Answer key

(1)

Assignment 11

Due: 11:59pm on Wednesday, April 30, 2014

You will receive no credit for items you complete after the assignment is due. Grading Policy

Conceptual Question 13.2

The gravitational force of a star on orbiting planet 1 is . Planet 2, which is twice as massive as planet 1 and orbits at twice the distance from the star, experiences gravitational force .

Part A

What is the ratio ?

ANSWER:

Correct

Conceptual Question 13.3

A 1500 satellite and a 2200 satellite follow exactly the same orbit around the earth.

Part A

What is the ratio of the force on the first satellite to that on the second satellite?

ANSWER:

Correct

F

1

F

2 F1 F₂ = 2 F₁ F2

kg

F₁ F2 = 0.682 F₁ F₂

(2)

Part B

What is the ratio of the acceleration of the first satellite to that of the second satellite?

ANSWER:

Correct

Problem 13.2

The centers of a 15.0 lead ball and a 90.0 lead ball are separated by 9.00 .

Part A

What gravitational force does each exert on the other? Express your answer with the appropriate units. ANSWER:

Correct

Part B

What is the ratio of this gravitational force to the weight of the 90.0 ball?

ANSWER: a₁ a₂ = 1 a₁ a2

kg

g

cm

1.11×10−8

N

g

1.26×10−8

(3)

Correct

Problem 13.6

The space shuttle orbits 310 above the surface of the earth.

Part A

What is the gravitational force on a 7.5 sphere inside the space shuttle?

Express your answer with the appropriate units. ANSWER:

Correct

± A Satellite in Orbit

A satellite used in a cellular telephone network has a mass of 2310 and is in a circular orbit at a height of 650 above the surface of the earth.

Part A

What is the gravitational force on the satellite?

Take the gravitational constant to be = 6.67×10−11 , the mass of the earth to be = 5.97×1024 , and the radius of the Earth to be = 6.38×106 .

Express your answer in newtons.

Hint 1. How to approach the problem

Use the equation for the law of gravitation to calculate the force on the satellite. Be careful about the units when performing the calculations.

km

kg

= 67.0

F

e on s

N

kg

km

F

grav

G

N ⋅

m

2

/k

_g

2

_m

e

kg

r

e

m

(4)

Hint 2. Law of gravitation

According to Newton's law of gravitation, , where is the gravitational constant, and are the masses of the two objects, and is the distance between the centers of mass of the two objects.

Hint 3. Calculate the distance between the centers of mass

What is the distance from the center of mass of the satellite to the center of mass of the earth? Express your answer in meters.

ANSWER:

Correct

Part B

What fraction is this of the satellite's weight at the surface of the earth? Take the free-fall acceleration at the surface of the earth to be = 9.80 .

All you need to do is to take the ratio of the gravitational force on the satellite to the weight of the satellite at ground level. There are two ways to do this, depending on how you define the force of gravity at the surface of the earth.

ANSWER:

F

= G

m

1

m

2

/

r

2

G

m

1

m

2

r

= 7.03×106

r

m

= 1.86×104

F

grav

N

g

m/s

2 0.824

(5)

Correct

Although it is easy to find the weight of the satellite using the constant acceleration due to gravity, it is instructional to consider the weight calculated using the law of gravitation:

. Dividing the gravitational force on the satellite by , we find that the ratio of the forces due to the earth's gravity is simply the square of the ratio of the earth's radius to the sum of the earth's radius and the height of the orbit of the satellite above the earth, . This will also be the fraction of the weight of, say, an astronaut in an orbit at the same altitude. Notice that an astronaut's weight is never zero. When people speak of "weightlessness" in space, what they really mean is "free fall."

Problem 13.8

Part A

What is the free-fall acceleration at the surface of the moon? Express your answer with the appropriate units.

ANSWER:

Correct

Part B

What is the free-fall acceleration at the surface of the Jupiter? Express your answer with the appropriate units.

ANSWER:

Correct

w

= G

m

e

m/

r

2e

F

grav

= G

m

e

m/( + h

r

e

)

2

w

[ /( + h)

r

e

r

e

]

2 = 1.62

g

_moon m_s₂ = 25.9

g

_Jupiter m s2

(6)

Enhanced EOC: Problem 13.14

A rocket is launched straight up from the earth's surface at a speed of 1.90×104 . You may want to review ( pages 362 - 365) .

For help with math skills, you may want to review: Mathematical Expressions Involving Squares Part A

What is its speed when it is very far away from the earth? Express your answer with the appropriate units.

What is conserved in this problem?

What is the rocket's initial kinetic energy in terms of its unknown mass, ? What is the rocket's initial gravitational potential energy in terms of its unknown mass, ?

When the rocket is very far away from the Earth, what is its gravitational potential energy?

Using conservation of energy, what is the rocket's kinetic energy when it is very far away from the Earth? Therefore, what is the rocket's velocity when it is very far away from the Earth?

ANSWER:

Correct

Problem 13.13

Part A

m/s

m

1.54×104 m_s

(7)

What is the escape speed from Venus?

Correct

Problem 13.17

The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 4.2 earth years.

Part A

What is the asteroid's orbital radius?

Correct

Part B

What is the asteroid's orbital speed?

Express your answer with the appropriate units. ANSWER: = 10.4

v

escape km_s = 3.89×1011

R

m

= 1.85×104

v

m_s

(8)

Correct

Problem 13.32

Part A

At what height above the earth is the acceleration due to gravity 15.0% of its value at the surface? Express your answer with the appropriate units.

ANSWER:

Correct

Part B

What is the speed of a satellite orbiting at that height? Express your answer with the appropriate units. ANSWER:

Correct

Problem 13.36

Two meteoroids are heading for earth. Their speeds as they cross the moon's orbit are 2 . 1.01×107

m

4920 m_s

(9)

Part A

The first meteoroid is heading straight for earth. What is its speed of impact? Express your answer with the appropriate units.

