (c). (d) How far to the west of the hole does the ball land?

Mayukh Banik
Mayukh Banik
Numerade Educator
20:35

Problem 57

A car rounds a banked curve as shown in Figure 6.5. 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 bank. (b) Find the minimum value for $\mu_s$ such that the minimum speed is zero. (c) What is the range of speeds possible if $R=100 \mathrm{~m}, \theta=10.0^{\circ}$, and $\mu_s=0.100$ (slippery conditions)?

Donald Albin
Donald Albin
Numerade Educator
06:09

Problem 58

A single bead can slide with negligible friction on a stiff wire that has been bent into a circular loop of radius 15.0 cm as shown in Figure P6.58. The circle is always in a vertical plane and rotates steadily about its vertical diameter with (a) 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. 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, taking the period of the circle's rotation as 0.850 s . (c) Describe how the solution to part (b) is fundamentally different from the solution to part (a). For any period or loop size, is there always an angle at which the bead can stand still relative to the loop? Are there ever more than two angles? Arnold Arons suggested the idea for this problem.

Ajay Singhal
Ajay Singhal
Numerade Educator
03:06

Problem 59

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 mm . For (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
Luis Mendoza
Numerade Educator
08:08

Problem 60

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.
$$
\begin{array}{lrrrrr}
t(\mathbf{s}) & \boldsymbol{d}(\mathbf{f t}) & t(\mathrm{~s}) & \boldsymbol{d}(\mathbf{f t}) & t(\mathbf{s}) & d(\mathbf{f t}) \\
\hline 0 & 0 & 7 & 652 & 14 & 1881 \\
1 & 16 & 8 & 808 & 15 & 2005 \\
2 & 62 & 9 & 971 & 16 & 2179 \\
3 & 138 & 10 & 1138 & 17 & 2353 \\
4 & 242 & 11 & 1309 & 18 & 2527 \\
5 & 366 & 12 & 1483 & 19 & 2701 \\
6 & 504 & 13 & 1657 & 20 & 2875 \\
\hline
\end{array}
$$

Donald Albin
Donald Albin
Numerade Educator
02:58

Problem 61

A model airplane of mass 0.750 kg flies with a speed of $35.0 \mathrm{~m} / \mathrm{s}$ in a horizontal circle at the end of a $60.0-\mathrm{m}$ control wire. Compute the tension in the wire, assuming it makes a constant angle of $20.0^{\circ}$ with the horizontal. The forces exerted on the airplane are the pull of the control wire, the gravitational force, and aerodynamic lift that acts at $20.0^{\circ}$ inward from the vertical as shown in Figure P6.61.

Donald Albin
Donald Albin
Numerade Educator
03:55

Problem 62

Galileo thought about whether acceleration should be defined as the rate of change of velocity over time or as the rate of change in velocity over distance. He chose the former, so let us use the name "vroomosity" for the rate of change of velocity in space. For motion of a particle on a straight line with constant acceleration, the equation $v=$ $v_i+a t$ gives its velocity $v$ as a function of time. Similarly, for a particle's linear motion with constant vroomosity $k$, the equation $v=v_i+k x$ gives the velocity as a function of the position $x$ if the particle's speed is $v_i$ at $x=0$. (a) Find the law describing the total force acting on this object, of mass $m$. Describe an example of such a motion, or explain why such a motion is unrealistic. Consider (b) the possibility of $k$ positive and also (c) the possibility of $k$ negative.

Katie Mcalpine
Katie Mcalpine
Numerade Educator