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MasteringPhysics: Assignment Print View

Cross Section for Asteroid Impact In this problem, you will estimate the cross section for an earth-asteroid collision. In all that follows, assume that the earth is fixed in space and that the radius of the asteroid is much less than the radius of the earth. The mass of the earth is , and the mass of the asteroid is

. Use

for the universal gravitational constant.

Part A Far away from the earth, the asteroid is moving with speed and has impact parameter , as shown in the figure. In this large-separation limit, the distance from the asteroid to the of the asteroid. earth is taken to be infinite. Find the total initial energy Hint A.1

Gravitational potential energy

The gravitational potential energy is proportional to

and approaches zero for large

values of . Express your answer in of

,

, ,

, and

.

ANSWER: =

Answer Requested

Part B

http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=1484586 (1 of 46) [12/13/2010 7:10:03 PM]


For large earth-asteroid separation, what is the magnitude of the asteroid's total angular momentum about the center of the earth? Hint B.1

Definition of angular momentum Hint not displayed

Express your answer in of

,

, ,

, and

.

ANSWER: = Answer Requested

Part C The maximum impact parameter for which collision is guaranteed,

, is obtained by

of the earth. This is setting the minimum earth-asteroid separation equal to the radius the configuration shown in the figure. In this case, it is clear that the velocity of the asteroid right before it hits the earth is tangent to the surface and therefore perpendicular to the position vector that points from the center of the earth to the asteroid. , what is the total energy of the asteroid the instant before it When crashes into the earth? Assume that the speed of the asteroid at closest approach is . Hint C.1

Potential energy Hint not displayed


,

,

,

, and

.


Part D



Again, suppose that . What is the angular momentum moment before it crashes into the earth's surface? Hint D.1

of the asteroid the

Direction of velocity before impact Hint not displayed


,

,

,

, and

.


Part E Use conservation of energy and angular momentum to find an expression for Hint E.1

.

Find the final velocity Hint not displayed

Express your answer as a function of

,

,

, and

.


Part F The collision cross section and is found by multiplying

represents the effective target area "seen" by the asteroid by

. If the asteroid comes into this area, it is

guaranteed to collide with the earth. A simple representation of the cross section is obtained when we write in of , the escape speed from the surface of the earth. First, find an expression for , and let , where is a constant of proportionality. Then combine this with your result for to write a simple-looking expression for Hint F.1

in of

and

.

Find the escape speed Hint not displayed



Express the collision cross section in of

and

.


Part G The point of origin of a typical asteroid might lie at a radius of about units;

) from the sun, the approximate location of the asteroid belt.

Calculate the effective target cross section the asteroid's orbit is cicular. Hint G.1

(astronomical

of the earth as seen by the asteroid. Assume

Useful constants Hint not displayed

Hint G.2

Find the orbital speed of the asteroid Hint not displayed

Hint G.3

Calculate a value for Hint not displayed

Give your answer as a multiple of the area of the disk of the earth,

ANSWER:

.

1.44 = Answer Requested

Therefore, because of the gravitational attraction of the asteroid by the earth, the effective target cross section seen by the asteroid is more than 40% larger then the . earth's geometric cross section of

Energy of a Spacecraft



Very far from earth (at ), a spacecraft has run out of fuel and its kinetic energy is zero. If only the gravitational force of the earth were to act on it (i.e., neglect the forces from the sun and other solar system objects), the spacecraft would eventually crash into the and its radius is . Neglect air resistance throughout earth. The mass of the earth is this problem, since the spacecraft is primarily moving through the near vacuum of space. Part A Find the speed Hint A.1

of the spacecraft when it crashes into the earth.

How to approach the problem Hint not displayed

Hint A.2

Total energy Hint not displayed

Hint A.3

Potential energy Hint not displayed

Express the speed in of constant

,

, and the universal gravitational

.

ANSWER: = Correct

Part B Now find the spacecraft's speed when its distance from the center of the earth is , where . Hint B.1

General approach Hint not displayed

Hint B.2

First step in finding the speed Hint not displayed

Express the speed in of

and

.



ANSWER: = All attempts used; correct answer displayed

Gravitational Acceleration inside a Planet Consider a spherical planet of uniform density . The distance from the planet's center to its surface (i.e., the planet's radius) is . An object is located a distance from the center of the planet, where

. (The object is located inside of the planet.)

Part A Find an expression for the magnitude of the acceleration due to gravity, planet. Hint A.1

, inside the

Force due to planet's mass outside radius Hint not displayed

Hint A.2

Find the force on an object at distance Hint not displayed

Hint A.3

from

Finding

Hint not displayed Express the acceleration due to gravity in of universal gravitational constant.