ANSWER:

Correct

Part B

The second misses the earth by 5500 . What is its speed at its closest point? Express your answer with the appropriate units.

ANSWER:

Incorrect; Try Again

Problem 14.2

An air-track glider attached to a spring oscillates between the 11.0 mark and the 67.0 mark on the track. The glider completes 11.0 oscillations in 32.0 .

Part A

What is the period of the oscillations?

Express your answer with the appropriate units. = 11.3

v

1 km_s

km

=

v

2

cm

s

(10)

ANSWER:

Correct

Part B

What is the frequency of the oscillations?

Correct

Part C

What is the angular frequency of the oscillations? Express your answer with the appropriate units. ANSWER:

Correct

Part D

What is the amplitude?

Express your answer with the appropriate units. 2.91

s

0.344

Hz

(11)

ANSWER:

Correct

Part E

What is the maximum speed of the glider?

Correct

Good Vibes: Introduction to Oscillations

Learning Goal:

To learn the basic terminology and relationships among the main characteristics of simple harmonic motion.

Motion that repeats itself over and over is called periodic motion. There are many examples of periodic motion: the earth revolving around the sun, an elastic ball bouncing up and down, or a block attached to a spring oscillating back and forth.

The last example differs from the first two, in that it represents a special kind of periodic motion called simple harmonic motion. The conditions that lead to simple harmonic motion are as follows: There must be a position of stable equilibrium.

There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force is given by , where is the displacement from equilibrium and is a constant that depends on the properties of the oscillating system.

The resistive forces in the system must be reasonably small.

In this problem, we will introduce some of the basic quantities that describe oscillations and the relationships among them.

Consider a block of mass attached to a spring with force constant , as shown in the figure. The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at . If the

28.0

cm

60.5 cm_s

F ⃗

= −k

x⃗

k

m

k

(12)

block is pulled to the right a distance and then released, will be the amplitude of the resulting oscillations.

Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.

Part A

After the block is released from , it will ANSWER:

Correct

As the block begins its motion to the left, it accelerates. Although the restoring force decreases as the block approaches equilibrium, it still pulls the block to the left, so by the time the equilibrium position is reached, the block has gained some speed. It will, therefore, pass the equilibrium position and keep moving, compressing the spring. The spring will now be pushing the block to the right, and the block will slow down, temporarily coming to rest at .

After is reached, the block will begin its motion to the right, pushed by the spring. The block will pass the equilibrium position and continue until it reaches , completing one cycle of motion. The motion will then repeat; if, as we've assumed, there is no friction, the motion will repeat indefinitely.

The time it takes the block to complete one cycle is called the period. Usually, the period is denoted and is measured in seconds.

The frequency, denoted , is the number of cycles that are completed per unit of time: . In SI units, is measured in inverse seconds, or hertz ( ).

A

x

= A

remain at rest.

move to the left until it reaches equilibrium and stop there. move to the left until it reaches and stop there.

move to the left until it reaches and then begin to move to the right.

x

= −A

x

= −A

x

= −A

x

= −A

x

= A

T

f

= 1/T

f

Hz

(13)

Part B

If the period is doubled, the frequency is ANSWER:

Correct

Part C

An oscillating object takes 0.10 to complete one cycle; that is, its period is 0.10 . What is its frequency ?

Express your answer in hertz. ANSWER:

Correct

unchanged. doubled. halved.

s

f

= 10

f

Hz

(14)

Part D

If the frequency is 40 , what is the period ? Express your answer in seconds.

ANSWER:

Correct

The following questions refer to the figure that graphically depicts the oscillations of the block on the spring.

Note that the vertical axis represents the x coordinate of the oscillating object, and the horizontal axis represents time.

Part E

Which points on the x axis are located a distance from the equilibrium position? ANSWER:

Hz

T

= 0.025

T

s

(15)

Correct

Part F

Suppose that the period is . Which of the following points on the t axis are separated by the time interval ? ANSWER:

Correct

Now assume for the remaining Parts G - J, that the x coordinate of point R is 0.12 and the t coordinate of point K is 0.0050 .

Part G

What is the period ?

Express your answer in seconds.

In moving from the point to the point K, what fraction of a full wavelength is covered? Call that fraction . Then you can set . Dividing by the fraction will give the R only Q only both R and Q

T

K and L K and M K and P L and N M and P

m

s

T

t

= 0

a

aT

= 0.005 s

a

(16)

period .

ANSWER:

Correct

Part H

How much time does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement? Express your answer in seconds.

ANSWER:

Correct

Part I

What distance does the object cover during one period of oscillation? Express your answer in meters.

ANSWER:

Correct

Part J

T

= 0.02

T

s

t

= 0.01

t

s

d

= 0.48

d

m

(17)

Express your answer in meters. ANSWER:

Correct

Problem 14.4

Part A

What is the amplitude of the oscillation shown in the figure?

Express your answer to three significant figures and include the appropriate units.

ANSWER:

Correct

= 0.36

d

m

= 20.0

A

cm

(18)

Part B

What is the frequency of this oscillation?

Express your answer to two significant figures and include the appropriate units. ANSWER:

Correct

Part C

What is the phase constant?

Incorrect; Try Again

Problem 14.10

An air-track glider attached to a spring oscillates with a period of 1.50 . At the glider is 4.60 left of the equilibrium position and moving to the right at 33.4 .

Part A

What is the phase constant?

Express your answer to three significant figures and include the appropriate units. ANSWER: = 0.25

f

Hz

=

ϕ

₀

s

t

= 0 s

cm

cm/s

(19)

Incorrect; Try Again

Part B

This question will be shown after you complete previous question(s).

