mp6

MasteringPhysics: As Assignment Pr Print Vi View

http://session.masteringphysics.com/myct/assignmentPrint?ass...

Assignment 6 Due: 11:59pm on Sunday, February 21, 2010 Note:

To understand how points are awarded, read your instructor's Grading Policy. Policy.

[Return to Standard Assignment View]

Spring and Projectile A child's toy consists of a block that attaches to a table with a suction cup, a spring connected to that block, a ball, and a launching ramp. The spring has a spring constant , the ball has a mass , and the ramp rises a height above the table, the surface of which is a height above the floor. Initially, the spring rests at its equilibrium length. length. The spring then is compressed a distance , where the ball is held at rest. The ball is then released, launching it up the ramp. When When the ball leaves the launching ramp its velocity vector makes makes an angle with respect to the horizontal. Throughout this problem, ignore friction and air resistance.

Part A Relative to the initial configuration (with the spring relaxed), when the spring has been compressed, the ball-spring system has ANSWER:

gained kinetic energy gained potential energy lost kinetic energy lost potential energy Correct

Part B As the spring expands (after the ball is released) the ball-spring system ANSWER:

gains kinetic energy and loses potential energy gains kinetic energy and gains potential energy loses kinetic energy and gains potential energy loses kinetic energy and loses potential energy Correct

Part C As the ball goes up the ramp, it ANSWER:


Part D As the ball falls to the floor (after having reached its maximum height), it ANSWER:


Part E Which of the graphs shown best represents the potential energy of the ball-spring system as a function of the ball's horizontal displacement? Take the "zero" on the distance axis to represent the point at which the spring is fully compressed. Keep in mind that the ball is not attached not attached to the spring, and neglect any recoil of the spring after the ball loses contact with it.

1 of 12

4/17/10 5:29 PM

MasteringPhysics: Assignment Print View

ANSWER:


C

Correct

Part F Calculate

, the speed of the ball when it leaves the launching ramp.

Hint F.1

General approach

Find an expression for the mechanical energy (kinetic plus potential) of the spring and ball when the spring is compressed. Then find an expression for the mechanical energy of the ball when it leaves the launching ramp. ( will be an unknown in this expression.) Since energy is conserved, you can set these two expressions equal to each other, and solve for . Hint F.2

Find the initial mechanical energy

Find the total mechanical energy of the ball-spring system when the spring is fully compressed. Take the gravitational potential energy to be zero at t he floor. Hint F.2.1

What contributes to the mechanical energy?

The total initial mechanical energy is the sum of the potential energy of the spring, the gravitational potential energy, plus any initial kinetic energy of the ball.

ANSWER: = Correct

Hint F.3

Find the mechanical energy at the end of the ramp

Find the total mechanical energy of the ball when it leaves the launching ramp. (At this point, assume that the spring is relaxed and has no stored potential energy.) Again, take the gravitational potential energy to be zero at the floor. Express your answer in terms of

and other given quantities.

ANSWER: = Correct

Hint F.4

Is energy conserved?

Because no nonconservative forces act on the system, energy is conserved: Express the speed of the ball in terms of

, ,

,

,

, and/or

.

ANSWER: = Correct

Part G With what speed will the ball hit the floor? Hint G.1

General approach

Find an expression for the initial mechanical energy (kinetic plus potential) of the spring and ball. Then find an expression for the mechanical energy of the ball when it hits the floor. ( unknown in this expression.) Since energy is conserved, you can set these two expressions equal to each other, and solve for . Hint G.2

will be an

Initial mechanical energy

For the initial mechanical energy, you can use either the expression you found for the mechanical energy of the ball at the top of the ramp or that for the total mechanical energy of the ball plus spring just before the ball was launched. These two expressions are equal. Hint G.3

Find the final mechanical energy

Find the total mechanincal energy

of the ball when it hits the floor.

Express your answer in terms of

and other given quantities.

ANSWER: = Correct

Hint G.4

Is energy conserved?

Only conservative forces (gravity, spring) are acting on the ball, so energy is conserved: Express the speed in terms of

, ,

,

,

, and/or

.

.