,

,

, and

, the

ANSWER: = Correct

Part B



Rewrite your result for in of the planet, times a function of R. Hint B.1

, the gravitational acceleration at the surface of

Acceleration at the surface Hint not displayed


,

, and

.

ANSWER: = Correct

Notice that

increases linearly with

, rather than being proportional to

. This

assures that it is zero at the center of the planet, as required by symmetry.

Part C Find a numerical value for meter. Use value of

, the average density of the earth in kilograms per cubic

for the radius of the earth,

at the surface of

Hint C.1

, and a

.


Calculate your answer to four significant digits.

ANSWER:

= 5497 Correct

Kepler's 3rd Law



A planet moves in an elliptical orbit around the sun. The mass of the sun is minimum and maximum distances of the planet from the sun are

and

. The , respectively.

Part A Using Kepler's 3rd law and Newton's law of universal gravitation, find the period of of the planet as it moves around the sun. Assume that the mass of the planet revolution is much smaller than the mass of the sun. for the gravitational constant. Use Hint A.1

Kepler's 3rd law Hint not displayed

Hint A.2

Find the semi-major axis Hint not displayed

Hint A.3

Find the period of a circular orbit Hint not displayed

Express the period in of

,

,

, and

.

ANSWER: = Correct

The Dyson Sphere The Dyson sphere is an hypothetical spherical structure centered around a star. Inspired by a science fiction story, physicist Freeman Dyson described such a structure for the first time in a scientific paper in 1959. His basic idea consisted of an artificial spherical structure of matter built around a star at a distance comparable to a planetary orbit, with the purpose of capturing the energy radiated by the star and reusing it for industrial purposes. Assume the mass of the sun to be 2.00×1030 . Part A



Consider a solid, rigid spherical shell with a thickness of 100

and a density of 3900

. The sphere is centered around the sun so that its inner surface is at a distance from the center of the sun. What is the net force that the sun would exert of 1.50×1011 on such a Dyson sphere were it to get displaced off-center by some small amount? Hint A.1


Express your answer numerically in newtons.

ANSWER:

0 Correct

Since there is no net attraction between a hollow sphere and a body inside, a Dyson sphere of this kind would be gravitationally unstable. If the sphere were hit by a meteor and were slightly shifted, the sun would exert no force on it to bring it back to its original position. The sphere would simply drift off and eventually hit the sun. Because of this gravitational instability, Dyson himself did not originally suggest a solid spherical shell; rather, he proposed a series of individual plates independently orbiting the sun.

Part B What is the net gravitational force Dyson sphere described in Part A? Hint B.1

on a unit mass located on the outer surface of the


Hint B.2

Find the gravitational force exerted by the Dyson sphere Hint not displayed

Hint B.3

Find the gravitational force exerted by the sun Hint not displayed

Express your answer in newtons.

ANSWER:

= 6.26×10−3 Correct

Part C



What is the net gravitational force Dyson sphere described in Part A? Hint C.1

on a unit mass located on the inner surface of the

How to approach the problem

Recall that there is no net attraction between a spherical shell and a point mass inside it; therefore, the only contribution to the net gravitational force exerted on a unit mass located on the inner surface of the Dyson sphere comes from the sun. Express your answer in newtons.

ANSWER:

5.93×10−3 = Answer Requested

The gravitational attraction of the sun would make the inner surface of the Dyson sphere described in Part A uninhabitable, because everything on the inner surface would slowly accelerate toward the sun. One way to solve this problem would be to create artificial gravity through rotation. Assume that the Dyson sphere rotates at a constant angular speed around an axis through its center so that earthlike gravity is re-created along the inner equator of the Dyson sphere. Take the radius of the Earth to be 6.38×106 and the mass of the Earth to be 5.97×1024 . Part D What is the linear speed Hint D.1

of a unit mass located at the inner equator of such a sphere?

How to approach the problem

Because of the constant rotation of the sphere, the mass at the inner equator moves along a circular path with constant angular speed; thus it has only a centripetal acceleration. There must be then a net force directed toward the center of the sphere. The only forces acting on the mass are the gravitational force of the sun and the normal force exerted by the surface of the sphere. To create the same gravitational conditions as on earth, the normal force exerted on the mass at the inner equator must be equal to the normal force exerted on a unit mass at earth's equator, since the normal force corresponds to the acceleration felt by a person on the inner surface of the Dyson sphere. Hint D.2

Find the net force at the inner surface of a rotating hollow sphere



Consider a spinning hollow sphere with a particle located at its center. Let be the magnitude of the gravitational force that the particle exerts on a unit mass located on the inner surface of the sphere and let be the magnitude of the normal force exerted by the surface of the sphere on the unit mass. What is the magnitude of the net force acting on the unit mass? ANSWER:

Answer not displayed

Hint D.3

Find the normal force acting on a unit mass on earth's surface

What is the magnitude of the normal force surface of the earth? Hint D.3.1

acting on a unit mass located on the

Acceleration of gravity Hint not displayed


ANSWER:

Hint D.4

= Answer not displayed

Equation for centripetal acceleration

Recall that the equation relating the centripetal acceleration about a point a distance away at a speed is given by

of an object spinning

. Express your answer in meters per second.