Part C

Part D

Problem 14.12

A 140 air-track glider is attached to a spring. The glider is pushed in 12.2 and released. A student with a stopwatch finds that 14.0 oscillations take 19.0 .

Part A

What is the spring constant?

=

ϕ

₀

(20)

Correct

Problem 14.14

The position of a 50 g oscillating mass is given by , where is in s. If necessary, round your answers to three significant figures. Determine:

Part A

The amplitude.

Express your answer to three significant figures and include the appropriate units. ANSWER:

Correct

Part B

The period.

Correct

Part C 3.00 _mN

x(t) = (2.0 cm)cos(10t − π/4)

t

2.00

cm

0.628

s

(21)

The spring constant.

Part D

The phase constant.

Incorrect; Try Again

Part E

Part F

(22)

Part H

Part I

Enhanced EOC: Problem 14.17

A spring with spring constant 16 hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down 4.0 and released. The ball makes 35 oscillations in 18 seconds.

You may want to review ( pages 389 - 391) . For help with math skills, you may want to review: Differentiation of Trigonometric Functions

Part A

What is its the mass of the ball?

Express your answer to two significant figures and include the appropriate units.

What is the period of oscillation? What is the angular frequency of the oscillations? How is the angular frequency related to the mass and spring constant? What is the mass?

N/m

cm

(23)

ANSWER:

Correct

Part B

What is its maximum speed?

What is the amplitude of the oscillations?

How is the maximum speed related to the amplitude of the oscillations and the angular frequency?

ANSWER:

Correct

Changing the Period of a Pendulum

A simple pendulum consisting of a bob of mass attached to a string of length swings with a period .

Part A

If the bob's mass is doubled, approximately what will the pendulum's new period be?

Hint 1. Period of a simple pendulum

The period of a simple pendulum of length is given by = 110

m

g

= 49

v

max cm_s

m

L

T

L

(24)

,

where is the acceleration due to gravity.

ANSWER:

Correct

Part B

If the pendulum is brought on the moon where the gravitational acceleration is about , approximately what will its period now be?

Recall the formula of the period of a simple pendulum. Since the gravitational acceleration appears in the denominator, the period must increase when the gravitational acceleration decreases. ANSWER:

T

= 2π

L g

−−

√

g

T

/2

T

2 √

2T

g/6

T

/6

T

/ 6

√

T

6 √

6T

(25)

Correct

Part C

If the pendulum is taken into the orbiting space station what will happen to the bob?

Recall that the oscillations of a simple pendulum occur when a pendulum bob is raised above its equilibrium position and let go, causing the pendulum bob to fall. The gravitational force acts to bring the bob back to its equilibrium position. In the space station, the earth's gravity acts on both the station and everything inside it, giving them the same acceleration. These objects are said to be in free fall.

ANSWER:

Correct

In the space station, where all objects undergo the same acceleration due to the earth's gravity, the tension in the string is zero and the bob does not fall relative to the point to which the string is attached.

Problem 14.20

A 175 ball is tied to a string. It is pulled to an angle of 8.0 and released to swing as a pendulum. A student with a stopwatch finds that 15 oscillations take 13 .

Part A

How long is the string?

Express your answer to two significant figures and include the appropriate units. It will continue to oscillate in a vertical plane with the same period.

It will no longer oscillate because there is no gravity in space.

It will no longer oscillate because both the pendulum and the point to which it is attached are in free fall. It will oscillate much faster with a period that approaches zero.

(26)

ANSWER:

Correct

Problem 14.22

Part A

What is the length of a pendulum whose period on the moon matches the period of a 2.1- -long pendulum on the earth? Express your answer to two significant figures and include the appropriate units.

ANSWER:

Correct

Problem 14.42

An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk ( = 0.17 ) driven back and forth in SHM at by an electromagnetic coil.

Part A

The maximum restoring force that can be applied to the disk without breaking it is 4.4×104 . What is the maximum oscillation amplitude that won't rupture the disk? Express your answer to two significant figures and include the appropriate units.

ANSWER: = 19

L

cm

m

= 0.35

l

moon

m

g

1.0 MHz

N

= 6.6

a

max

µm

(27)

Correct

Part B

What is the disk's maximum speed at this amplitude?

Correct

Score Summary:

Your score on this assignment is 81.4%.

You received 117.25 out of a possible total of 144 points. = 41

(28)

Assignment 6

Due: 11:59pm on Friday, March 7, 2014

Conceptual Question 7.7

A small car is pushing a large truck. They are speeding up.

Part A

Is the force of the truck on the car larger than, smaller than, or equal to the force of the car on the truck? ANSWER:

Correct

Conceptual Question 7.12

The figure shows two masses at rest. The string is massless and the pulley is frictionless. The spring scale reads in . Assume that = 4 .

The force of the truck on the car is larger than the force of the car on the truck. The force of the truck on the car is equal to the force of the car on the truck. The force of the truck on the car is smaller than the force of the car on the truck.

(29)

Part A

What is the reading of the scale?

Express your answer to one significant figure and include the appropriate units. ANSWER:

Correct

Problem 7.1

A weightlifter stands up at constant speed from a squatting position while holding a heavy barbell across his shoulders.

Part A

Draw a free-body diagram for the barbells.

Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded.

ANSWER: = 4

(30)

Correct

Part B

Draw a free-body diagram for the weight lifter.

Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded.

(31)

Correct

Problem 7.6

Block A in the figure is sliding down the incline. The rope is massless, and the massless pulley turns on frictionless bearings, but the surface is not frictionless. The rope and the pulley are among the interacting objects, but you'll have to decide if they're part of the system.

