Homework #3

Homework #3

Due: 11:58pm on Monday, April 21, 2014

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

Two Blocks and Two Pulleys

Learning Goal:

To be able to calculate the tension in a string and the acceleration of each of two blocks in a two-pulley system.

As shown, a block with mass is attached to a massless ideal string. The string wraps around a massless pulley and then wraps around a second massless

pulley that is attached to a block with mass and ultimately attaches to a wall. The whole system is frictionless.

A coordinate system is given in the figure.

Part A - Tension in the string

Given that is the magnitude of the horizontal acceleration of the block with mass , what is , the tension in the string?

Express the tension in terms of and .

Hint 1. Which physical principle to use

Use Newton's second law,

m1m2

a2 m2 T

m2 a2

=F2 m2a2

where is the sum of the forces acting on the block with mass . Recall that the whole system is frictionless.

Hint 2. Identify the force diagram for the block with mass

Which figure correctly illustrates the forces acting on the block with mass ?

The vectors , , , and denote the normal force, the gravitational force, the tension in

the string, and the frictional force, respectively.

ANSWER:

Correct

ANSWER:

Correct

F2 m2

m2

m2

n W T fk

Figure 1

Figure 2

Figure 3

= T (0.5 )m2 a2

Part B - Acceleration of suspended block

Given , the tension in the string, calculate , the magnitude of the vertical acceleration of the block with mass .

Express the acceleration's magnitude, , in terms of , , and .

T a1 m1

a1 m1 g T

Hint 1. Which physical principle to use

Use Newton's second law,

where is the sum of the forces acting on the block with mass .

Hint 2. Identify the force diagram for the block of mass

Which figure correctly illustrates the forces acting on the block with mass ?

The vectors , , and denote the gravitational force, the tension in the string, and the

inertial force, respectively.

ANSWER:

ANSWER:

=F1 m1a1

F1 m1

m1

m1

W T Fi

Figure 1

Figure 2

Figure 3

= a1 g Tm1

Correct

Part C - Relative acceleration between blocks

Given the magnitude of the acceleration of the block with mass , find , the magnitude of the horizontal acceleration of the block with mass .

Express in terms of .

Hint 1. How to solve the problem using the string constraint (with calculus)

Define and as the vertical coordinate (as a function of time) of the block with mass and the horizontal coordinate (as a function of time)

of the block with mass , respectively. The length of the string, , is

where is a constant that accounts for the length of the wound portions of the string and the

length of string between the y axis and the wall. By differentiating this equation twice with

respect to time, you will obtain the relationship between and . The variables and

vanish during differentiation.

Hint 2. How to solve the problem using your intuition (without calculus)

While the block of mass descends a height , the other block moves only half of . Hence, at each instant, , where and are the

speeds of the blocks with masses and , respectively.

ANSWER:

a1 m1 a2 m2

a2 a1

(t)y1 (t)x2 m1

m2 l

l = 2 (t) + (t) + Cx2 y1C

a1 a2 C l

m1 h h = 2v1 v2 v1 v2m1 m2

= a2a12

Correct

Part D

Using the result from Part C in the equation for from Part A, express as a function of .

Express your answer in terms of and .

ANSWER:

Correct

Part E

Having solved the previous parts, you have all the pieces needed to calculate , the magnitude of the acceleration of the block with mass . Enter an

expression for .

Express the acceleration's magnitude in terms of , , and .

Hint 1. How to approach this problem

In Part B, using Newton's second law, you derived a relationship between and the tension in the string, . In Part D, you found as a function of .

Eliminate from this system of two linear equations and solve for .

ANSWER:

Correct

T T a1

m2 a1

= T (.25 )m2 a1

a1 m1

a1

a1 m1 m2 g

a1 T T a1

T a1

= a14 gm1+4m2 m1

Video Solution Problem for Chapter 13 Sections 1-4 - Particle Kinetics with Friction.

Watch the following video and answer the question.

Part A

A 1.5- brick is released from rest and slides down the inclined roof. If the coefficient of friction between the roof and the brick is , determine the

speed at which the brick strikes the gutter .

ANSWER:

Correct

Equation of Motion for a System of Particles

lb A = 0.3

G

v = 15.23 ft/s

v = 2.68 ft/s

v = 3.00 ft/s

v = 5.61 ft/s

Learning Goal:

To be able to set up and analyze the free-body diagrams and equations of motion for a system of particles.