ANSWER:

1.21×106 = Answer Requested

The stresses generated by such rotation would be so intense that no material would be able to sustain them, another reason for which such a Dyson sphere would not be http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=1484586 (11 of 46) [12/13/2010 7:10:03 PM]


physically feasible. Nevertheless, it remains popular among many science-fiction authors!

Gravitational Force of Three Identical Masses Three identical masses of 700 each are placed on the x axis. One mass is at , one is at the origin, and one is at = 390 .

= -130

Part A What is the magnitude of the net gravitational force the other two masses? = 6.67×10−11 Take the gravitational constant to be Hint A.1

on the mass at the origin due to .


Hint A.2

Calculate the gravitational force from the first mass Hint not displayed

Hint A.3

Determine the direction of the gravitational force from the first mass Hint not displayed

Hint A.4

Calculate the gravitational force from the second mass Hint not displayed

Hint A.5

Determine the direction of the gravitational force from the second mass Hint not displayed


ANSWER:

= 1.72×10−5 Correct

Part B



What is the direction of the net gravitational force on the mass at the origin due to the other two masses? ANSWER:

+x direction -x direction Correct

The closer together two masses are, the stronger is the gravitational attraction between them. Thus, the mass at the origin is more strongly attracted to the mass at = -130 than it is to the mass at = 390 . Thus, the net force on the mass at the origin is in the -x direction.

Weight on a Neutron Star Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun but a much smaller diameter. Part A If you weigh 665

on the earth, what would be your weight on the surface of a neutron

star that has the same mass as our sun and a diameter of 17.0 Take the mass of the sun to be = 6.67×10−11 be

= 9.810

Hint A.1

= 1.99×1030

?

, the gravitational constant to be

, and the acceleration due to gravity at the earth's surface to .


Hint A.2

Law of universal gravitation Hint not displayed

Hint A.3

Calculate your mass Hint not displayed

Hint A.4

Calculate the distance between you and the star Hint not displayed

Express your weight

in newtons.



ANSWER:

This is over

1.25×1014 = All attempts used; correct answer displayed times your weight on earth! You probably shouldn't venture there....

Matching Initial Position and Velocity of Oscillator Learning Goal: Understand how to determine the constants in the general equation for simple harmonic motion, in of given initial conditions. A common problem in physics is to match the particular initial conditions - generally given as and velocity at - once you have obtained the general solution. an initial position You have dealt with this problem in kinematics, where the formula 1. has two arbitrary constants (technically constants of integration that arise when finding the position given that the acceleration is a constant). The constants in this case are the initial position and velocity, so "fitting" the general solution to the initial conditions is very simple. For simple harmonic motion, it is more difficult to fit the initial conditions, which we take to be , the position of the oscillator at , and , the velocity of the oscillator at . There are two common forms for the general solution for the position of a harmonic oscillator as a function of time : 2. 3.

and ,

where , , , and are constants, is the oscillation frequency, and is time. Although both expressions have two arbitrary constants--parameters that can be adjusted to fit the solution to the initial conditions--Equation 3 is much easier to use to accommodate and . (Equation 2 would be appropriate if the initial conditions were specified as the total energy and the time of the first zero crossing, for example.) Part A



Find

and

Hint A.1

in of the initial position and velocity of the oscillator. The only good way to start Hint not displayed

Hint A.2

Using kinematic relationships Hint not displayed

Hint A.3

Initial position Hint not displayed

Hint A.4

Initial velocity Hint not displayed

Give your answers in of comma.

,

, and

. Separate your answers with a

ANSWER: ,

= Correct

A Pivoting Rod on a Spring A slender, uniform metal rod of mass and length is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring, assumed massless and with force constant , is attached to the lower end of the rod, with the other end of the spring attached to a rigid .



Part A We start by analyzing the torques acting on the rod when it is deflected by a small angle from the vertical. Consider first the torque due to gravity. Which of the following statements most accurately describes the effect of gravity on the rod? Choose the best answer.