(32)

Part A

Draw a free-body diagram for the block A.

The orientation of your vectors will be graded. The exact length of your vectors will not be graded. ANSWER:

(33)

Part B

Draw a free-body diagram for the block B.

The orientation of your vectors will be graded. The exact length of your vectors will not be graded. ANSWER:

Correct

A Space Walk

(34)

An astronaut is taking a space walk near the shuttle when her safety tether breaks. What should the astronaut do to get back to the shuttle?

Newton's 3rd law tells us that forces occur in pairs. Within each pair, the forces, often called action and reaction, have equal magnitude and opposite direction. Which of the actions suggested in the problem will result in the force pushing the astronaut back to the shuttle?

ANSWER:

Correct

As the astronaut throws the tool away from the shuttle, she exerts a force in the direction away from the shuttle. Then, by Newton's 3rd law, the tool will exert an opposite force on her. Thus, as she throws the tool, a force directed toward the shuttle will act on the astronaut. Newton's 2nd law stipulates that the astronaut would acquire an acceleration toward the shuttle.

Part B

Assuming that the astronaut can throw any tool with the same force, what tool should be thrown to get back to the shuttle as quickly as possible? You should consider how much mass is left behind as the object is thrown as well as the mass of the object itself.

Recall that the force acting on the astronaut is equal in magnitude and opposite in direction to the force that she exerts on the tool.

Hint 2. Newton's 2nd law

Newton's 2nd law states that . If force is held constant, then acceleration is inversely proportional to mass. For example, when the same force is applied to objects of different mass, the object with the largest mass will experience the smallest acceleration.

ANSWER:

Attempt to "swim" toward the shuttle. Take slow steps toward the shuttle.

Take a tool from her tool belt and throw it away from the shuttle.

Take the portion of the safety tether still attached to her belt and throw it toward the shuttle.

(35)

Correct

The force that acts on the astronaut must equal in magnitude the force that she exerts on the tool. Therefore, if she exerts the same force on any tool, the force acting on her will be

independent of the mass of the tool. However, the acceleration that the astronaut would acquire is inversely proportional to her mass since she is acted upon by a constant force. If she throws the tool with the largest mass, the remaining mass (the astronaut plus her remaining tools) would be smallest—and the acceleration the greatest!

Part C

If the astronaut throws the tool with a force of 16.0 , what is the magnitude of the acceleration of the astronaut during the throw? Assume that the total mass of the astronaut after she throws the tool is 80.0 .

Express your answer in meters per second squared.

Hint 1. Find the force acting on the astronaut

What is the magnitude of the force acting on the astronaut as she throws the tool? Express your answer in newtons.

ANSWER:

Hint 2. Newton's 2nd law

An object of mass acted upon by a net force has an acceleration given by .

ANSWER:

The tool with the smallest mass. The tool with the largest mass.

Any tool, since the mass of the tool would make no difference.

N

a

kg

F

= 16.0

F

N

m

F

a

F

= ma

= 0.200

a

m/s

2

(36)

Correct

Problem 7.10

Blocks with masses of 2 , 4 , and 6 are lined up in a row on a frictionless table. All three are pushed forward by a 60 force applied to the 2 block.

Part A

How much force does the 4 block exert on the 6 block?

Express your answer to one significant figure and include the appropriate units. ANSWER:

Correct

Part B

How much force does the 4 block exert on the 2 block?

Correct

Problem 7.9

A 1000 car pushes a 2100 truck that has a dead battery. When the driver steps on the accelerator, the drive wheels of the car push against the ground with a force of 4500 . Rolling friction can

kg

N

kg

= 30

F

N

kg

= 50

F

N

kg

N

(37)

be neglected.

Part A

What is the magnitude of the force of the car on the truck?

Correct

Part B

What is the magnitude of the force of the truck on the car?

Correct

Atwood Machine Special Cases

An Atwood machine consists of two blocks (of masses and ) tied together with a massless rope that passes over a fixed, perfect (massless and frictionless) pulley. In this problem you'll investigate some special cases where physical variables describing the Atwood machine take on limiting values. Often, examining special cases will simplify a problem, so that the solution may be found from inspection or from the results of a problem you've already seen.

For all parts of this problem, take upward to be the positive direction and take the gravitational constant, , to be positive. = 3000

F

N

= 3000

F

N

m

1

m

2

g

(38)

Part A

Consider the case where and are both nonzero, and . Let be the magnitude of the tension in the rope connected to the block of mass , and let be the magnitude of the tension in the rope connected to the block of mass . Which of the following statements is true?

ANSWER:

Correct

Part B

Now, consider the special case where the block of mass is not present. Find the magnitude, , of the tension in the rope. Try to do this without equations; instead, think about the physical consequences.

If the block of mass is not present, and the rope connecting the two blocks is massless, will the motion of the block of mass be any different from free fall?

Hint 2. Which physical law to use

Use Newton's 2nd law on the block of mass .

m

1

m

2

m

2

>

m

1

T

1

m

1

T

2

m

2

is always equal to .

is greater than by an amount independent of velocity.

is greater than but the difference decreases as the blocks increase in velocity. There is not enough information to determine the relationship between and .

T

1

T

2

T

2

T

1

T

2

T

1

T

1

T

2

m

1

T

m

1

m

2

m

2

(39)

ANSWER:

Correct

Part C

For the same special case (the block of mass not present), what is the acceleration of the block of mass ?

Express your answer in terms of , and remember that an upward acceleration should be positive. ANSWER:

Correct

Part D

Next, consider the special case where only the block of mass is present. Find the magnitude, , of the tension in the rope.

ANSWER:

Correct

Part E

For the same special case (the block of mass not present) what is the acceleration of the end of the rope where the block of mass would have been attached? Express your answer in terms of , and remember that an upward acceleration should be positive.

