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MasteringPhysics: Assignment Print View

Introduction to Collisions

Learning Goal: To understand how to find the velocities of particles after a collision.There are two main types of collisions that you will study: elastic and perfectly inelastic. In an elastic collision, kinetic energy is conserved. In a perfectly inelastic collision, the particles stick together and thus have the same velocity after the collision. There is actually a range of collision types, with elastic and perfectly inelastic at the extreme ends. These extreme cases are easier to solve than the in-between cases.In this problem, we will look at one of these in-between cases after first working through some basic calculations related to elastic and perfectly inelastic collisions.

Let two particles of equal mass collide. Particle 1 has initial velocity , directed to the right, and particle 2 is initially stationary.

Part AIf the collision is elastic, what are the final velocities and of particles 1 and 2?

Hint A.1 How to approach the problemHint not displayed

Hint A.2 Conservation of momentumHint not displayed

Hint A.3 Conservation of energyHint not displayed

Give the velocity of particle 1 followed by the velocity of particle 2, separated by a comma. Express each velocity in terms of .

ANSWER: =

Correct

Part B

http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=1469625 (1 of 68) [12/13/2010 7:05:48 PM]


Now suppose that the collision is perfectly inelastic. What are the velocities and of the two particles after the collision?

Hint B.1 How to approach the problemHint not displayed

Hint B.2 Conservation of momentumHint not displayed

Give the velocity of particle 1 followed by the velocity of particle 2, separated by a comma. Express the velocities in terms of .

ANSWER: =

Correct

Part C

Now assume that the mass of particle 1 is , while the mass of particle 2 remains . If the collision is elastic, what are the final velocities and of particles 1 and 2?

Hint C.1 How to approach the problemHint not displayed

Hint C.2 Conservation of momentumHint not displayed

Hint C.3 Conservation of energyHint not displayed


ANSWER: =

Correct

Note that in both the conservation of momentum equation and the conservation of energy equation, cancels out. This is a general feature of many collision situations: The ratio of the two masses is important, but the absolute masses are not.



Part D

Let the mass of particle 1 be and the mass of particle 2 be . If the collision is perfectly inelastic, what are the velocities of the two particles after the collision?

Hint D.1 How to approach the problemHint not displayed

Hint D.2 Conservation of momentumHint not displayed


ANSWER: =

Correct

This applet shows two disks colliding. The orange disk has always the same initial velocity. You can change the ratio of the masses of the two disks and the elasticity of the collision. You should try the four different settings corresponding to Parts A through D. An elastic collision has elasticity , and a perfectly inelastic collision has elasticity .

Part EWhat qualitative change takes place as the ratio of the mass of the blue disk to the mass

of the orange disk, , increases from 0.3 to 4.0? Set the elasticity to 1.0 for a perfectly

elastic collision.

ANSWER: The final speed of the orange disk decreases as the ratio of masses increases. As the ratio increases past 1.0, the final velocity of the orange disk changes direction. The difference in final velocities between the disks decreases. The difference in final velocities between the disks increases.

Correct

Most real collisions are somewhere between elastic and perfectly inelastic. This is



indicated by the elasticity of the collision, which measures the difference in the velocities of the particles after the collision compared with the difference in velocities before the collision. For instance, in a perfectly inelastic collision, the two particles stick together after colliding. The elasticity of such a collision is , because the difference in velocities between the particles is 0 after they collide.Technically, the elasticity is defined by the relation , where and are the initial and final velocities of particle 1, and and are the initial and final velocities of particle 2. In this problem, the formula is simplified by our definition of

and the hypothesis . So, using for the final velocity of particle 1 and

for the final velocity of particle 2, we obtain the simpler formula .This final form will be most useful to you in solving Part F.

Part F

If the two particles with equal masses collide with elasticity , what are the final velocities of the particles? Assume that particle 1 has initial velocity and particle 2 is initially at rest. Look at the applet to be sure that your answer is reasonable.

Hint F.1 How to approach the problemHint not displayed


ANSWER: =

Correct

Notice that if you look back at your answers to Parts A and C, the diference between and is always , as you would expect from setting in the definition of

elasticity. It is possible, though it takes some algebra, to prove that the definition of elasticity with implies conservation of energy.This applet is the same as the previous one, but now you are given a graph of the momentum for each disk at the bottom. Run a few of the collisions that you have studied in this problem so that you can see how the momenta of the two disks change with differing elasticities and mass ratios.Also in this applet you can have the two disks collide off-center. While this looks much more complicated, the law of conservation of momentum still always applies. With a modification to make it more precise for two- and three-dimensional collisions, the definition of elasticity still applies as well.

PSS 8.1 Conservation of Momentum



Learning Goal: To practice Problem-Solving Strategy 8.1 Conservation of Momentum.

Protecting his nest, a 600- peregrine falcon rams a marauding 1.5- raven in midair. The

falcon is moving at 20.0 , and the raven at 9.00 at the moment of impact. The falcon strikes the raven at right angles to the raven's direction of flight and rebounds straight back with a speed of 5.00 . By what angle does the impact change the raven's direction of motion?

Problem-Solving Strategy: Conservation of momentum IDENTIFY the relevant concepts: First, decide whether momentum is conserved – that is, whether the vector sum of the external forces acting on the system is zero. If it isn't, you can't use conservation of momentum. SET UP the problem using the following steps:

1. Define a coordinate system. Often it's easiest to point the x axis in the direction of one of the initial velocities. Make sure you are using an inertial frame of reference.

2. Draw "before" and "after" sketches, treating each body as a particle. Draw vectors for known velocities, and indicate all other given information. We usually use capital letters to label each particle and subscripts 1 and 2 for initial and final velocities, respectively.

3. Identify your target variable(s). EXECUTE the solution as follows:

1. Write an equation in symbols equating the total initial x component of momentum (i.e., before the interaction) to the total final x component of momentum (i.e., after the interaction), using for each particle. Do the same for the initial and final y components, if needed. Be careful with signs!

2. Solve your equation(s) for the desired unknown variables. You may need to convert from velocity components to magnitudes and directions, or vice versa.

3. In some problems, energy considerations give additional relationships among the various velocities. EVALUATE your answer: Does your answer make physical sense?

IDENTIFY the relevant concepts Each bird is acted on by four external forces: The air exerts lift, thrust, and drag forces on the wings and body, and the earth exerts weight. The forces that the birds exert on each other are internal to the system. During the collision, the forces that the birds exert on each other are much larger than any net external force on either bird. Since external forces acting on the birds are negligibly small compared to the internal forces of the collision, you can treat the birds' momentum as conserved during the collision.