ANSWER:

Under the action of gravity alone the rod would move to a horizontal position. But for small deflections from the vertical the torque due to gravity is sufficiently small to be ignored. Under the action of gravity alone the rod would move to a vertical position. But for small deflections from the vertical the restoring force due to gravity is sufficiently small to be ignored. There is no torque due to gravity on the rod. Correct

Assume that the spring is relaxed (exerts no torque on the rod) when the rod is vertical. The rod is displaced by a small angle from the vertical. Part B



Find the torque

due to the spring. Assume that

is small enough that the spring

remains effectively horizontal and you can approximate Hint B.1

(and

).

Find the change in spring length Hint not displayed

Hint B.2

Find the moment arm Hint not displayed

Express the torque as a function of

and other parameters of the problem.

ANSWER: = Correct

Since the torque is opposed to the deflection and increases linearly with it, the system will undergo angular simple harmonic motion.

Part C What is the angular frequency Hint C.1

of oscillations of the rod?

How to find the oscillation frequency Hint not displayed

Hint C.2

Solve the angular equation of motion Hint not displayed

Hint C.3

Determine the moment of inertia of the rod Hint not displayed

Express the angular frequency in of parameters given in the introduction.

ANSWER: = Correct



Note that if the spring were simply attached to a mass were concentrated at its ends,

would be

, or if the mass of the rod

. The frequency is greater in this

case because mass near the pivot point doesn't move as much as the end of the spring. What do you suppose the frequency of oscillation would be if the spring were attached near the pivot point?

A Wobbling Bridge On June 10, 2000, the Millennium Bridge, a new footbridge over the River Thames in London, England, was opened to the public. However, after only two days, it had to be closed to traffic for safety reasons. On the opening day, in fact, so many people were crossing it at the same time that unexpected sideways oscillations of the bridge were observed. Further investigations indicated that the oscillation was caused by lateral forces produced by the synchronization of steps taken by the pedestrians. Although the origin of this cadence synchronization was new to the engineers, its effect on the structure of the bridge was very well known. The combined forces exerted by the pedestrians as they were walking in synchronization had a frequency very close to the natural frequency of the bridge, and so resonance occurred.

and natural angular frequency . When the Consider an oscillating system of mass system is subjected to a periodic external (driving) force, whose maximum value is , the amplitude of the driven oscillations is and angular frequency is , where is the force constant of the system and is the damping constant. We will use this simple model to study the oscillations of the Millennium Bridge. Part A http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=1484586 (18 of 46) [12/13/2010 7:10:03 PM]


Assume that, when we walk, in addition to a fluctuating vertical force, we exert a periodic . Given that the mass of the lateral force of amplitude 25 at a frequency of about 1 per linear meter, how many people were walking along the bridge is about 2000 144- -long central span of the bridge at one time, when an oscillation amplitude of 75 was observed in that section of the bridge? Take the damping constant to be such that the amplitude of the undriven oscillations would decay to of its original value in a time Hint A.1

, where

is the period of the undriven, undamped system.


Hint A.2

Find the maximum value of the driving force when resonance occurs Hint not displayed

Hint A.3

Find the damping constant Hint not displayed

Hint A.4

Find the total number of synchronized pedestrians Hint not displayed

Hint A.5

Find the mass of the bridge Hint not displayed

Hint A.6

Find the angular frequency Hint not displayed

Express your answer numerically to three significant figures.

ANSWER:

1810 number of people = Answer Requested

Video footage of the crowd on the bridge taken on the opening day confirmed that up to 2000 people were walking on the bridge at one time! Note that the synchronization of the pedestrians' gait observed on the Millennium Bridge is somewhat different from the organized marching of an army of soldiers, even though they can both cause similar effects. The pedestrians in London did not deliberately walk in step; rather, they subconsciously synchronized their pace to the bridge's sideways, left-to-right swaying motions. The more the bridge shook, the more http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=1484586 (19 of 46) [12/13/2010 7:10:03 PM]


people involuntarily walked in step with each other, which caused the bridge to shake even more.

Part B What would the amplitude of oscillation of the Millennium Bridge have been on the opening day if the damping effects had been three times more effective? Hint B.1


Hint B.2

What formula to consider Hint not displayed

Express your answer numerically in millimeters to three significant figures.

ANSWER:

25 = Answer Requested

As you found out, a resonance response can be considerably reduced by increasing the damping. To prevent resonance from occuring again, engineers installed a series of dampers underneath the deck of the bridge and, after several tests, the Millennium Bridge was succefully reopened to the public.

Damped Egg on a Spring A 50.0- hard-boiled egg moves on the end of a spring with force constant

.