Consider the mass and pulley system shown. Mass = 40 and mass = 10 . The angle of the inclined plane is given, and the coefficient of kinetic

friction between mass and the inclined plane is . Assume the pulleys are massless and frictionless.

Part A - Finding the acceleration of the mass on the inclined plane

What is the acceleration of mass on the inclined plane? Take positive acceleration to be up the ramp.

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

Hint 1. Determining the acceleration

Use the free-body diagram of to find the tension in the rope between the pulley and mass . Use this tension in the free-body diagram of to

find the net force acting on parallel to the plane. Establish the relation between the acceleration of the two masses. Finally, use Newton's second law

to find the acceleration.

Hint 2. Identify the tension in the rope

What is the tension in the rope between the pulley supporting the hanging mass and mass ?

Hint 1. The free-body diagram of the hanging mass

Consider the free-body diagram of mass .

m1 kg m2 kgm2 = 0.12k

m2

m1 m2 m2

m2

m2

m1

ANSWER:

Hint 3. Calculate the normal force on the mass on the inclined plane

What is the magnitude of the normal force on mass ?


Hint 1. The free-body diagram of the mass on the inclined plane

Consider the free-body diagram of mass .

T = (g a)m1

T = (g a)12m1

T = (g + a)m1

T = (g+ a)12m1

m2

m2

ANSWER:

Hint 4. Calculate the friction force on the mass on the inclined plane

What is the magnitude of the friction force on mass ?


ANSWER:

Hint 5. Identify the relation between the accelerations of the two masses

Which of the following equations correctly states the relation between the magnitudes of the accelerations of the two masses?

ANSWER:

ANSWER:

Answer Requested

90.6 N = N

m2

10.9 f = N

=a212a1

= 2a2 a1

=a223a2

=a232a2

=a2 a1

7.38 =a2ms2

Part B - Finding the speed of the mass moving up the ramp after a given time

If the system is released from rest, what is the speed of mass after 3 ?


Hint 1. Calculating the final speed

Because the force on mass is constant, the kinematics equations for constant acceleration can be used to find the speed at a given time.

Hint 2. Select the appropriate kinematics equation

Which of the kinematics equations can be used to solve the problem?

ANSWER:

ANSWER:

Correct

Part C - Finding the distance moved by the hanging mass

When mass moves a distance 4.44 up the ramp, how far downward does mass move?


m2 s

m2

x = + ( )tx0 v+v02v = at + v0

x = + t+ ax0 v012

t2

= + 2a(x )v2 v20 x0

22.1 v(3 s) = ms

m2 m m1

Hint 1. How to approach the problem

Set up and solve an equation that relates the two distances. Pick a datum that is fixed in space, then relate all changing rope lengths to that datum.

Notice that the length of the segment between the ceiling bracket and the datum remains constant

when the system is in motion; the same is true for the segment parallel to the ramp between the

top pulley and the datum (provided the mass does not move up far enough to reach the datum).

Hint 2. Find the equation of the segment lengths

What is the equation for the total length of the segments of the rope that vary when the system moves?

Express your answer in terms of the segment lengths and .

ANSWER:

Correct

Note that the change in segment length is equal to the distance moved by the hanging mass.

ANSWER:

Answer Requested

Fundamental Problem 13.4

s1 s2

l = 2 +s1 s2

s1

2.22 m

The 1.1- car is being towed by a winch.

Part A

If the winch exerts a force of on the cable, where is the displacement of the car in meters, determine the speed of the car when = 10

, starting from rest. Neglect rolling resistance of the car.


ANSWER:

Correct

Problem 13.11

The safe has a weight of 188 and is supported by the rope and pulley arrangement shown.

Mg

T = 100(s + 1) N s s

m

= 3.30 vms

S lb

Part A

If the end of the rope is given to a boy of weight 87 , determine his acceleration if in the confusion he doesn't let go of the rope. Neglect the mass of the

pulleys and rope.


ANSWER:

Correct

Problem 13.17

The double inclined plane supports two blocks and , each having a weight of 18 .

lb

= 1.68 aBfts2

A B lb

Part A

If the coefficient of kinetic friction between the blocks and the plane is 0.11, determine the acceleration of each block.


ANSWER:

Correct

Problem 13.18

A 40- suitcase slides from rest 20 down the smooth ramp.

Part A

= 3.47 a fts2

lb ft

Determine the point where it strikes the ground at .


ANSWER:

Correct

Part B

How long does it take to go from to ?