ANSWER: = Correct

2 of 12

4/17/10 5:29 PM



Circling Ball A ball of mass

is attached to a string of length

. It is being swung in a vertical circle with enough speed so that the string remains taut throughout the ball's motion. Assume that the ball travels

freely in this vertical circle with negligible loss of total mechanical energy. At the top and bottom of the vertical circle, the ball's speeds are and , and the corresponding tensions in the string are and . and have magnitudes and .

Part A Find

, the difference between the magnitude of the tension in the string at the bottom relative to that at the top of the circle.

Hint A.1

How to approach this problem

Identify the forces that act on the ball as it moves along the circular path. Then, write equations for the sum of the forces on the ball at the top and the bottom of the path. Next, use Newton's second law to relate these net forces to the acceleration of the ball. Notice that the ball does not move with uniform speed so the acceleration of the ball at the top of the circle is different from the acceleration at the bottom of the circle. To finish the problem, you may want to use energy conservation to relate the speed of the ball at the bottom of the circle to the speed at the top. Hint A.2

Find the sum of forces at the bottom of the circle

What is the magnitude of the net force in the y direction acting on the ball at the bottom of the circle? Express your answer in terms of the variables given in the problem. You may use ANSWER:

to represent the acceleration of gravity, 9.8

.

= Correct

Hint A.3 Find

Find the acceleration at the bottom of the circle

, the magnitude of the vertical acceleration of the ball at the bottom of its circle.


and possibly other given quantities.

ANSWER: = Answer Requested

Hint A.4

Find the tension at the bottom of the circle

Find the magnitude of the tension Hint A.4.1

in the string when the ball is at the bottom of the circle.

What physical principle to use

Apply Newton's 2nd law in the y direction to obtain Express your answer in terms of

,

,

.

, and the speed

of the ball at the bottom of the circle.


Hint A.5

Find the sum of forces at the top of the circle

What is the magnitude of the net force in the y direction acting on the ball at the top of its circle? Express your answer in terms of the variables given in the problem. You may use ANSWER:

to represent the acceleration of gravity, 9.8

.

= Answer Requested

Hint A.6 Find

Find the acceleration at the top of the circle

, the magnitude of the vertical acceleration of the ball at the top of its circle.


and possibly other given quantities.


Hint A.7

3 of 12

Find the tension at the top of the circle

4/17/10 5:29 PM


Find the magnitude of the tension Hint A.7.1


in the string when the ball is at the top of the circle.

Relationship to solution for

Follow the same steps you used to find Express your answer in terms of

,

(see Hint 3), noting carefully where various directions (signs) are reversed. ,

, and the speed

of the ball at the top of the circle.


Hint A.8

Find the relationship between

and

The total mechanical energy of the system is the same when the ball is at the top and bottom of the vertical circle. Use conservation of energy to write an expression for Your answer may also include

,

, and

in terms of

.

.


Express the difference in tension in terms of ANSWER:

and . The quantities

and

should not appear in your final answer.

= Correct

The method outlined in the hints is really the only practical way to do this problem. If done properly, finding the difference between the tensions,

, can be accomplished fairly simply and

elegantly.

A Mass-Spring System with Recoil and Friction An object of mass

is traveling on a horizontal surface. There is a coefficient of kinetic friction

between the object and the surface. The object has speed

spring. The object compresses the spring, stops, and then recoils and travels in the opposite direction. When the object reaches

when it reaches

and encounters a

on its return trip, it stops.

Part A Find

, the spring constant.

Hint A.1

Why does the object stop? Hint not displayed

Hint A.2

How does friction affect the system? Hint not displayed

Hint A.3

Energy stored in a spring Hint not displayed

Hint A.4

Compute the compression of the spring Hint not displayed

Hint A.5

Putting it all together Hint not displayed

Hint A.6

The value of Hint not displayed

Hint A.7

Find

for this part of the motion Hint not displayed

Hint A.8

Find

for this part of the motion Hint not displayed

Express

in terms of

,

,

, and

.

ANSWER: = Correct

Drag on a Skydiver A skydiver of mass

jumps from a hot air balloon and falls a distance

before reaching a terminal velocity of magnitude

. Assume that the magnitude of the acceleration due to gravity is .