It is released with an amplitude 0.300 . A damping force acts on the egg. After it oscillates for 5.00 , the amplitude of the motion has decreased to 0.100 . Part A



Calculate the magnitude of the damping coefficient . Hint A.1

How damped is it? Hint not displayed

Hint A.2

What formula to use Hint not displayed

Hint A.3

Find the amplitude Hint not displayed

Hint A.4

Solving for

in Hint not displayed

Express the magnitude of the damping coefficient numerically in kilograms per second, to three significant figures.

ANSWER:

= 2.20×10−2 kg/s Correct

Measuring the Acceleration Due to Gravity with a Speaker To measure the magnitude of the acceleration due to gravity in an unorthodox manner, a student places a ball bearing on the concave side of a flexible speaker cone . The speaker cone acts as a simple harmonic and oscillator whose amplitude is whose frequency

can be varied.

The student can measure both and with a strobe light. Take the equation of motion of the oscillator as , where and the y axis points upward. Part A



If the ball bearing has mass , find , the magnitude of the normal force exerted by the speaker cone on the ball bearing as a function of time. Hint A.1

Determine the total force on the ball bearing Hint not displayed

Hint A.2

Find the acceleration of the ball bearing Hint not displayed

Your result should be in of constant .

,

(or

),

,

, a phase angle

, and the

ANSWER: = Correct

Part B The frequency is slowly increased. Once it es the critical value , the student hears the ball bounce. There is now enough information to calculate . What is ? Hint B.1

Determine the force on the ball bearing when it loses Hint not displayed

Hint B.2

Find the value of

when the ball loses Hint not displayed

Hint B.3

Relation between

and Hint not displayed

Express the magnitude of the acceleration due to gravity in of .

ANSWER: = Correct


and


Oscillations of a Balanced Object Two identical thin rods, each of mass and length , are ed at right angles to form an L-shaped object. This object is balanced on top of a sharp edge . If the object is displaced slightly, it oscillates. Assume that the magnitude of the acceleration due to gravity is .

Part A Find

, the angular frequency of oscillation of the object.

Hint A.1

Determine the angular frequency of a physical pendulum Hint not displayed

Hint A.2

Calculate Hint not displayed

Hint A.3

Find the moment of inertia Hint not displayed

Your answer for the angular frequency may contain the given variables and as well as .

ANSWER: = Correct



Period of a Mass-Spring System Ranking Task Different mass crates are placed on top of springs of uncompressed length . The crates are released and the springs compress to a length crates back up to their original positions.

and stiffness

before bringing the

Part A Rank the time required for the crates to return to their initial positions from largest to smallest. Hint A.1

Formula for the period Hint not displayed

Hint A.2

Determining the mass Hint not displayed

Hint A.3

Determining Hint not displayed

Rank from largest to smallest. To rank items as equivalent, overlap them.



ANSWER:

View All attempts used; correct answer displayed

Springs in Series In this problem you will study two cases of springs connected in series that will enable you to draw a general conclusion. Two springs in series Consider two massless springs connected in series. Spring 1 has a spring constant

,

and spring 2 has a spring constant . A constant force of magnitude is being applied to the right. When the two springs are connected in this way, they form a system equivalent to a single spring of spring constant .

Part A



What is the effective spring constant Hint A.1

of the two-spring system?

Free-body diagram Hint not displayed

Hint A.2

Free-body diagram for spring 2 Hint not displayed

Hint A.3

Determine the extension of spring 2 Hint not displayed

Hint A.4

Determine the extension of spring 1 Hint not displayed

Hint A.5

Determine the total extension of the two springs Hint not displayed

Hint A.6

Replace

and Hint not displayed

Hint A.7

Determine the extension of the equivalent system Hint not displayed

Hint A.8

Solving for Hint not displayed

Express the effective spring constant in of

and

.

ANSWER: = Correct

Three springs in series http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=1484586 (26 of 46) [12/13/2010 7:10:03 PM]


Now consider three springs set up in series as shown. The spring constants are and

,

,

, and the force acting to the

right again has magnitude

.

Part B Find the spring constant

of the three-spring system.


,

, and

.

ANSWER: = Correct

You have now found the pattern for the general form of the overall spring constant of a set of springs connected in series. This result will be similar to the one for the total capacitance of a set of capacitors attached in series that you will see when you study electric circuits.

Weighing Lunch



For lunch you and your friends decide to stop at the nearest deli and have a sandwich made fresh for you with 0.300 of Italian ham. The slices of ham are weighed on a plate of mass 0.400

placed atop a vertical spring of negligible mass and force constant of 200

. The slices of ham are dropped on the plate all at the same time from a height of 0.250 . They make a totally inelastic collision with the plate and set the scale into vertical simple harmonic motion (SHM). You may assume that the collision time is extremely small. Part A What is the amplitude of oscillation Hint A.1

of the scale after the slices of ham land on the plate?