SET UP the problem using the following steps

Part A



You must define your coordinate axes. Which of the choices shown in the figure is the most convenient set of axes for this problem?

ANSWER:

Correct

Part BNow, use the diagram below to draw before-and-after sketches for the collision. The dots represent the raven (R) and the falcon (F). Assume that the raven is initially flying in the x direction. Add vectors representing the initial and final velocities of both birds (four vectors total). You will have to estimate the raven's final velocity; any reasonable guess will be accepted.

Draw the vectors starting at the appropriate black dots. The location, orientation and length of the vectors will be graded.

ANSWER:

View All attempts used; correct answer displayed



When you draw an actual sketch for a problem like this, you should include known values (in this case, the birds' masses and speeds) and the quantities you may need, including any components. Your real diagram might look like this:

Note that and are the x and y components of the raven's final velocity.

EXECUTE the solution as follows

Part C

By what angle does the falcon change the raven's direction of motion?

Hint C.1 Determine how to approach the problem Hint not displayed

Hint C.2 The signs of the velocities Hint not displayed

Hint C.3 Find an expression for the x component of the raven's final velocity

Hint not displayed

Hint C.4 Find an expression for the y component of the raven's final velocity

Hint not displayed

Express your answer in degrees to three significant figures.

ANSWER: = 48.0

Correct



EVALUATE your answer

Part D

Your answer says that the raven is deflected through 48 by a falcon that weighs a bit less than half as much but that is moving roughly twice as fast before the collision. Is this answer physically plausible? To decide, classify the following idealized collisions according to the angle by which particle A would be deflected.

Drag the appropriate items to their respective bins.

ANSWER:

View Correct

We see that your answer is physically reasonable. The magnitude of the falcon's initial momentum is actually slightly less than that of the raven, so if the birds stuck together, the deflection would be less than 45 . The raven's actual deflection is greater than 45 because the falcon bounces back.

The Impulse-Momentum Theorem

Learning Goal: To learn about the impulse-momentum theorem and its applications in some common cases.

Using the concept of momentum, Newton's second law can be rewritten as

, (1)

where is the net force acting on the object, and is the rate at which the object's

momentum is changing.If the object is observed during an interval of time between times and , then integration of both sides of equation (1) gives



. (2)

The right side of equation (2) is simply the change in the object's momentum . The

left side is called the impulse of the net force and is denoted by . Then equation (2) can

be rewritten as.

This equation is known as the impulse-momentum theorem. It states that the change in an object's momentum is equal to the impulse of the net force acting on the object. In the case of a constant net force acting along the direction of motion, the impulse-momentum theorem can be written as

. (3)

Here , , and are the components of the corresponding vector quantities along the chosen coordinate axis. If the motion in question is two-dimensional, it is often useful to apply equation (3) to the x and y components of motion separately.

The following questions will help you learn to apply the impulse-momentum theorem to the cases of constant and varying force acting along the direction of motion. First, let us consider a particle of mass moving along the x axis. The net force is acting on the particle along the x axis. is a constant force.

Part A

The particle starts from rest at . What is the magnitude of the momentum of the particle at time ? Assume that .

Express your answer in terms of any or all of , , and .

ANSWER: =

Correct

Part B



The particle starts from rest at . What is the magnitude of the velocity of the particle at time ? Assume that .

Express your answer in terms of any or all of , , and .

ANSWER: =

Correct

Part CThe particle has momentum of magnitude at a certain instant. What is , the magnitude of its momentum seconds later?

Express your answer in terms of any or all of , , , and .

ANSWER: =

Correct

Part DThe particle has momentum of magnitude at a certain instant. What is , the magnitude of its velocity seconds later?

Express your answer in terms of any or all of , , , and .

ANSWER: =

Correct

Let us now consider several two-dimensional situations.A particle of mass is moving in the positive x direction at speed . After a certain constant force is applied to the particle, it moves in the positive y direction at speed .

Part E



Find the magnitude of the impulse delivered to the particle.

Hint E.1 How to approach the problemHint not displayed

Hint E.2 Find the change in momentumHint not displayed

Express your answer in terms of and . Use three significant figures in the numerical coefficient.

ANSWER: =

Correct

Part F

Which of the vectors below best represents the direction of the impulse vector ?

ANSWER: 1 2 3 4 5 6 7 8

Correct



Part G

What is the angle between the positive y axis and the vector as shown in the figure?

ANSWER: 26.6 degrees 30 degrees 60 degrees 63.4 degrees

Correct

Part H

If the magnitude of the net force acting on the particle is , how long does it take the particle to acquire its final velocity, in the positive y direction?

Express your answer in terms of , , and . If you use a numerical coefficient, use three significant figures.

ANSWER: =

Correct

So far, we have considered only the situation in which the magnitude of the net force acting on the particle was either irrelevant to the solution or was considered constant. Let us now consider an example of a varying force acting on a particle.



Part I

A particle of mass kilograms is at rest at seconds. A varying force

is acting on the particle between seconds and

seconds. Find the speed of the particle at seconds.

Hint I.1 Use the impulse-momentum theoremHint not displayed

Hint I.2 What is the correct antiderivative?Hint not displayed

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

ANSWER: = 43.0

Correct

A Ball Hits a Wall Elastically

A ball of mass moving with velocity strikes a vertical wall. The angle between the ball's initial velocity vector and the wall is

as shown on the diagram, which depicts the situation as seen from above. The duration of the collision between the ball and the wall is , and this collision is completely elastic. Friction is negligible, so the ball does not start spinning. In this idealized collision, the force exerted on the ball by the wall is parallel to the x axis.

Part A



What is the final angle that the ball's velocity vector makes with the negative y axis?

Hint A.1 How to approach the problemRelate the vector components of the ball's initial and final velocities. This will allow you to determine in terms of .

Hint A.2 Find the y component of the ball's final velocityWhat is , the component of the final velocity of the ball?

Hint A.2.1 How to approach this partHint not displayed

Express your answer in terms of quantities given in the problem introduction and/or and , the and components of the ball's initial velocity.

ANSWER: =

Correct

Hint A.3 Find the component of the ball's final velocity

What is , the component of the ball' final velocity?

Hint A.3.1 How to approach this problemHint not displayed

Express your answer in terms of quantities given in the problem introduction and/or and , the and components of the ball's initial velocity.

