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MasteringPhysics: Assignment Print View

Cross Section for Asteroid ImpactIn 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 earth is taken to be infinite. Find the total initial energy of the asteroid.

Hint A.1 Gravitational potential energy

The gravitational potential energy is proportional to and approaches zero for large values of .

Express your answer in terms 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 momentumHint not displayed


ANSWER: =

Answer Requested

Part C

The maximum impact parameter for which collision is guaranteed, , is obtained by setting the minimum earth-asteroid separation equal to the radius of the earth. This is 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. When , what is the total energy of the asteroid the instant before it crashes into the earth? Assume that the speed of the asteroid at closest approach is .

Hint C.1 Potential energyHint not displayed


ANSWER:

=

Answer Requested

Part D



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

Hint D.1 Direction of velocity before impactHint not displayed


ANSWER: =

Answer Requested

Part E

Use conservation of energy and angular momentum to find an expression for .

Hint E.1 Find the final velocityHint not displayed

Express your answer as a function of , , , and .

ANSWER:

=

Answer Requested

Part F

The collision cross section represents the effective target area "seen" by the asteroid

and is found by multiplying 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 terms 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 in terms of and .

Hint F.1 Find the escape speedHint not displayed



Express the collision cross section in terms of and .

ANSWER:

=

Answer Requested

Part G

The point of origin of a typical asteroid might lie at a radius of about (astronomical

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

Calculate the effective target cross section of the earth as seen by the asteroid. Assume the asteroid's orbit is cicular.

Hint G.1 Useful constants Hint not displayed

Hint G.2 Find the orbital speed of the asteroidHint 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 earth. The mass of the earth is and its radius is . Neglect air resistance throughout this problem, since the spacecraft is primarily moving through the near vacuum of space.

Part AFind the speed of the spacecraft when it crashes into the earth.

Hint A.1 How to approach the problemHint not displayed

Hint A.2 Total energyHint not displayed

Hint A.3 Potential energyHint not displayed

Express the speed in terms of , , and the universal gravitational constant .

ANSWER:

=

Correct

Part BNow find the spacecraft's speed when its distance from the center of the earth is

, where .

Hint B.1 General approachHint not displayed

Hint B.2 First step in finding the speedHint not displayed

Express the speed in terms 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, , inside the planet.

Hint A.1 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 Finding from

Hint not displayed

Express the acceleration due to gravity in terms of , , , and , the universal gravitational constant.

ANSWER:

=

Correct

Part B



Rewrite your result for in terms of , the gravitational acceleration at the surface of the planet, times a function of R.

Hint B.1 Acceleration at the surfaceHint not displayed

Express your answer in terms of , , 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 CFind a numerical value for , the average density of the earth in kilograms per cubic meter. Use for the radius of the earth, , and a

value of at the surface of .

Hint C.1 How to approach the problemHint not displayed

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 . The

minimum and maximum distances of the planet from the sun are and , respectively.

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

Hint A.1 Kepler's 3rd lawHint not displayed

Hint A.2 Find the semi-major axisHint not displayed

Hint A.3 Find the period of a circular orbitHint not displayed

Express the period in terms 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 of 1.50×1011 from the center of the sun. What is the net force that the sun would exert on such a Dyson sphere were it to get displaced off-center by some small amount?


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 on a unit mass located on the outer surface of the Dyson sphere described in Part A?

Hint B.1 How to approach the problemHint not displayed

Hint B.2 Find the gravitational force exerted by the Dyson sphereHint not displayed

Hint B.3 Find the gravitational force exerted by the sunHint not displayed

Express your answer in newtons.

ANSWER: = 6.26×10−3 Correct

Part C



What is the net gravitational force on a unit mass located on the inner surface of the Dyson sphere described in Part A?

Hint C.1 How to approach the problemRecall 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.


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 DWhat is the linear speed of a unit mass located at the inner equator of such a sphere?

Hint D.1 How to approach the problemBecause 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 acting on a unit mass located on the surface of the earth?

Hint D.3.1 Acceleration of gravityHint not displayed


ANSWER: = Answer not displayed

Hint D.4 Equation for centripetal accelerationRecall that the equation relating the centripetal acceleration of an object spinning about a point a distance away at a speed is given by

.