Part A

4 of 12

4/17/10 5:29 PM


What is the work Hint A.1

done on the skydiver, over the distance


, by the drag force of the air?

How to approach the problem Hint not displayed

Hint A.2

Find the change in potential energy Hint not displayed

Hint A.3

Find the change in kinetic energy Hint not displayed

Express the work in terms of

,

,

, and the magnitude of the acceleration due to gravity .

ANSWER: = Correct

Part B Find the power Hint B.1

supplied by the drag force after the skydiver has reached terminal velocity

.


Hint B.2

Magnitude of the drag force Hint not displayed

Hint B.3

Relative direction of the drag force and velocity Hint not displayed

Express the power in terms of quantities given in the problem introduction. ANSWER:

= Correct

Problem 7.63 A skier starts at the top of a very large, frictionless snowball, with a very small initial speed, and skis straight down the side (the figure ).

Part A At what point does she lose contact with the snowball and fly off at a tangent? That is, at the instant she loses contact with the snowball, what angle

does a radial line from the center of the

snowball to the skier make with the vertical? ANSWER:

= 48.2 Correct

Problem 7.83 A cutting tool under microprocessor control has several forces acting on it. One force is constant is

= 2.65

, a force in the negative -direction whose magnitude depends on the position of the tool. The

. Consider the displacement of the tool from the origin to the point

2.10

,

2.10

.

Part A Calculate the work done on the tool by

ANSWER:

if this displacement is along the straight line

that connects these two points.

= -12.9 All attempts used; correct answer displayed

Part B Calculate the work done on the tool by

5 of 12

if the tool is first moved out along the x-axis to the point

2.10

,

and then moved parallel to the y-axis to

2.10

,

2.10

.

4/17/10 5:29 PM


ANSWER:


= -17.2 Correct

Part C Compare the work done by

ANSWER:

along these two paths. Is

conservative or nonconservative?

The force is conservative. The force is not conservative. Correct

Part D Explain. ANSWER:

My Answer:

Problem 7.86 A particle moves along the x-axis while acted on by a single conservative force parallel to the x-axis. The force corresponds to the potential-energy function graphed in the figure . The particle is released from rest at point .

Part A What is the direction of the force on the particle when it is at point

ANSWER:

?

positive negative Correct

Part B At point

?

ANSWER:

positive negative Correct

Part C At what value of

is the kinetic energy of the particle a maximum?

Express your answer using two significant figures. ANSWER:

= 0.75 Correct

Part D What is the force on the particle when it is at point

?

Express your answer using two significant figures. ANSWER:

0

= 0.0×10 Correct

Part E What is the largest value of

reached by the particle during its motion?

Express your answer using two significant figures.

6 of 12

4/17/10 5:29 PM


ANSWER:


= 2.2 Correct

Part F What value or values of

correspond to points of stable equilibrium?

Express your answer numerically using two significant figures. If there is more than one point, enter each point separated by a comma. ANSWER:

= 0.75,1.9 Correct

Part G Of unstable equilibrium? Express your answer numerically using two significant figures. If there is more than one point, enter each point separated by a comma. ANSWER:

= 1.4 Correct

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

Express your answer in terms of any or all of ANSWER:

=

,

of the momentum of the particle at time ? Assume that

.

, and .

Correct

Part B The particle starts from rest at

. What is the magnitude

Express your answer in terms of any or all of

,

of the velocity of the particle at time ? Assume that

.

, and .

ANSWER: = Correct

Part C The particle has momentum of magnitude

at a certain instant. What is


7 of 12

,

,

, and

, the magnitude of its momentum

seconds later?

.

4/17/10 5:29 PM



ANSWER: = Correct

Part D The particle has momentum of magnitude

at a certain instant. What is


,

,

, and

, the magnitude of its velocity

seconds later?

.

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 Hint E.1

delivered to the particle.


Hint E.2

Find the change in momentum Hint not displayed

Express your answer in terms of ANSWER:

and

. Use three significant figures in the numerical coefficient.

= 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

ANSWER:

8 of 12

between the positive y axis and the vector

as shown in the figure?

26.6 degrees 30 degrees 60 degrees 63.4 degrees

4/17/10 5:29 PM



Correct

Part H If the magnitude of the net force acting on the particle is Express your answer in terms of

,

, how long does it take the particle to acquire its final velocity,

in the positive y direction?