ANSWER: =

Correct

The wall exerts a force on the ball in the direction. However, because energy is conserved in this collision, the final speed of the ball must be equal to its initial speed. Since there is no force on the ball in the y direction, the magnitude of the component of the ball's velocity is constant. Therefore, the magnitude of the component of the velocity must be constant as well. However, the sign of the velocity will change as the ball moves first toward, then away from, the wall.



Hint A.4 Putting it togetherOnce you find the vector components of the final velocity in terms of the initial velocity, use the geometry of similar triangles to determine in terms of .

Express your answer in terms of quantities given in the problem introduction.

ANSWER: =

Correct

Part B

What is the magnitude of the average force exerted on the ball by the wall?

Hint B.1 What physical principle to useHint not displayed

Hint B.2 Change in momentum of the ballHint not displayed

Express your answer in terms of variables given in the problem introduction and/or .

ANSWER:

=

Correct

A Game of Frictionless Catch



Chuck and Jackie stand on separate carts, both of which can slide without friction. The combined mass of Chuck and his cart, , is identical to the combined mass of Jackie and her cart. Initially, Chuck and Jackie and their carts are at rest.Chuck then picks up a ball of mass and throws it to Jackie, who catches it. Assume that the ball travels in a straight line parallel to the ground (ignore the effect of gravity). After Chuck throws the ball, his speed relative to the ground is . The speed of the thrown ball relative to the ground is .Jackie catches the ball when it reaches her, and she and her cart begin to move. Jackie's speed relative to the ground after she catches the ball is .When answering the questions in this problem, keep the following in mind:1. The original mass of Chuck and his cart does not include the mass of the ball. 2. The speed of an object is the magnitude of its velocity. An object's speed will always be a nonnegative quantity.

Part AFind the relative speed between Chuck and the ball after Chuck has thrown the ball.


Express the speed in terms of and .

ANSWER: =

Correct

Make sure you understand this result; the concept of "relative speed" is important. In general, if two objects are moving in opposite directions (either toward each other or away from each other), the relative speed between them is equal to the sum of their speeds with respect to the ground. If two objects are moving in the same direction, then the relative speed between them is the absolute value of the difference of the their two speeds with respect to the ground.

Part B



What is the speed of the ball (relative to the ground) while it is in the air?


Hint B.2 Initial momentum of Chuck, his cart, and the ballHint not displayed

Hint B.3 Find the final momentum of Chuck, his cart, and the thrown ball

Hint not displayed

Express your answer in terms of , , and .

ANSWER: =

Correct

Part CWhat is Chuck's speed (relative to the ground) after he throws the ball?



ANSWER: =

Correct

Part D



Find Jackie's speed (relative to the ground) after she catches the ball, in terms of .


Hint D.2 Initial momentumHint not displayed

Hint D.3 Find the final momentumHint not displayed

Express in terms of , , and .

ANSWER: =

Correct

Part EFind Jackie's speed (relative to the ground) after she catches the ball, in terms of .

Hint E.1 How to approach the problemHint not displayed


ANSWER: =

Correct

A Girl on a Trampoline



A girl of mass kilograms springs from a trampoline with an initial upward velocity of

meters per second. At height meters above the trampoline, the girl grabs a

box of mass kilograms.

For this problem, use meters per second per second for the magnitude of the acceleration due to gravity.

Part AWhat is the speed of the girl immediately before she grabs the box?


Hint A.2 Initial kinetic energyHint not displayed

Hint A.3 Potential energy at height

Hint not displayed

Express your answer numerically in meters per second.

ANSWER: = 4.98

Correct

Part B



What is the speed of the girl immediately after she grabs the box?


Hint B.2 Total initial momentumHint not displayed

Express your answer numerically in meters per second.

ANSWER: = 3.98

Correct

Part CIs this "collision" elastic or inelastic?

Hint C.1 Definition of an inelastic collisionHint not displayed

ANSWER: elasticinelastic

Correct

In inelastic collisions, some of the system's kinetic energy is lost. In this case the kinetic energy lost is converted to heat energy in the girl's muscles as she grabs the box, and sound energy.

Part D



What is the maximum height that the girl (with box) reaches? Measure with respect to the top of the trampoline.


Hint D.2 Finding

Hint not displayed

Hint D.3 Finding

Hint not displayed

Express your answer numerically in meters.

ANSWER: = 2.81

Correct

An Exciting Encounter

An atom of mass is initially at rest, in its ground state. A moving (nonrelativistic) electron of mass collides with the atom. The atom+electron system can exist in an excited state in which the electron is absorbed into the atom. The excited state has an extra, "internal," energy relative to the atom's ground state.



Part A

Find the kinetic energy that the electron must have in order to excite the atom.


Hint A.2 Find the final kinetic energy in terms of the initial kinetic energy of the electron

Hint not displayed


ANSWER:

=

All attempts used; correct answer displayed

Part BWe can use the result from Part A to study a process of interest in atomic physics: a collision of two atoms that causes one of the atoms to ionize (lose an electron). In this case, is the energy needed to ionize one of the atoms, called the ionization energy. The most efficient way to ionize an atom in a collision with another atom is for the collision to be completely inelastic (atoms stick together after the collision). If the collision were perfectly elastic, then translational kinetic energy would be conserved, and there would be no energy left over for exciting the atom. If the collision were partially elastic, then some of the initial kinetic energy would be converted into internal energy, but not as much as in a perfectly inelastic collision. In practice, interatomic collisions are never perfectly inelastic, but analyzing this case can give a lower bound on the amount of kinetic energy needed for ionization.Is it possible to ionize an atom of , initially at rest, by a collision with an atom of

that has kinetic energy of 4.0 electron volts? The ionization energy of the cesium atom is 3.9 electron volts.You can take the mass of the oxygen atom to be 16 atomic mass units and that of the cesium atom to be 133 atomic mass units. It doesn't matter what mass units you choose, as long as you are consistent. For this question, it is most convenient to use atomic mass units, since these are the numbers you are provided with.




Hint B.2 What is the correspondence with Part A?Hint not displayed

Hint B.3 The masses of cesium and oxygenHint not displayed

ANSWER: yes no

Correct

Part C

What is the least possible initial kinetic energy the oxygen atom could have and still excite the cesium atom?


Express your answer in electron volts, to one decimal place.

ANSWER: =

4.4 All attempts used; correct answer displayed

A Superball Collides Inelastically with a Table

As shown in the figure , a superball with mass equal to 50 grams is dropped from a



height of . It collides with a table, then bounces up to a height of . The duration of the collision (the time during which the superball is in contact with the table) is . In this problem, take the positive y direction to be upward, and use for the magnitude of the acceleration due to gravity. Neglect air resistance.