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



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 = -130 , one is at the origin, and one is at = 390 .

Part A

What is the magnitude of the net gravitational force on the mass at the origin due to the other two masses? Take the gravitational constant to be = 6.67×10−11 .


Hint A.2 Calculate the gravitational force from the first massHint 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 massHint 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 = 1.99×1030 , the gravitational constant to be

= 6.67×10−11 , and the acceleration due to gravity at the earth's surface to

be = 9.810 .


Hint A.2 Law of universal gravitationHint not displayed

Hint A.3 Calculate your massHint not displayed

Hint A.4 Calculate the distance between you and the starHint not displayed

Express your weight in newtons.



ANSWER: =

1.25×1014 All attempts used; correct answer displayed

This is over 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 terms of given initial conditions.

A common problem in physics is to match the particular initial conditions - generally given as an initial position and velocity at - once you have obtained the general solution. 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. and

3. ,

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 in terms of the initial position and velocity of the oscillator.

Hint A.1 The only good way to startHint not displayed

Hint A.2 Using kinematic relationshipsHint not displayed

Hint A.3 Initial positionHint not displayed

Hint A.4 Initial velocityHint not displayed

Give your answers in terms of , , and . Separate your answers with a comma.

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 support.



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 (and ).

Hint B.1 Find the change in spring lengthHint not displayed

Hint B.2 Find the moment armHint 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 CWhat is the angular frequency of oscillations of the rod?

Hint C.1 How to find the oscillation frequencyHint not displayed

Hint C.2 Solve the angular equation of motionHint not displayed

Hint C.3 Determine the moment of inertia of the rodHint not displayed

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

ANSWER:

=

Correct



Note that if the spring were simply attached to a mass , or if the mass of the rod

were concentrated at its ends, would be . 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.

Consider an oscillating system of mass and natural angular frequency . When the system is subjected to a periodic external (driving) force, whose maximum value is and angular frequency is , the amplitude of the driven oscillations 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



Assume that, when we walk, in addition to a fluctuating vertical force, we exert a periodic lateral force of amplitude 25 at a frequency of about 1 . Given that the mass of the bridge is about 2000 per linear meter, how many people were walking along the 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 , 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 constantHint not displayed

Hint A.4 Find the total number of synchronized pedestriansHint not displayed

Hint A.5 Find the mass of the bridgeHint not displayed

Hint A.6 Find the angular frequencyHint not displayed

Express your answer numerically to three significant figures.

ANSWER: number of people =

1810 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



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.2 What formula to considerHint 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 useHint not displayed

Hint A.3 Find the amplitudeHint 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 Correct kg/s

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 oscillator whose amplitude is and 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 bearingHint not displayed

Hint A.2 Find the acceleration of the ball bearingHint not displayed

Your result should be in terms of , (or ), , , a phase angle , and the constant .

ANSWER: =

Correct

Part B

The frequency is slowly increased. Once it passes 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 contactHint not displayed

Hint B.2 Find the value of when the ball loses contact

Hint not displayed

Hint B.3 Relation between and

Hint not displayed

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

ANSWER: =

Correct



Oscillations of a Balanced Object

Two identical thin rods, each of mass and length , are joined 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 AFind , the angular frequency of oscillation of the object.

Hint A.1 Determine the angular frequency of a physical pendulumHint not displayed

Hint A.2 Calculate

Hint not displayed

Hint A.3 Find the moment of inertiaHint 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 and stiffness . The crates are released and the springs compress to a length before bringing the

crates back up to their original positions.

Part ARank the time required for the crates to return to their initial positions from largest to smallest.

Hint A.1 Formula for the periodHint not displayed

Hint A.2 Determining the massHint 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 of the two-spring system?

Hint A.1 Free-body diagramHint not displayed

Hint A.2 Free-body diagram for spring 2Hint not displayed

Hint A.3 Determine the extension of spring 2Hint not displayed

Hint A.4 Determine the extension of spring 1Hint not displayed

Hint A.5 Determine the total extension of the two springsHint not displayed

Hint A.6 Replace and Hint not displayed

Hint A.7 Determine the extension of the equivalent systemHint not displayed

Hint A.8 Solving for

Hint not displayed

Express the effective spring constant in terms of and .