Hint A.2

Find the position of the plate and the ham immediately after the collision Hint not displayed

Hint A.3

Find the speed of the plate and the ham immediately after the collision Hint not displayed

Hint A.4

How to find

by matching initial conditions Hint not displayed

Hint A.5

Find

using energy conservation Hint not displayed

Express your answer numerically in meters and take free-fall acceleration . to be = 9.80

ANSWER:

5.80×10−2 = All attempts used; correct answer displayed

Part B http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=1484586 (28 of 46) [12/13/2010 7:10:03 PM]


What is the period of oscillation Hint B.1

of the scale?

Period of oscillation in SHM Hint not displayed

Express your answer numerically in seconds.

ANSWER:

= 0.372 Correct

Gravity on Another Planet After landing on an unfamiliar planet, a space explorer constructs a simple pendulum of length 53.0 . The explorer finds that the pendulum completes 102 full swing cycles in a time of 142 . Part A What is the value of the acceleration of gravity on this planet? Hint A.1


Hint A.2

Calculate the period Hint not displayed

Hint A.3

Equation for the period Hint not displayed

Express your answer in meters per second per second.

ANSWER:

= 10.8 Correct

The Fish Scale



The scale of a spring balance reading from 0 to 190

has a length of 11.0

. A fish

hanging from the bottom of the spring oscillates vertically at a frequency of 2.25

.

Part A Ignoring the mass of the spring, what is the mass Hint A.1

of the fish?


Hint A.2

Calculate the spring constant Hint not displayed

Hint A.3

Calculate the angular frequency Hint not displayed

Hint A.4

Formula for the angular frequency of a mass on a spring Hint not displayed

Express your answer in kilograms.

ANSWER:

= 8.64 Correct

Vibrating Hydrogen Molecule When displaced from equilibrium by a small amount, the two hydrogen atoms in an molecule are acted on by a restoring force

with

= 500

.

Part A



Calculate the oscillation frequency of the molecule. Use as the "effective mass" of the system, where in the mass of a hydrogen atom. Hint A.1

Formula for the oscillation frequency Hint not displayed

Take the mass of a hydrogen atom as 1.008

, where

.

Express your answer in hertz.

ANSWER:

= 1.23×1014 Correct

Ant on a Tightrope A large ant is standing on the middle of a circus tightrope that is stretched with tension . The rope has mass per unit length . Wanting to shake the ant off the rope, a tightrope walker moves her foot up and down near the end of the tightrope, generating a sinusoidal transverse wave of wavelength and amplitude . Assume that the magnitude of the acceleration due to gravity is . Part A What is the minimum wave amplitude such that the ant will become momentarily weightless at some point as the wave es underneath it? Assume that the mass of the ant is too small to have any effect on the wave propagation. Hint A.1

Weight and weightless Hint not displayed

Hint A.2


Hint A.3

Find the maximum acceleration of the string Hint not displayed

Hint A.4

Putting it all together Hint not displayed



Express the minimum wave amplitude in of

,

,

, and

.

ANSWER: = Correct

Fundamental Wavelength and Frequency Ranking Task A combination work of art/musical instrument is illustrated. Six pieces of identical piano wire (cut to different lengths) are hung from the same , and masses are hung from the free end of each wire. Each wire is 1, 2, or 3 units long, and each s 1, 2, or 4 units of mass. The mass of each wire is negligible compared to the total mass hanging from it. When a strong breeze blows, the wires vibrate and create an eerie sound.

Part A Rank each wire-mass system on the basis of its fundamental wavelength. Hint A.1

Identify the fundamental wavelength Hint not displayed




ANSWER:

View Correct

Part B Rank each wire-mass system on the basis of its wave speed. Hint B.1

Factors that determine wave speed Hint not displayed

Hint B.2

Tension in the wires Hint not displayed


ANSWER:

View Correct

Part C



Rank each wire-mass system on the basis of its fundamental frequency. Hint C.1

Find an equation for the fundamental frequency Hint not displayed


ANSWER:

View Correct

Wave and Particle Velocity Vector Drawing A long string is stretched and its left end is oscillated upward and downward. Two points on the string are labeled A and B. Part A At the instant shown, orient velocity at points A and B. Hint A.1

and

to correctly represent the direction of the wave

Distinguishing between wave velocity and particle velocity

A wave is a collective disturbance that, typically, travels through some medium, in this case along a string. The velocity of the individual particles of the medium are quite distinct from the velocity of the wave as it es through the medium. In fact, in a transverse wave such as a wave on a string, the wave velocity and particle velocities are perpendicular. Hint A.2

Wave velocity

A wave on a stretched string travels away from the source of the wave along the length of the string. At each of the points A and B, rotate the given vector to indicate the direction of the wave velocity.