Part AFind the y component of the momentum, , of the ball immediately before the collision.


Hint A.2 Find the speed of the ballHint not displayed

Express your answer numerically, to two significant figures.

ANSWER: = -0.27

Correct

Part B



Find the y component of the momentum of the ball immediately after the collision, that is, just as it is leaving the table.


Hint B.2 Find the speed of the ballHint not displayed


ANSWER: = 0.22

Correct

Part C

Find the y component of the time-averaged force , in newtons, that the table exerts on the ball.



ANSWER: = 33

Correct

Part D

Find , the y component of the impulse imparted to the ball during the collision.



ANSWER: = 0.49

Correct



Part E

Find , the change in the kinetic energy of the ball during the collision, in joules.

Hint E.1 Find the kinetic energy before the collisionHint not displayed


ANSWER: = -0.25

Correct

Ballistic Pendulum

In a ballistic pendulum an object of mass is fired with an initial speed at a pendulum bob. The bob has a mass , which is suspended by a rod of length and negligible mass. After the collision, the pendulum and object stick together and swing to a maximum angular displacement as shown .

Part A



Find an expression for , the initial speed of the fired object.


Hint A.2 Determine which physical laws and principles applyHint not displayed

Hint A.3 Describe the collisionHint not displayed

Hint A.4 Describe the swingHint not displayed

Hint A.5 Relating the two physical processesHint not displayed

Express your answer in terms of some or all of the variables , , , and and the acceleration due to gravity, .

ANSWER:

=

Correct

The ballistic pendulum was invented during the Napoleonic Wars to aide the British Navy in making better cannons. It has since been used by ballisticians to measure the velocity of a bullet as it leaves the barrel of a gun. In Part B you will use your expression for to compare the initial speeds of bullets fired from 9- and .44-caliber handguns.

Part B



An experiment is done to compare the initial speed of bullets fired from different handguns: a 9 and a .44 caliber. The guns are fired into a 10- pendulum bob of length . Assume that the 9- bullet has a mass of 6 and the .44-caliber bullet has a mass of 12 . If the 9- bullet causes the pendulum to swing to a maximum angular displacement of 4.3 and the .44-caliber bullet causes a displacement of 10.1 , find the ratio of the initial speed of the 9- bullet to the speed of the .44-caliber bullet, .


Express your answer numerically.

ANSWER: = 0.847 Correct

Police officers in the United States commonly carry 9- handguns because they are easier to handle, having a shorter barrel than typical .44-caliber guns. Not only does the .44-caliber bullet have more mass than the 9- one, its passage through a longer gun barrel means that it also moves faster as it leaves the barrel, which makes the .44-caliber Magnum a particularly powerful handgun. A .44-caliber bullet can travel at speeds over 1000 (1600 ).

Colliding Cars

In this problem we will consider the collision of two cars initially moving at right angles. We assume that after the collision the cars stick together and travel off as a single unit. The collision is therefore completely inelastic.Two cars of masses and collide at an intersection. Before the collision, car 1 was traveling eastward at a speed of , and car 2 was traveling northward at a speed of .



After the collision, the two cars stick together and travel off in the direction shown.

Part A

First, find the magnitude of , that is, the speed of the two-car unit after the collision.

Hint A.1 Conservation of momentumHint not displayed

Hint A.2 x and y components of momentumHint not displayed

Hint A.3 A vector and its componentsHint not displayed

Hint A.4 Velocity and momentumHint not displayed

Express in terms of , , and the cars' initial speeds and .

ANSWER:

=

Correct



Part B

Find the tangent of the angle .

Express your answer in terms of the momenta of the two cars, and .

ANSWER: =

Correct

Part C

Suppose that after the collision, ; in other words, is . This means that before the collision:

ANSWER: The magnitudes of the momenta of the cars were equal. The masses of the cars were equal. The velocities of the cars were equal.

Correct

Collision at an Angle

Two cars, both of mass , collide and stick together. Prior to the collision, one car had been traveling north at speed , while the second was traveling at speed at an angle south of east (as indicated in the figure). After the collision, the two-car system travels at speed at an angle east of north.



Part AFind the speed of the joined cars after the collision.

Hint A.1 Determine the conserved quantitiesHint not displayed

Hint A.2 The component of the final velocity in the east-west directionHint not displayed

Hint A.3 Find the north-south component of the final momentumHint not displayed

Hint A.4 Math helpHint not displayed

Express your answer in terms of and .

ANSWER: =

Correct

Part B



What is the angle with respect to north made by the velocity vector of the two cars after the collision?

Hint B.1 A formula for

Hint not displayed

Express your answer in terms of . Your answer should contain an inverse trigonometric function.

ANSWER:

=

Correct

Elastic Collision in One Dimension

Block 1, of mass , moves across a frictionless surface with speed . It collides elastically with block 2, of mass , which is at rest ( ). After the collision, block 1 moves with speed , while block 2 moves with speed . Assume that

, so that after the collision, the two objects move off in the direction of the first object before the collision.

Part A



This collision is elastic. What quantities, if any, are conserved in this collision?

Hint A.1 What to think aboutHint not displayed

ANSWER: kinetic energy only momentum only kinetic energy and momentum

Correct

Part BWhat is the final speed of block 1?


Hint B.2 Apply conservation of momentumHint not displayed

Hint B.3 Apply conservation of energyHint not displayed

Hint B.4 Putting it togetherHint not displayed


ANSWER:

=

Correct

Part C



What is the final speed of block 2?

Hint C.1 Using the result from the previous partHint not displayed


ANSWER: =

Correct

Filling the Boat

A boat of mass 250 is coasting, with its engine in neutral, through the water at

speed 3.00 when it starts to rain. The rain is falling vertically, and it accumulates in the

boat at the rate of 10.0 .

Part A

What is the speed of the boat after time 2.00 has passed? Assume that the water resistance is negligible.


Hint A.2 Find the momentum of the boat before it starts to rainHint not displayed

Hint A.3 Find the mass of the boat after it has started to rainHint not displayed

Express your answer in meters per second.

ANSWER: 2.78 Correct

Part B



Now assume that the boat is subject to a drag force due to water resistance. Is the component of the total momentum of the system parallel to the direction of motion still conserved?

ANSWER: yesno

Correct

The boat is subject to an external force, the drag force due to water resistance, and therefore its momentum is not conserved.