= 0

T

m

1

m

2

g

= -9.80

a

2

m

1

T

= 0

T

m

2

m

2

g

(40)

ANSWER:

Correct

Part F

Next, consider the special case . What is the magnitude of the tension in the rope connecting the two blocks?

Use the variable in your answer instead of or . ANSWER:

Correct

Part G

For the same special case ( ), what is the acceleration of the block of mass ?

ANSWER:

Correct

Part H

Finally, suppose , while remains finite. What value does the the magnitude of the tension approach?

Hint 1. Acceleration of block of mass

= 9.80

a

2

=

= m

m

1

m

2

m

1

m

2 =

T

mg

=

= m

m

1

m

2

m

2 = 0

a

2

→∞

m

1

m

2

m

1

(41)

As becomes large, the finite tension will have a neglible effect on the acceleration, . If you ignore , you can pretend the rope is gone without changing your results for . As , what value does approach?

ANSWER:

Hint 2. Acceleration of block of mass

As , what value will the acceleration of the block of mass approach?

ANSWER:

Hint 3. Net force on block of mass

What is the magnitude of the net force on the block of mass .

Express your answer in terms of , , , and any other given quantities. Take the upward direction to be positive. ANSWER:

ANSWER:

Correct

Imagining what would happen if one or more of the variables approached infinity is often a good way to investigate the behavior of a system.

m

1

T

a

1

T

a

1

→∞

m

1

a

1 = -9.80

a

1

m

2

→∞

m

1

m

2 = 9.80

a

2

m

2

F

net

m

2

T m

2

g

=

F

net

T

−

m

2

g

=

T

2 m

2

g

(42)

Problem 7.17

A 5.9 rope hangs from the ceiling.

Part A

What is the tension at the midpoint of the rope?

Correct

Problem 7.23

The sled dog in figure drags sleds A and B across the snow. The coefficient of friction between the sleds and the snow is 0.10.

Part A

If the tension in rope 1 is 100 , what is the tension in rope 2?

kg

= 29

T

N

= 180

T

2

N

(43)

Correct

Enhanced EOC: Problem 7.31

Two packages at UPS start sliding down the ramp shown in the figure. Package A has a mass of 4.50 and a coefficient of kinetic friction of 0.200. Package B has a mass of 11.0 and a coefficient of kinetic friction of 0.150.

You may want to review ( pages 177 - 181) . For help with math skills, you may want to review: Vector Components

Part A

How long does it take package A to reach the bottom? Express your answer with the appropriate units.

Start by drawing force identification diagrams for package A and package B separately. What are the four forces acting on each block? Which of the forces are related by Newton's third law?

Draw separate free-body diagrams for block A and for block B. What is a good coordinate system to use to describe the motion of the blocks down the ramp? Label your coordinate system on the free-body diagram.

In your coordinate system, compute the x and y components of each force on block A. What are the x and y components of the net force on block A? What are the x and y components of the net force on block B?

Given that the coefficient of friction of block A is greater than the coefficient of friction of block B, do you think the blocks will stay together as they slide down the ramp? Assuming that they do stay together, how is the acceleration of the two blocks related? (We can check this assumption later.)

Using the components of the forces and Newton's second law, what is the acceleration of the blocks? What is the initial velocity of the blocks? Given the initial velocity and the acceleration,

(44)

how long does it take block A to go the given distance?

To check that the blocks do indeed stay together, solve for the force of block B on block A. If the force is directed toward the bottom of the ramp, then the blocks stay together.

ANSWER:

Correct

Problem 7.33

The 1.0 kg block in the figure is tied to the wall with a rope. It sits on top of the 2.0 kg block. The lower block is pulled to the right with a tension force of 20 N. The coefficient of kinetic friction at both the lower and upper surfaces of the 2.0 kg block is = 0.420.

1.48

s

(45)

Part A

What is the tension in the rope holding the 1.0 kg block to the wall? Express your answer with the appropriate units.

ANSWER:

Correct

Part B

What is the acceleration of the 2.0 kg block? Express your answer with the appropriate units. ANSWER:

Correct

Problem 7.38

The 100 kg block in figure takes 5.60 to reach the floor after being released from rest. 4.12

N

1.77 m s2

(46)

Part A

What is the mass of the block on the left?

Correct

Problem 7.41

Figure shows a block of mass m resting on a 20 slope. The block has coefficients of friction 0.82 and 0.51 with the surface. It is connected via a massless string over a massless, frictionless pulley to a hanging block of mass 2.0 .

Part A

What is the minimum mass that will stick and not slip? 98.7

kg

∘

kg

(47)

Correct

If you need to use the rounded answer you submitted here in a subsequent part, instead use the full precision answer and only round as a final step before submitting an answer.

Part B

If this minimum mass is nudged ever so slightly, it will start being pulled up the incline. What acceleration will it have? Express your answer to three significant figures and include the appropriate units.

ANSWER:

Correct

Problem 7.46

A house painter uses the chair and pulley arrangement of the figure to lift himself up the side of a house. The painter's mass is 75 and the chair's mass is 12 . = 1.80

m

kg

= 1.35

a

m_s2

(48)

Part A

With what force must he pull down on the rope in order to accelerate upward at 0.22 ?

Correct

Score Summary:

Your score on this assignment is 98.6%.

You received 104.5 out of a possible total of 106 points.

m/s

2

= 440

(49)

Assignment 7

Due: 11:59pm on Friday, March 21, 2014

Conceptual Question 8.5

The figure shows two balls of equal mass moving in vertical circles.

Part A

Is the tension in string A greater than, less than, or equal to the tension in string B if the balls travel over the top of the circle with equal speed? ANSWER:

Correct

The tension in string A is less than the tension in string B. The tension in string A is equal to the tension in string B. The tension in string A is greater than the tension in string B.