ANSWER:

=

Correct

Three springs in series



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.

Express your answer in terms of , , 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 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



What is the period of oscillation of the scale?

Hint B.1 Period of oscillation in SHMHint 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 AWhat is the value of the acceleration of gravity on this planet?


Hint A.2 Calculate the periodHint not displayed

Hint A.3 Equation for the periodHint 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 AIgnoring the mass of the spring, what is the mass of the fish?


Hint A.2 Calculate the spring constantHint not displayed

Hint A.3 Calculate the angular frequencyHint not displayed

Hint A.4 Formula for the angular frequency of a mass on a springHint 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 frequencyHint 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 passes 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 weightlessHint not displayed


Hint A.3 Find the maximum acceleration of the stringHint not displayed

Hint A.4 Putting it all togetherHint not displayed



Express the minimum wave amplitude in terms 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 support, and masses are hung from the free end of each wire. Each wire is 1, 2, or 3 units long, and each supports 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 ARank each wire-mass system on the basis of its fundamental wavelength.

Hint A.1 Identify the fundamental wavelengthHint not displayed




ANSWER:

View Correct

Part BRank each wire-mass system on the basis of its wave speed.

Hint B.1 Factors that determine wave speedHint not displayed

Hint B.2 Tension in the wiresHint 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 frequencyHint 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 and to correctly represent the direction of the wave velocity at points A and B.

Hint A.1 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 passes 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 velocityA 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 and to correctly represent the direction of the velocity of points A and B.

Hint B.1 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

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


Hint A.2 Wave equation for a stringHint not displayed

Hint A.3 Find the general speed of the waveHint 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 terms of and constants such as (the magnitude of the acceleration due to gravity), , etc.

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 AHow 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 segmentHint not displayed

Hint A.3 Some math helpHint not displayed

Express the time in terms 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 . The steel wire has a length of 0.700 and a mass of 5.25 .

Part A

What is the frequency of the string's fundamental mode of vibration?


Hint A.2 Find the mass per unit lengthHint 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 stringHint not displayed

Express your answer exactly.

ANSWER: = 70 Correct

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

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 members, sequentially, do the same. Each member records the initial beat frequency between her low E string and Kira's low E string.

Part ARank each member on the basis of the frequency of her low E string.

Hint A.1 Beat frequencyHint not displayed

Hint A.2 Find the frequency of Aiko's E stringHint not displayed


ANSWER:

View Correct



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

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

Part BRank 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 stringHint 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 AGiven 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 in meters.

ANSWER: = 3.16 Correct m

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 functionsHint not displayed

Express the distance in terms of . Be careful about your signs!



ANSWER:

=

All attempts used; correct answer displayed

m

Interference of Sound Waves

Two loudspeakers, A and B, are driven by the same amplifier and emit sinusoidal waves in phase. The frequency of the waves emitted by each speaker is 172 . You are 8.00

from speaker A. Take the speed of sound in air to be 344 .

Part AWhat is the closest you can be to speaker B and be at a point of destructive interference?


Hint A.2 Find the wavelength of the sound waveHint not displayed

Hint A.3 Find the condition for destructive interferenceHint 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 same amplifier so that they emit sound waves in phase at 688 . Take the speed of sound

in air to be 344 .

Part AIf 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.2 Constructive and destructive interferenceHint not displayed

Hint A.3 Find the wavelength of the soundHint not displayed

ANSWER: yesno

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 you need to walk forward to be at a point where you cannot hear the speakers?


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 in air.

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 and the molar mass of helium

to be 4.00 .


Hint A.2 Find the length of the pipeHint not displayed

Hint A.3 Find the speed of sound in heliumHint 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 BNow 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 questionHint 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 . This bat hears its own sound plus the sound reflected by the wall.

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.2 Find the frequency of the wave bouncing off the wallHint not displayed

Hint A.3 Find the frequency of the echo that the bat hearsHint not displayed

Hint A.4 Find the expression for the beat frequencyHint not displayed

Hint A.5 Working the mathHint not displayed

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

ANSWER: = 1.54

Correct


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