Part C

The drag is proportional to the square of the speed of the boat, in the form . What is the acceleration of the boat just after the rain starts? Take the positive axis along the direction of motion.


Hint C.2 Find the time rate of change of momentum of the boatHint not displayed

Express your answer in meters per second per second.

ANSWER:−1.80×10−2 Correct

Impulse and Change in Velocity



A glob of very soft clay is dropped from above onto a digital scale. The clay sticks to the scale on impact. A graph of the clay's velocity vs. time, , is given, with the upward direction defined as positive.The experiment is then repeated, but instead of using the clay glob, a superball with identical mass is dropped from the same height onto the scale.Both the clay and the superball hit the scale 2.9 after they are dropped. Assume that the duration of the collision is the same in both cases and the force exerted by the scale on the clay and the force exerted by the scale on the superball are constant.

Part A

Sketch the graph of the superball's velocity vs. time, , from the instant it is dropped

( ) until it bounces to its maximum height ( ). Assume that the superball undergoes an elastic collision with the scale, and that the scale's recoil velocity is negligible. The light colored graph already present in the answer window is .

Hint A.1 Velocity vs. time before the collisionHint not displayed

Hint A.2 Find the speed of the ball as it leaves the scaleHint not displayed

Hint A.3 Determine the sign of the ball's velocity after the collisionHint not displayed

Hint A.4 Velocity vs. time graph after the collisionHint not displayed

Hint A.5 Find the time of the ball's collision with the scaleHint not displayed



ANSWER:


Part BBased on your graph, is the change in velocity of the superball during its collision with the scale greater than, less than, or equal to the change in velocity of the clay during its collision with the scale?

ANSWER: The change in velocity of the superball is greater than the change in velocity of the clay.The change in velocity of the superball is less than the change in velocity of the clay.The change in velocity of the superball is equal to the change in velocity of the clay.

Correct

Part CIs the force exerted by the scale on the superball greater than, less than, or equal to the force exerted by the scale on the clay?

Hint C.1 Relating change in velocity to forceHint not displayed

ANSWER: The force exerted by the scale on the superball is greater than the force exerted by the scale on the clay.The force exerted by the scale on the superball is less than the force exerted by the scale on the clay.The force exerted by the scale on the superball is equal to the force exerted by the scale on the clay.

Correct



Impulse and Momentum Ranking Task

Six automobiles are initially traveling at the indicated velocities. The automobiles have different masses and velocities. The drivers step on the brakes and all automobiles are brought to rest.

Part ARank these automobiles based on their momentum before the brakes are applied, from largest to smallest.

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

ANSWER:

View Answer Requested

Part BRank these automobiles based on the magnitude of the impulse needed to stop them, from largest to smallest.

Hint B.1 Relating impulse to momentumThe impulse applied to an object is equal to the object’s change in momentum. Therefore, the impulse needed to stop them should be equal to the difference between the initial momentum and the final momentum. (The final momentum is zero since the car is brought to a stop.)

Hint B.2 Apply the impulse-momentum theorem



All of the cars are brought to rest. What is the final momentum of each automobile?

Hint B.2.1 How to find momentumHint not displayed

Enter your answer in kilogram meters per second to three significant figures.

ANSWER: = 0

Correct


ANSWER:


Part CRank the automobiles based on the magnitude of the force needed to stop them, from largest to smallest.

Hint C.1 How to approach the problemYou know the impulse needed to stop the cars. However, this impulse could be a very large force exerted over a fraction of a second, a very small force exerted over several minutes, or any situation in between, so long as the force multiplied by the time gives the proper impulse. You must know the time intervals over which the stopping forces are exerted to determine the magnitudes of the stopping forces from the impulses.




ANSWER:


The more momentum an object has, the more impulse is needed to stop it. However, this impulse can be provided via a large force acting over a short time interval or a relatively small force acting over a relatively long time interval. If you are driving down the highway at 55 , you can stop your car by either lightly pressing on the brakes and traveling a long time before stopping, or pressing more firmly on the brakes and stopping more quickly. In both cases, your braking system has applied the same amount of impulse to your car.

Rocket Car

A rocket car is developed to break the land speed record along a salt flat in Utah. However, the safety of the driver must be considered, so the acceleration of the car must not exceed

(or five times the acceleration of gravity) during the test. Using the latest materials and technology, the total mass of the car (including the fuel) is 6000 kilograms, and the mass of the fuel is one-third of the total mass of the car (i.e., 2000 killograms). The car is moved to the starting line (and left at rest), at which time the rocket is ignited. The rocket fuel is expelled at a constant speed of 900 meters per second relative to the car, and is burned at a constant rate until used up, which takes only 15 seconds. Ignore all effects of friction in this problem.

Part AFind the acceleration of the car just after the rocket is ignited.

Hint A.1 How to approach the problem

The equation for the acceleration due to rocket propulsion is , where is

the exhaust speed. To use this equation, first find an expression for the rate of mass loss of the car.

Hint A.2 find the rate of mass changeHint not displayed



Express your answer to two significant figures.

ANSWER: =

20 Answer Requested

The driver of this car is experiencing just over , or two times the acceleration one normally feels due to gravity, at the start of the trip. This is not much different from the acceleration typically experienced by thrill seekers on a roller coaster, so the driver is in no danger on this score.

Part BFind the final acceleration of the car as the rocket is just about to use up its fuel supply.

Hint B.1 What has changed?What has changed from the time of the initial ignition of the rocket to the moment when the fuel is used up?

ANSWER: the exhaust speed of the rocket relative to the car the total mass of the car (including the fuel) the rate of mass change of the car

Correct

Hint B.2 Find the final massHint not displayed


ANSWER: =

30 Answer Requested

The driver of this car is experiencing just over , or three times the acceleration one normally feels due to gravity, by the end of the trip. This is the maximum acceleration achieved during the trip, and it is still very safe for the driver, who can easily withstand over with training.

Part C



Find the final velocity of the car just as the rocket is about to use up its fuel supply.

Hint C.1 Find the change in speedWrite an expression for the change in speed of the car from start to finish: . You will need to make use of the differential equation for rocket motion

,

if you don't know the equation for velocity of a rocket.

Hint C.1.1 How to solve the differential equationHint not displayed

Express your answer in terms of the exhaust speed , the initial mass of the car (plus fuel) , and the final mass of the car .

ANSWER: = Answer not displayed


ANSWER: =

360 Answer Requested

At the end of the trip, the driver is going a bit over Mach 1, or one times the speed of sound. This problem was based loosely on the breaking of the sound barrier by the ThrustSSC team in October 1997.