(50)

Part B

Is the tension in string A greater than, less than, or equal to the tension in string B if the balls travel over the top of the circle with equal angular velocity? ANSWER:

Correct

A Mass on a Turntable: Conceptual

A small metal cylinder rests on a circular turntable that is rotating at a constant rate, as illustrated in the diagram.

Part A

Which of the following sets of vectors best describes the velocity, acceleration, and net force acting on the cylinder at the point indicated in the diagram? The tension in string A is less than the tension in string B.

The tension in string A is equal to the tension in string B. The tension in string A is greater than the tension in string B.

(51)

Hint 1. The direction of acceleration can be determined from Newton's second law

According to Newton's second law, the acceleration of an object has the same direction as the net force acting on that object.

ANSWER:

Correct

Part B

Let be the distance between the cylinder and the center of the turntable. Now assume that the cylinder is moved to a new location from the center of the turntable. Which of the following statements accurately describe the motion of the cylinder at the new location?

Check all that apply. a b c d e

R

R/2

(52)

Hint 1. Find the speed of the cylinder

Find the speed of the cylinder at the new location. Assume that the cylinder makes one complete turn in a period of time . Express your answer in terms of and .

ANSWER:

Hint 2. Find the acceleration of the cylinder

Find the magnitude of the acceleration of the cylinder at the new location. Assume that the cylinder makes one complete turn in a period of time . Express your answer in terms of and .

Hint 1. Centripetal acceleration

Recall that the acceleration of an object that moves in a circular path of radius with constant speed has magnitude given by

.

Note that both the velocity and radius of the trajectory change when the cylinder is moved.

ANSWER: ANSWER:

v

T

R

T

=

v

πR_T

a

T

R

T

r

v

a

=

v_r2 =

a

2π2R T2

(53)

Correct

Accelerating along a Racetrack

A road race is taking place along the track shown in the figure . All of the cars are moving at constant speeds. The car at point F is traveling along a straight section of the track, whereas all the other cars are moving along curved segments of the track.

Part A

Let be the velocity of the car at point A. What can you say about the acceleration of the car at that point?

Hint 1. Acceleration along a curved path

The speed of the cylinder has decreased. The speed of the cylinder has increased.

The magnitude of the acceleration of the cylinder has decreased. The magnitude of the acceleration of the cylinder has increased. The speed and the acceleration of the cylinder have not changed.

(54)

Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object's acceleration is always perpendicular to the velocity vector at each point along the curved path and is directed toward the center of curvature of the path.

ANSWER:

Correct

Part B

Let be the velocity of the car at point C. What can you say about the acceleration of the car at that point?

ANSWER:

v⃗

The acceleration is parallel to .

The acceleration is perpendicular to and directed toward the inside of the track. The acceleration is perpendicular to and directed toward the outside of the track. The acceleration is neither parallel nor perpendicular to .

The acceleration is zero.

v⃗

A

v⃗

_A

v⃗

_A

v⃗

A

v⃗

C

v⃗

(55)

Correct

Part C

Let be the velocity of the car at point D. What can you say about the acceleration of the car at that point?

ANSWER:

Correct

The acceleration is perpendicular to and pointed toward the inside of the track. The acceleration is perpendicular to and pointed toward the outside of the track. The acceleration is neither parallel nor perpendicular to .

v⃗

C

v⃗

C

v⃗

C

v⃗

_C

v⃗

D

v⃗

D

v⃗

_D

v⃗

D

(56)

Part D

Let be the velocity of the car at point F. What can you say about the acceleration of the car at that point?

Hint 1. Acceleration along a straight path

The velocity of an object that moves along a straight path is always parallel to the direction of the path, and the object has a nonzero acceleration only if the magnitude of its velocity changes in time.

ANSWER:

Correct

Part E

Assuming that all cars have equal speeds, which car has the acceleration of the greatest magnitude, and which one has the acceleration of the least magnitude?

Use A for the car at point A, B for the car at point B, and so on. Express your answer as the name the car that has the greatest magnitude of acceleration followed by the car with the least magnitude of accelation, and separate your answers with a comma.

Recall that the magnitude of the acceleration of an object that moves at constant speed along a curved path is inversely proportional to the radius of curvature of the path.

ANSWER:

v⃗

F

v⃗

F

v⃗

F

v⃗

F

(57)

Correct

Part F

Assume that the car at point A and the one at point E are traveling along circular paths that have the same radius. If the car at point A now moves twice as fast as the car at point E, how is the magnitude of its acceleration related to that of car E.

Hint 1. Find the acceleration of the car at point E

Let be the radius of the two curves along which the cars at points A and E are traveling. What is the magnitude of the acceleration of the car at point E? Express your answer in terms of the radius of curvature and the speed of car E.

Hint 1. Uniform circular motion

The magnitude of the acceleration of an object that moves with constant speed along a circular path of radius is given by

.

ANSWER:

Hint 2. Find the acceleration of the car at point A

If , what is the acceleration of the car at point A? Let be the radius of the two curves along which the cars at points A and E are traveling.

Express your answer in terms of the speed of the car at E and the radius .

r

a

_E

r

v

_E

a

v

r

a

=

v_r2 =

a

_E vE2 r

= 2

v

A

v

E

a

A

r

v

_E

r

(58)

Hint 1. Uniform circular motion

The magnitude of the acceleration of an object that moves with constant speed along a circular path of radius is given by

.

ANSWER:

Correct

Problem 8.5

A 1300 car takes a 50- -radius unbanked curve at 13 .