, 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 Find the speed Hint I.1

kilograms is at rest at

of the particle at

seconds. A varying force

is acting on the particle between

seconds and

seconds.

seconds.

Use the impulse-momentum theorem Hint 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 Relation Between Momentum and Kinetic Energy Part A A cardinal ( Richmondena cardinalis) of mass 3.70 ×10−2 magnitude

and a baseball of mass 0.144

have the same kinetic energy. What is the ratio of the cardinal's magnitude

of momentum to the

of the baseball's momentum?

Hint A.1

How to approach the problem

Recall that the kinetic energy of an object (of mass

and speed ) is given by

, and the magnitude of the momentum by

. Combining these equations into a single expression

can then be used to eliminate , giving an expression of the kinetic energy in terms of the momentum instead of the velocity. We can then use this relation, along with the assumptions, to find the ratio of the momenta

Hint A.2

in terms of the masses.

Find a relationship between kinetic energy and momentum

Select the general expression for the kinetic energy

of an object with mass

and momentum

.

ANSWER:

Correct

Use the fact that the kinetic energies of the cardinal and the baseball are the same to find an equation for the ratio

ANSWER:

.

= 0.507 Correct

Part B A man weighing 740 Hint B.1

and a woman weighing 490

have the same momentum. What is the ratio of the man's kinetic energy

to that of the woman

?


9 of 12

4/17/10 5:29 PM


Hint B.2


Find a relationship between momentum and kinetic energy Hint not displayed

ANSWER:

= 0.662 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, Chuck and Jackie and their carts are at rest. Chuck then picks up a ball of mass ball, his speed relative to the ground is

, is identical to the combined mass of Jackie and her cart. Initially,

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 . 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 A Find the relative speed Hint A.1

between Chuck and the ball after Chuck has thrown the ball.


Express the speed in terms of ANSWER:

=

and

.

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 Hint B.1

of the ball (relative to the ground) while it is in the air? How to approach the problem Hint not displayed

Hint B.2

Initial momentum of Chuck, his cart, and the ball Hint not displayed

Hint B.3

Find the final momentum of Chuck, his cart, and the thrown ball Hint not displayed


,

, and

.

ANSWER: = Correct

Part C What is Chuck's speed Hint C.1

(relative to the ground) after he throws the ball?



,

, and

.

ANSWER: = Correct

Part D Find Jackie's speed Hint D.1

(relative to the ground) after she catches the ball, in terms of

.


Hint D.2

10 of 12

Initial momentum

4/17/10 5:29 PM



Hint not displayed Hint D.3

Find the final momentum Hint not displayed

Express

in terms of

,

, and

.

ANSWER: = Correct

Part E Find Jackie's speed Hint E.1

(relative to the ground) after she catches the ball, in terms of

.


Express

in terms of

,

, and

.

ANSWER: = Correct

Exercise 8.20 Block

in the figure has mass 1.00

, and block

has mass 3.00

. The blocks are forced together, compressing a spring

between

them; then the system is released from rest on a level, frictionless surface. The spring, which has negligible mass, is not fastened to either block and drops to the surface after it has expanded. Block acquires a speed of 1.40 .

Part A What is the final speed of block

ANSWER:

?

= 4.20 Correct

Part B How much potential energy was stored i n the compressed spring? ANSWER:

= 11.8 Correct

Problem 7.74 A 2.00-

package is released on a

and the incline are

and

incline, 4.00

from a long spring with force constant 120

that is attached at the bottom of the incline . The coefficients of friction between the package

. The mass of the spring is negligible.

Part A What is the speed of the package just before it reaches the spring?

11 of 12

4/17/10 5:29 PM


ANSWER:


= 7.30 Correct

Part B What is the maximum compression of the spring? ANSWER:

= 1.06 Correct

Part C The package rebounds back up the incline. How close does it get to its initial position? ANSWER:

= 1.32 Correct

Score Summary: Your score on this assignment is 93.3%. You received 111.9 out of a possible total of 120 points.

12 of 12

4/17/10 5:29 PM

mp6

Recommend Documents