Surprising Exploding Firework

A mortar fires a shell of mass at speed . The shell explodes at the top of its trajectory (shown by a star in the figure) as designed. However, rather than creating a shower of

colored flares, it breaks into just two pieces, a smaller piece of mass and a larger piece

of mass . Both pieces land at exactly the same time. The smaller piece lands perilously

close to the mortar (at a distance of zero from the mortar). The larger piece lands a distance from the mortar. If there had been no explosion, the shell would have landed a distance

from the mortar. Assume that air resistance and the mass of the shell's explosive charge are negligible.



Part A

Find the distance from the mortar at which the larger piece of the shell lands.

Hint A.1 Find the position of the center of mass in terms of Hint not displayed

Hint A.2 Find the position of the center of mass in terms of

Hint not displayed

Express in terms of .

ANSWER: =

Correct

Sinking the 9-Ball



Jeanette is playing in a 9-ball pool tournament. She will win if she sinks the 9-ball from the final rack, so she needs to line up her shot precisely. Both the cue ball and the 9-ball have mass , and the cue ball is hit at an initial speed of . Jeanette carefully hits the cue ball into the 9-ball off center, so that when the balls collide, they move away from each other at the same angle from the direction in which the cue ball was originally traveling (see figure). Furthermore, after the collision, the cue ball moves away at speed , while the 9-ball moves at speed . For the purposes of this problem, assume that the collision is perfectly elastic, neglect friction, and ignore the spinning of the balls.

Part A

Find the angle that the 9-ball travels away from the horizontal, as shown in the figure.

Hint A.1 Determine the conserved quantitiesHint not displayed

Hint A.2 Apply conservation of kinetic energyHint not displayed

Hint A.3 Apply conservation of momentum in the y directionHint not displayed

Hint A.4 Apply conservation of momentum in the x directionHint not displayed

Hint A.5 Putting it all together



Hint not displayed

Express your answer in degrees to three significant figures.

ANSWER: = 45.0

Correct

Note that the angle between the final velocities of the two balls is . It turns out that in any elastic collision between two objects of equal mass, one of which is initially at rest, the angle between the final velocities of the two objects will be ninety degrees.

Trading Momenta in a Collision

Two particles move perpendicular to each other until they collide. Particle 1 has mass and momentum of magnitude , and particle 2 has mass and momentum of magnitude . Note: Magnitudes are not drawn to scale in any of the figures.

Part A



Suppose that after the collision, the particles "trade" their momenta, as shown in the figure. That is, particle 1 now has magnitude of momentum , and particle 2 has magnitude of momentum ; furthermore, each particle is now moving in the direction in

which the other had been moving. How much kinetic energy, , is lost in the collision?


Hint A.2 Find the relationship between energy and momentumHint not displayed

Hint A.3 Find the initial kinetic energyHint not displayed

Hint A.4 Find the final kinetic energyHint not displayed


ANSWER:

=

Correct

Part BConsider an alternative situation: This time the particles collide completely inelastically. How much kinetic energy is lost in this case?


Hint B.2 Definition of completely inelastic Hint not displayed

Hint B.3 Initial kinetic energyHint not displayed



Hint B.4 Find the final kinetic energyHint not displayed


ANSWER:

=

Correct

Traffic Accident Analysis

Consider the following two-car accident: Two cars of equal mass collide at an intersection. Driver E was traveling eastward, and driver N, northward. After the collision, the two cars remain joined together and slide, with locked wheels, before coming to rest. Police on the scene measure the length of the skid marks to be 9 meters. The coefficient of friction between the locked wheels and the road is equal to 0.9.

Each driver claims that his speed was less than 14 meters per second (50 mph). A third driver, who was traveling closely behind driver E prior to the collision, supports driver E's claim by asserting that driver E's speed could not have been greater than 12 meters per second. Take the following steps to decide whether driver N's statement is consistent with the third driver's contention.

Part A



Let the speeds of drivers E and N prior to the collision be denoted by and , respectively. Find , the square of the speed of the two-car system the instant after the collision.

Hint A.1 How to approach the problem

In general, you can find either by using conservation of energy or by finding the

individual components of the velocity using conservation of momentum, depending on which quantity is conserved. What is conserved in this collision?

ANSWER: energy only momentum only both energy and momentum

Correct

As long as there are no external forces, momentum is always conserved. Note: Conservation of momentum yields two equations, one for each component of the velocity. Furthermore, the condition that the two velocities after the collision are equal yields two constraints (one for each component). So you have four equations in four variables. However these latter constraints are trivial, so practically speaking you only have to write the conservation of momentum equations.

Hint A.2 Find the x and y components of velocityHint not displayed

Hint A.3 The magnitude of the velocity vectorHint not displayed

Express your answer terms of and .

ANSWER:

=

Answer Requested

Part B



What is the kinetic energy of the two-car system immediately after the collision?

Hint B.1 Definition of kinetic energy

The kinetic energy of an object of mass and speed is given by the formula

.

Hint B.2 The mass of the systemHint not displayed


ANSWER:

=

Answer Requested

Part C

Write an expression for the work done on the cars by friction.

Hint C.1 Definition of work

For a constant applied linear force , the work required to move an object through a

straight-line displacement is given by , where is the angle

between the direction of the force and the direction of the displacement.

Hint C.2 Magnitude of the frictional forceHint not displayed

Hint C.3 Direction of the frictional forceHint not displayed

Express your answer symbolically in terms of the mass of a single car, the magnitude of the acceleration due to gravity , the coefficient of sliding friction , and the distance through which the two-car system slides before coming to rest.



ANSWER: =

Answer Requested

Part DUsing the information given in the problem introduction and assuming that the third driver is telling the truth, determine whether driver N has reported his speed correctly. Specifically, if driver E had been traveling with a speed of exactly 12 meters per second before the collision, what must driver N's speed have been before the collision?

Hint D.1 How to approach the problem

Use the work-energy theorem, , to relate the kinetic energy of the

two-car system immediately after the collision (now for this part of the motion), to the nonconservative work done by friction in bringing the two cars finally to rest.

Hint D.2 The final kinetic energyHint not displayed

Express your answer numerically, in meters per second, to the nearest integer. Take , the magnitude of the acceleration due to gravity, to be 9.81 meters per second per second.