ANSWER:

View Correct

Part B At the instant shown, orient the given vectors direction of the velocity of points A and B. Hint B.1

and

to correctly represent the

Distinguishing between wave velocity and particle velocity Hint not displayed

Hint B.2

Determining velocity from a snapshot Hint not displayed

Hint B.3

Find the change in point A’s position Hint not displayed

Hint B.4

Find the change in point B’s position Hint not displayed

At each of the points A and B, rotate the given vector to indicate the direction of the velocity.

ANSWER:

View Correct



Wave in a Dangling Rope A uniform rope of length

and negligible stiffness hangs from a solid fixture in the ceiling .

Part A . What is the time it takes The free lower end of the rope is struck sharply at time the resulting wave on the rope to travel to the ceiling, be reflected, and return to the lower end of the rope? Hint A.1


Hint A.2

Wave equation for a string Hint not displayed

Hint A.3

Find the general speed of the wave Hint not displayed

Hint A.4

Find the speed of the wave at Hint not displayed

Hint A.5

Find the time



Hint not displayed Express your answer in of and constants such as of the acceleration due to gravity), , etc.

(the magnitude

ANSWER: = Correct

Notice the similarities between this result and the period of a simple ideal pendulum of length

(which has a period of (

). Not surprisingly, these two times

are closely related. In the first case, the time does not depend on the mass of the rope; in the second, the time does not depend on the mass of the pendulum.

Wave Propagation in a String of Varying Density Consider a string of total length

, made up of three segments of equal length. The mass

per unit length of the first segment is

, that of the second is

, and that of the third

The third segment is tied to a wall, and the string is stretched by a force of magnitude applied to the first segment;

is much greater than the total weight of the string.

Part A How long will it take a transverse wave to propagate from one end of the string to the other? Hint A.1

How do the segments differ? Hint not displayed

Hint A.2

Example: speed in the second segment Hint not displayed

Hint A.3

Some math help Hint not displayed

Express the time

in of

,

, and

.


.


ANSWER: = Correct

The changes in density along the string are sudden, and the wave will experience them as boundaries. This will cause a fraction of the wave (energy, amplitude) to be reflected, while the rest is transmitted at each boundary. Although this will not affect the time it takes for the wave to reach the end of the string (thus it is not directly relevant to this question), the wave's amplitude will be reduced. Also, after the main wave has arrived, we may observe later arrivals of waves that have reflected back and forth between the boundaries before finally reaching the end of the string.

Harmonics of a Piano Wire A piano tuner stretches a steel piano wire with a tension of 765 and a mass of 5.25 . length of 0.700

. The steel wire has a

Part A What is the frequency Hint A.1

of the string's fundamental mode of vibration?


Hint A.2

Find the mass per unit length Hint not displayed

Hint A.3

Equation for the fundamental frequency of a string under tension Hint not displayed

Express your answer numerically in hertz using three significant figures.

ANSWER:

= 228 Correct

Part B



What is the number of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to = 16 kHz? Hint B.1

Harmonics of a string Hint not displayed

Express your answer exactly.

ANSWER:

= 70 Correct

When solving this problem, you may have found a noninteger value for harmonics can only be integer multiples of the fundamental frequency.

, but

Beat Frequency Ranking Task An all female guitar septet is getting ready to go on stage. The lead guitarist, Kira,who is always in tune, plucks her low E string and the other six , sequentially, do the between her low E string and same. Each member records the initial beat frequency Kira's low E string. Part A Rank each member on the basis of the frequency of her low E string. Hint A.1

Beat frequency Hint not displayed

Hint A.2

Find the frequency of Aiko's E string Hint not displayed


ANSWER:

View Correct



Because the beat frequency between Kira's guitar and Diane's guitar is 0 guitars play the exact same note and are in tune.

, these

To tune an instrument using beats, more information than just the beat frequency is , each member, except needed. In addition to recording the initial beat frequency Diane, also records the change in the frequency increases the tension in her low E string.