Part A

What is the size of the friction force on the car?

v

r

a

=

v2 r =

a

_A 4vE2 r

The magnitude of the acceleration of the car at point A is twice that of the car at point E. The magnitude of the acceleration of the car at point A is the same as that of the car at point E. The magnitude of the acceleration of the car at point A is half that of the car at point E.

The magnitude of the acceleration of the car at point A is four times that of the car at point E.

(59)

Correct

Problem 8.10

It is proposed that future space stations create an artificial gravity by rotating. Suppose a space station is constructed as a 1600- -diameter cylinder that rotates about its axis. The inside surface is the deck of the space station.

Part A

What rotation period will provide "normal" gravity? Express your answer with the appropriate units. ANSWER:

Correct

Problem 8.7

In the Bohr model of the hydrogen atom, an electron orbits a proton at a distance of . The proton pulls on the electron with an electric force of .

Part A

How many revolutions per second does the electron make? Express your answer with the appropriate units. ANSWER: = 4400

f

_s

N

m

= 56.8

T

s

(mass m = 9.1 ×

10

−31

kg)

5.3 ×

10

−11

m

8.2 ×

10

−8

N

(60)

Correct

Problem 8.14

The weight of passengers on a roller coaster increases by 56 as the car goes through a dip with a 38 radius of curvature.

Part A

What is the car's speed at the bottom of the dip?

Correct

Problem 8.18

While at the county fair, you decide to ride the Ferris wheel. Having eaten too many candy apples and elephant ears, you find the motion somewhat unpleasant. To take your mind off your stomach, you wonder about the motion of the ride. You estimate the radius of the big wheel to be 14 , and you use your watch to find that each loop around takes 24 .

Part A

What is your speed?

Express your answer to two significant figures and include the appropriate units. ANSWER: 6.56×1015 rev_s

%

m

= 14

v

m_s

m

s

= 3.7

v

m_s

(61)

Correct

Part B

What is the magnitude of your acceleration?

Correct

Part C

What is the ratio of your weight at the top of the ride to your weight while standing on the ground? Express your answer using two significant figures.

ANSWER:

Correct

Part D

What is the ratio of your weight at the bottom of the ride to your weight while standing on the ground? Express your answer using two significant figures.

ANSWER: = 0.96

a

m_s₂ = 0.90 wtop F_G

(62)

Correct

Enhanced EOC: Problem 8.46

A heavy ball with a weight of 120 is hung from the ceiling of a lecture hall on a 4.4- -long rope. The ball is pulled to one side and released to swing as a pendulum, reaching a speed of 5.6 as it passes through the lowest point.

You may want to review ( pages 201 - 204) . For help with math skills, you may want to review: Solutions of Systems of Equations

Part A

What is the tension in the rope at that point?

Start by drawing a free-body diagram indicating the forces acting on the ball when it is at its lowest point.

Choose a coordinate system. What is the direction of the acceleration in your chosen coordinate system? What is the magnitude of the acceleration for the mass, which is moving in a circular path?

What is Newton's second law applied to the mass at the bottom of its swing? Make sure to use your coordinate system when determining the signs of all the forces and the acceleration. What is the tension in the rope at this point?

ANSWER: = 1.1 w_bottom FG

N

m

m/s

= 210

T

N

(63)

Correct

Problem 8.43

In an amusement park ride called The Roundup, passengers stand inside a 16.0 -diameter rotating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane, as shown in the figure .

Part A

Suppose the ring rotates once every 4.80 . If a rider's mass is 54.0 , with how much force does the ring push on her at the top of the ride?

Correct

Part B

m

s

kg

211

N

(64)

Suppose the ring rotates once every 4.80 . If a rider's mass is 54.0 , with how much force does the ring push on her at the bottom of the ride?

Correct

Part C

What is the longest rotation period of the wheel that will prevent the riders from falling off at the top? Express your answer with the appropriate units.

ANSWER:

Correct

Conceptual Question 9.9

A 2 object is moving to the right with a speed of 1 when it experiences an impulse of 6 .

Part A

What is the object's speed after the impulse?

Express your answer as an integer and include the appropriate units. ANSWER:

s

kg

1270

N

5.68

s

kg

i^

m/s

i^

N s

= 4

v

m_s

(65)

Correct

Part B

What is the object's direction after the impulse? ANSWER:

Correct

Conceptual Question 9.10

A 2 object is moving to the right with a speed of 2 when it experiences an impulse of -6 .

Part A

What is the object's speed after the impulse?

Express your answer as an integer and include the appropriate units. ANSWER:

Correct

Part B

What is the object's direction after the impulse? to the right

to the left

kg

i^

m/s

i^

N s

= 1

(66)

ANSWER:

Correct

Problem 9.5

Part A

In the figure , what value of gives an impulse of 6.4 ?

ANSWER:

Correct

to the right to the left

F

max

N s

= 1.6×103

F

max

N

(67)

Impulse on a Baseball

Learning Goal:

To understand the relationship between force, impulse, and momentum.

The effect of a net force acting on an object is related both to the force and to the total time the force acts on the object. The physical quantity impulse is a measure of both these effects. For a constant net force, the impulse is given by

.

The impulse is a vector pointing in the same direction as the force vector. The units of are or .

Recall that when a net force acts on an object, the object will accelerate, causing a change in its velocity. Hence the object's momentum ( ) will also change. The impulse-momentum theorem describes the effect that an impulse has on an object's motion:

.

So the change in momentum of an object equals the net impulse, that is, the net force multiplied by the time over which the force acts. A given change in momentum can result from a large force over a short time or a smaller force over a longer time.

In Parts A, B, C consider the following situation. In a baseball game the batter swings and gets a good solid hit. His swing applies a force of 12,000 to the ball for a time of .