ANSWER: =

22 Answer Requested

m/s

If you believe the report by the third driver that the speed of driver E's car was less than or equal to 12 meters per second, then driver N's speed just obtained is the minimum speed that driver N could have had before the collision. So, even if you do not know that driver E's car was traveling at exactly 12 meters per second before the collision, it is still evident that the driver of car N was not reporting his speed accurately. Also, we have assumed that neither driver brakes before or during the collision. Including this factor makes the analysis somewhat more involved in real situations.

Two Worlds on a String



Two balls, A and B, with masses and are connected by a taut, massless string, and are moving along a horizontal frictionless plane. The distance between the centers of the two balls is . At a certain instant, the velocity of ball B has magnitude and is directed perpendicular to the string and parallel to the horizontal plane, and the velocity of ball A is zero.

Part A

Find , the tension in the string.

Hint A.1 Descibe the nature of the motionHint not displayed

Hint A.2 The key ideaHint not displayed

Hint A.3 Find the velocity of the center of massHint not displayed

Hint A.4 Find the rotational speedHint not displayed

Hint A.5 Find the radius of rotationHint not displayed



Hint A.6 Acceleration of ball BHint not displayed

Express in terms of , , , and .

ANSWER:

=

Correct

Note that your answer is "symmetric" between the parameters and . This is as it should be: The tension should be the same regardless of whether or initially moves. Only their relative velocity matters.

Collisions in One Dimension

On a frictionless horizontal air table, puck A (with mass 0.255 ) is moving toward puck B

(with mass 0.371 ), which is initially at rest. After the collision, puck A has velocity 0.124

to the left, and puck B has velocity 0.648 to the right.

Part AWhat was the speed of puck A before the collision?


Hint A.2 The initial momentumHint not displayed

Hint A.3 Find the final momentum of puck AHint not displayed

Hint A.4 Find the final momentum of puck BHint not displayed

ANSWER: = 0.819

Correct



Part B

Calculate , the change in the total kinetic energy of the system that occurs during the collision.


Hint B.2 Find the initial kinetic energy of puck AHint not displayed

Hint B.3 Find the final kinetic energy of puck AHint not displayed

Hint B.4 Find the final kinetic energy of puck BHint not displayed

ANSWER: = −5.62×10−3

Correct

Conservation of Momentum in Two Dimensions Ranking Task

Part AThe figures below show bird's-eye views of six automobile crashes an instant before they occur. The automobiles have different masses and incoming velocities as shown. After impact, the automobiles remain joined together and skid to rest in the direction shown by

. Rank these crashes according to the angle , measured counterclockwise as shown, at which the wreckage initially skids.

Hint A.1 Conservation of momentum in two dimensionsHint not displayed

Hint A.2 Determining the angleHint not displayed




ANSWER:


Spinning the Wheels: An Introduction to Angular Momentum

Learning Goal: To learn the definition and applications of angular momentum including its relationship to torque.

By now, you should be familiar with the concept of momentum, defined as the product of an object's mass and its velocity:

.You may have noticed that nearly every translational concept or equation seems to have an analogous rotational one. So, what might be the rotational analogue of momentum?Just as the rotational analogue of force , called the torque , is defined by the formula

,

the rotational analogue of momentum , called the angular momentum , is given by the

formula,

for a single particle. For an extended body you must add up the angular momenta of all of the pieces. There is another formula for angular momentum that makes the analogy to momentum particularly clear. For a rigid body rotating about an axis of symmetry, which will be true for all parts in this problem, the measure of inertia is given not by the mass but by the rotational inertia (i.e., the moment of inertia) . Similarly, the rate of rotation is given by the

body's angular speed, . The product gives the angular momentum of a rigid body

rotating about an axis of symmetry. (Note that if the body is not rotating about an axis of symmetry, then the angular momentum and the angular velocity may not be parallel.)

Part A



Which of the following is the SI unit of angular momentum?

ANSWER:

Correct

Part B

An object has rotational inertia . The object, initially at rest, begins to rotate with a constant angular acceleration of magnitude . What is the magnitude of the angular momentum of the object after time ?



ANSWER: =

Correct

Part C

A rigid, uniform bar with mass and length rotates about the axis passing through the midpoint of the bar perpendicular to the bar. The linear speed of the end points of the bar is . What is the magnitude of the angular momentum of the bar?


Hint C.2 Rotational inertia of the barHint not displayed

Hint C.3 Finding the magnitude of the angular velocity



Hint not displayed

Express your answer in terms of , , , and appropriate constants.

ANSWER: =

Correct

You may recall that, according to Newton's 2nd law, the rate of change of momentum of an object equals the net force acting on the object:

.

Similarly, the rate of change of angular momentum of an object equals the net torque acting on the object:

.

Therefore, if the net torque acting on an object (or a system of objects) is zero (i.e., the system is "closed"), then the rate of change of angular momentum is also zero. In other words, the net angular momentum of a closed system is constant (conserved).This statement is known as the law of conservation of angular momentum. Just like the laws of conservation of energy and momentum, the law of conservation of angular momentum plays a major role in mechanics.

Part DThe uniform bar shown in the diagram has a length of 0.80 m. The bar begins to rotate from rest in the horizontal plane about the axis passing through its left end. What will be the magnitude of the angular momentum of the bar 6.0 s after the motion has begun? The forces acting on the bar are shown.




Express your answer in to two significant figures.

ANSWER: = 4.8

Correct

Part EEach of the four bars shown can rotate freely in the horizontal plane about its left end. For which diagrams is the net torque equal to zero? Type in alphabetical order the letters corresponding to the correct diagrams. For instance, if you think that only diagrams A, B, and C answer the question, type ABC.

ANSWER: BC Correct

If the sum of the forces on a body is zero, then the net torque is independent of the point about which the torque is calculated. If the net force on the body is not zero, as is true for most of the beams in this part, then the torque will depend on the point about which you calculate the torque.

Part F



Consider the figures for Part E. For which diagrams is the angular momentum constant?

Hint F.1 Determining when angular momentum is constantHint not displayed

Type alphabetically the letters corresponding to the correct diagrams. For instance, if you think that only diagrams A, B, and C answer the question, type ABC.

ANSWER: BC Correct

Angular momentum is conserved when the net torque is zero. This is analogous to the statement from linear dynamics that momentum is conserved when the net force is zero.

Part GEach of the disks in the figure has radius . Each disk can rotate freely about the axis passing through the center of the disk perpendicular to the plane of the figure, as shown. For which diagrams is the angular momentum constant? In your calculations, use the information provided in the diagrams. Type alphabetically the letters corresponding to the correct diagrams. For instance, if you think that only diagrams A, B, and C answer the question, type ABC.