(increase or decrease) when she

Part B Rank each member on the basis of the initial frequency of her low E string. Hint B.1

Determine the relationship between tension and beat frequency Hint not displayed

Hint B.2

Determine the initial frequency of Aiko's E string Hint not displayed


ANSWER:

View All attempts used; correct answer displayed

Breathe Quietly to Avoid Detection



The sound of normal breathing is not very loud, with an intensity of about 11 dB at a distance of 1 m away from the face of the breather. Note that in this problem sound intensity in decibels is denoted ; intensity in is denoted . Part A Given that a person with normal hearing can barely detect a sound with intensity of 1 dB at a frequency of 1 kHz (the sensitivity of the human ear peaking near 1 kHz), how far away could this person detect another person breathing normally? Hint A.1

Find the decibel change corresponding to a change in the distance from the source Hint not displayed

Hint A.2

Determine the numerical value for Hint not displayed

Express the distance

ANSWER:

in meters.

= 3.16 m Correct

Part B In general, if a sound has intensity of dB at 1 m from the source, at what distance from the source would the decibel level decrease to 0 dB? Since the limit of hearing is 1 dB this would mean you could no longer hear it. Hint B.1

Find the decibel decrease corresponding to an increase in distance from the source Hint not displayed

Hint B.2

Solving equations involving logarithmic functions Hint not displayed

Express the distance in of

. Be careful about your signs!



ANSWER: =

m

All attempts used; correct answer displayed

Interference of Sound Waves Two loudspeakers, A and B, are driven by the same amplifier and emit sinusoidal waves in . You are 8.00 phase. The frequency of the waves emitted by each speaker is 172 from speaker A. Take the speed of sound in air to be 344

.

Part A What is the closest you can be to speaker B and be at a point of destructive interference? Hint A.1


Hint A.2

Find the wavelength of the sound wave Hint not displayed

Hint A.3

Find the condition for destructive interference Hint not displayed

Express your answer in meters.

ANSWER:

1.00 Correct

Two Loudspeakers in an Open Field



Imagine you are in an open field where two loudspeakers are set up and connected to the . Take the speed of sound same amplifier so that they emit sound waves in phase at 688 in air to be 344

.

Part A If you are 3.00 from speaker A directly to your right and 3.50 from speaker B directly to your left, will the sound that you hear be louder than the sound you would hear if only one speaker were in use? Hint A.1


Hint A.2

Constructive and destructive interference Hint not displayed

Hint A.3

Find the wavelength of the sound Hint not displayed

ANSWER:

yes no Correct

Because the path difference is equal to the wavelength of the sound, the sound originating at the two speakers will interfere constructively at your location and you will perceive a louder sound.

Part B



What is the shortest distance cannot hear the speakers? Hint B.1

you need to walk forward to be at a point where you


Hint B.2

Find the path-length difference at a point of destructive interference Hint not displayed

Hint B.3

Find your distance from speaker A Hint not displayed

Express your answers in meters to three significant figures.

ANSWER:

= 5.62 Correct

A Pipe Filled with Helium A certain organ pipe, open at both ends, produces a fundamental frequency of 256 air.

in

Part A If the pipe is filled with helium at the same temperature, what fundamental frequency will it produce? Take the molar mass of air to be 28.8 to be 4.00 Hint A.1

and the molar mass of helium

. How to approach the problem Hint not displayed

Hint A.2

Find the length of the pipe Hint not displayed

Hint A.3

Find the speed of sound in helium Hint not displayed



Express your answer in hertz.

ANSWER:

= 750 Correct

Because helium is less dense than air and has a lower molar mass, sound waves propagate faster in helium than in air. Thus, the frequencies produced in the pipe when it is filled with helium are higher than those produced in the same pipe filled with air.

Part B Now consider a pipe that is stopped (i.e., closed at one end) but still has a fundamental frequency of 256 in air. How does your answer to Part A, , change? Hint B.1

How to approach the question Hint not displayed

ANSWER:

increases. decreases. stays the same. Correct

The fundamental frequency of the pipe in helium is given by . This relationship is independent of the length of the pipe or whether the pipe is open or stopped. A relationship of this type, known as a scaling law, is very powerful because it allows you to solve problems without knowing all of the values that would normally be relevant. In this case, you can determine the frequency in helium knowing only the frequency in air and the ratio of the speed of sound in helium to that in air.

The Beat Heard by a Bat



A bat flies toward a wall, emitting a steady sound with a frequency of 25.0 hears its own sound plus the sound reflected by the wall.

. This bat

Part A How fast should the bat fly,

, to hear a beat frequency of 215

Take the speed of sound to be 344 Hint A.1

?

.


Hint A.2

Find the frequency of the wave bouncing off the wall Hint not displayed

Hint A.3

Find the frequency of the echo that the bat hears Hint not displayed

Hint A.4

Find the expression for the beat frequency Hint not displayed

Hint A.5

Working the math Hint not displayed

Express your answer numerically in meters per second to three significant figures.

ANSWER:

= 1.54 Correct


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