Part A

Assuming that this force is constant, what is the magnitude of the impulse on the ball? Enter your answer numerically in newton seconds using two significant figures. ANSWER:

Correct

We often visualize the impulse by drawing a graph of force versus time. For a constant net force such as that used in the previous part, the graph will look like the one shown in the figure.

ΣF ⃗

J ⃗

=

∆t

J ⃗

F ⃗

J ⃗

N ⋅ s

kg ⋅ m/s

= m

p⃗

v⃗

∆ =

p⃗

J ⃗

=

F ⃗

∆t

N

0.70 ×

10

−3

s

J

= 8.4

J

N ⋅ s

(68)

Part B

The net force versus time graph has a rectangular shape. Often in physics geometric properties of graphs have physical meaning. ANSWER:

Correct

The assumption of a constant net force is idealized to make the problem easier to solve. A real force, especially in a case like the one presented in Parts A and B, where a large force is applied for a short time, is not likely to be constant.

A more realistic graph of the force that the swinging bat applies to the baseball will show the force building up to a maximum value as the bat comes into full contact with the ball. Then as the ball loses contact with the bat, the graph will show the force decaying to zero. It will look like the graph in the figure.

For this graph, the

length height area slope

(69)

Part C

If both the graph representing the constant net force and the graph representing the variable net force represent the same impulse acting on the baseball, which geometric properties must the two graphs have in common?

ANSWER:

maximum force area

(70)

Correct

When the net force varies over time, as in the case of the real net force acting on the baseball, you can simplify the problem by finding the average net force acting on the baseball during time . This average net force is treated as a constant force that acts on the ball for time . The impulse on the ball can then be found as .

Graphically, this method states that the impulse of the baseball can be represented by either the area under the net force versus time curve or the area under the average net force versus time curve. These areas are represented in the figure as the areas shaded in red and blue respectively.

The impulse of an object is also related to its change in momentum. Once the impulse is known, it can be used to find the change in momentum, or if either the initial or final momentum is known, the other momentum can be found. Keep in mind that . Because both impulse and momentum are vectors, it is essential to account for the direction of each vector, even in a one-dimensional problem.

Part D

Assume that a pitcher throws a baseball so that it travels in a straight line parallel to the ground. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. Define the direction the pitcher originally throws the ball as the +x direction.

ANSWER:

F ⃗

avg

∆t

J ⃗

=

F ⃗

avg

∆t

= ∆ = m(

−

)

J ⃗

p⃗

v⃗

_f

v⃗

_i

(71)

Correct

Part E

Now assume that the pitcher in Part D throws a 0.145- baseball parallel to the ground with a speed of 32 in the +x direction. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. What is the ball's velocity just after leaving the bat if the bat applies an impulse of to the baseball?

Enter your answer numerically in meters per second using two significant figures. ANSWER:

Correct

The negative sign in the answer indicates that after the bat hits the ball, the ball travels in the opposite direction to that defined to be positive.

Problem 9.9

A 2.6 object is moving to the right with a speed of 1.0 when it experiences the force shown in the figure. The impulse on the ball caused by the bat will be in the

positive negative x direction.

kg

m/s

−8.4 N ⋅ s

= -26

v⃗

m/s

kg

m/s

(72)

Part A

What is the object's speed after the force ends?

Correct

Part B

What is the object's direction after the force ends? ANSWER:

Correct

Enhanced EOC: Problem 9.27

A tennis player swings her 1000 g racket with a speed of 11.0 . She hits a 60 g tennis ball that was approaching her at a speed of 19.0 . The ball rebounds at 41.0 . You may want to review ( pages 226 - 232) .

For help with math skills, you may want to review: = 0.62

v

m_s

to the right to the left

(73)

Solving Algebraic Equations Part A

How fast is her racket moving immediately after the impact? You can ignore the interaction of the racket with her hand for the brief duration of the collision. Express your answer with the appropriate units.

Given that you can ignore the interaction of the racket with her hand during the collision, what is conserved during the collision?

Draw a picture indicating the direction of the racket and ball before the collision and a separate picture for after the collision. Place a coordinate system on your pictures, indicating the positive x direction.

Keeping in mind that velocity can be either positive or negative in your coordinate system, what is the initial momentum of the ball–racket system? What is the final momentum of the ball–racket system in terms of the velocity of the racket after the collision?

Using conservation of momentum, what are the velocity and speed of the racket after the collision?

ANSWER:

Correct

Part B

If the tennis ball and racket are in contact for 8.00 , what is the average force that the racket exerts on the ball? Express your answer with the appropriate units.

How is the impulse on the ball related to the change in momentum of the ball? What is the change in momentum of the ball? How are the impulse on the ball and the collision time related to the average force on the ball?

7.40 m_s

(74)

ANSWER:

Correct

Problem 9.14

A 2.00×104 railroad car is rolling at 6.00 when a 6000 load of gravel is suddenly dropped in.

Part A

What is the car's speed just after the gravel is loaded? Express your answer with the appropriate units. ANSWER:

Correct

Problem 9.17

A 330 bird flying along at 5.0 sees a 9.0 insect heading straight toward it with a speed of 34 (as measured by an observer on the ground, not by the bird). The bird opens its mouth wide and enjoys a nice lunch.

Part A

What is the bird's speed immediately after swallowing?

450

N

kg

m/s

kg

4.62 m_s

(75)

Correct

Problem 9.20

A 50.0 archer, standing on frictionless ice, shoots a 200 arrow at a speed of 200 .

Part A

What is the recoil speed of the archer?

Correct

Problem 9.25

A 40.0 ball of clay traveling east at 4.50 collides and sticks together with a 50.0 ball of clay traveling north at 4.50 .

Part A

What is the speed of the resulting ball of clay? Express your answer with the appropriate units. ANSWER: = 4.0