ANSWER: AD Correct

Part H



Three disks are spinning independently on the same axle without friction. Their respective rotational inertias and angular speeds are (clockwise); (counterclockwise); and

(clockwise). The disks then slide together and stick together, forming one piece with a single angular velocity. What will be the direction and the rate of rotation of the single piece?

Hint H.1 How to approach the problemHint not displayed

Hint H.2 Find the rotational inertiaHint not displayed

Express your answer in terms of one or both of the variables and and appropriate constants. Use a minus sign for clockwise rotation.

ANSWER: =

Correct

A Toy Gyroscope

The rotor (flywheel) of a toy gyroscope has mass 0.140 kilograms. Its moment of inertia about its axis is kilogram meters squared. The mass of the frame is 0.0250 kilograms. The gyroscope is supported on a single pivot with its center of mass a horizontal distance 4.00 centimeters from the pivot. The gyroscope is precessing in a horizontal plane at the rate of one revolution in 2.20 seconds.



Part A

Find the upward force exerted by the pivot.

Hint A.1 Precession in a gyroscopeHint not displayed


Hint A.3 Balance of forcesHint not displayed

Enter your answer in newtons to four significant figures.

ANSWER: = 1.617

Correct

Part BFind the angular speed at which the rotor is spinning about its axis, expressed in revolutions per minute.


Hint B.2 How to calculate the angular momentumHint not displayed

Hint B.3 Calculate the precession angular speedHint not displayed

Hint B.4 Calculate the torqueHint not displayed

Hint B.5 Using the angular momentumHint not displayed



Enter your answer in revolutions per minute to four significant figures.

ANSWER: = 1802

Correct

Change in Angular Velocity Ranking Task

A merry-go-round of radius , shown in the figure, is rotating at constant angular speed. The friction in its bearings is so small that it can be ignored. A sandbag of mass is dropped onto the merry-go-round, at a position designated by . The sandbag does not slip or roll upon contact with the merry-go-round.

Part ARank the following different combinations of and on the basis of the angular speed of the merry-go-round after the sandbag "sticks" to the merry-go-round.


Hint A.2 Determining the change in moment of inertiaHint not displayed




ANSWER:

View Correct

Hockey Stick and Puck

A hockey stick of mass and length is at rest on the ice (which is assumed to be frictionless). A puck with mass hits the stick a distance from the middle of the stick. Before the collision, the puck was moving with speed in a direction perpendicular to the stick, as indicated in the figure. The collision is completely inelastic, and the puck remains attached to the stick after the collision.

Part A



Find the speed of the center of mass of the stick+puck combination after the collision.

Hint A.1 Which conservation law to useHint not displayed

Hint A.2 Calculate the initial momentum of the systemHint not displayed

Hint A.3 Calculate the final momentum of the systemHint not displayed

Express in terms of the following quantities: , , , and .

ANSWER: =

Correct

Part BAfter the collision, the stick and puck will rotate about their combined center of mass. How far is this center of mass from the point at which the puck struck? In the figure, this distance is .

Hint B.1 Distance from middle of stick to center of mass of stick+puckHint not displayed



ANSWER: =

Correct

Note that if , the previous expression approaches D; that is, for a very massive stick, the center of mass of the combination is at the center of the stick.

Part C

What is the angular momentum of the system before the collision, with respect to the center of mass of the final system?

Hint C.1 Formula for angular momentumHint not displayed

Express in terms of the given variables.

ANSWER: =

Correct

This is why, in the previous part, we were interested in the distance from the center of mass to the point of impact.

Part DWhat is the angular velocity of the stick+puck combination after the collision? Assume that the stick is uniform and has a moment of inertia about its center.


Hint D.2 Express angular momentum in terms of moment of inertia and velocity

Hint not displayed

Hint D.3 Calculate the moment of inertiaHint not displayed



Hint D.4 Putting it all togetherHint not displayed

Your answer for should not contain the variable .

ANSWER: =

Correct

Part EWhich of the following statements are TRUE? 1) Kinetic energy is conserved. 2) Linear momentum is conserved. 3) Angular momentum of the stick+puck is conserved about the center of mass of the combined system. 4) Angular momentum of the stick+puck is conserved about the (stationary) point where the collision occurs.

Hint E.1 About conservation of angular momentumHint not displayed

ANSWER: 1 only 2 only 3 only 4 only 1 & 2 1 & 4 2 & 4 1 2 & 3 2 3 & 4

Correct

Note that there are no external torques on the system, so angular momentum is conserved about all points. However, typically we consider conservation of about the center of mass of the whole system because it is useful to talk about the future motion as the linear motion of the center of mass (of the new system)+ rotation about the center of mass (of the new system).



Precessing Tilted Gyroscope

A gyroscope consists of a flywheel of mass , which has a moment of inertia for rotation about its axis. It is mounted on a rod of negligible mass, which is supported at one end by a frictionless pivot attached to a vertical post, as shown in the diagram. The distance between the center of the wheel and the pivot is . The wheel rotates about its axis with angular velocity , where positive refers to counterclockwise rotation as seen by an observer looking at the face of the wheel that is opposite the pivot. The rod is tilted upward, making an angle with respect to the horizontal. Gravity acts downward with a force of magnitude

. Adopt a coordinate system with the z axis pointing upward and the x and y axes in the horizontal plane. The gyroscope is moving, but at , the rod is in the yz plane.

Part A



Assuming that the only significant contribution to the angular momentum comes from the spinning of the flywheel about its center, what is the angular momentum vector about

the pivot at ?

Hint A.1 A formula for angular momentumHint not displayed

Hint A.2 Find the direction of the angular velocity vectorHint not displayed

Specify the components of with respect to the axes shown in the

diagram. Write the components in order , , separated by commas.

ANSWER: , , =

Correct

Part B

At , what is the torque acting on the wheel about the pivot?

Hint B.1 What is the direction of the torque?Hint not displayed

Hint B.2 Find the x component of the torqueHint not displayed

Express your answer in terms of components , , , separated by commas.

ANSWER: , , =

Correct

Part C



The gyroscope is observed to precess about the vertical axis, with an angular velocity of precession , defined as positive for counterclockwise precession as seen from above. Find in terms of the given quantities.

Hint C.1 Relevant laws for rotational dynamicsHint not displayed

Hint C.2A relation between and

Hint not displayed

ANSWER: =

Correct

Thus the rate of precession is independent of ! The reason for this is that varying changes both and the torque due to gravity by the same factor, .


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