ANSWER:

Correct

Problem 13.20

The 370- mine car is hoisted up the incline using the cable and motor . For a short time, the force in the cable is = (3200 ) , where is in seconds.

C

= 5.30 R ft

A C

= 1.82 t s

kg M F t2 N t

Part A

If the car has an initial velocity = 2 when = 0, determine its velocity when = 3 .


ANSWER:

Correct

Video Solution Problem for Chapter 13 Section 5 - Particle Kinetics--Normal/Tangential Components.

Watch the following video and answer the question.

v1 m/s t t s

= 66.0 vms

Part A

A toboggan and rider have a total mass of and travel down along the (smooth) slope defined by the equation . At the instant , the

toboggan's speed is . At this point, determine the rate of increase in speed and the normal force which the toboggan exerts on the slope. Neglect the size

of the toboggan and rider for the calculation.

ANSWER:

Correct

100 kg y = 0.2x2 x = 8 m

4 m/s

= 9.36 m/ , N = 310 Nat s2

= 8.32 m/ , N = 537 Nat s2

= 8.32 m/ , N = 520 Nat s2

= 9.36 m/ , N = 293 Nat s2

Problem 13.65

Part A

Determine the constant speed of the passengers on the amusement-park ride if it is observed that the supporting cables are directed at = 30 from the

vertical. Each chair including its passenger has a mass of 80 .


ANSWER:

Correct

Part B

Also, what are the components of force in the , , and directions which the chair exerts on a 50- passenger during the motion?

Enter the n, t, and b components of the force separated by commas. Express your answer using three significant figures.

ANSWER:

kg

= 6.30 vms

n t b kg

, , = 283,0,491 Fn Ft Tb N

Correct

Problem 13.72

The ball has a mass of 30 and a speed = 4 at the instant it is at its lowest point, = 0 .

Part A

Determine the tension in the cord at the instant = 20 . Neglect the size of the ball.


ANSWER:

Correct

Part B

Determine rate at which the ball's speed is decreasing at the instant = 20 .

kg v m/s

= 361 T N


ANSWER:

Correct

Problem 13.82

The 890 motorbike travels with a constant speed of 80 up the hill.

Part A

Determine the normal force the surface exerts on its wheels when it reaches point A. Neglect its size.

Express your answer with the appropriate units.

ANSWER:

= 3.36 atms2

kg km/h

= 8.55 N kN

Correct

Equations of Motion: Cylindrical Coordinates

Learning Goal:

To be able to analyze the motion of a particle using the equations of motion in cylindrical coordinates.

Cylindrical, or polar, coordinates are useful for describing the motion of particles that involve angular positions and radial distances. When the forces acting on a

particle are along the unit vectors , , and , the scalar equations of motion are

where is the particle's mass and , , and are the accelerations in the r (radial), (transverse), and z directions, respectively. If the particle only moves in

the r plane, then only the first two equations are used to describe the motion.

Part A

A particle P moves along a curve. Its position is described by the three cylindrical coordinates: r, , and z. Although the components of the velocity and

acceleration (in rectangular coordinates) are the first and second time derivatives of the rectangular coordinates x, y, and z, this is not the case in cylindrical

coordinates. Match the appropriate component equations to the appropriate targets to complete the equations for the position vector, , the velocity vector, ,

and the acceleration vector, . The terms , , and are the unit vectors in the r, , and z directions, respectively. The terms , , and are the first time

derivatives of , , and , respectively. The terms , , and are the second time derivatives of , , and , respectively.

Drag the appropriate labels to their respective targets.

ANSWER:

Answer Requested

Part B

As shown, a cam has a shape that is described by the function , where = 2.40 .

ur u uz

= mFr ar

= mF a

= mFz azm ar a az

rP v

a ur u uz r z

r z r z r z

r = (2.00 cos )r0 r0 ft

A slotted bar is attached to the origin and rotates in the horizontal plane with a constant angular velocity of 0.425 . The bar moves a roller weighing

32.3 along the cam's perimeter. A spring holds the roller in place; the spring's spring constant is 0.527 . The friction in the system is negligible. When =

124 , what are and , the magnitudes of the cylindrical components of the total force acting on the roller?

Express your answers numerically in pounds to three significant figures separated by a comma.


Use the definition of acceleration (in cylindrical coordinates) to find and , the acceleration components in the r and directions. Use them in the

equations for the force components.

Hint 2. Find an expression for

What is , the acceleration component in the direction, in terms of some or all of the following variables: the angular velocity of the bar, ; the spring

constant, ; the constant ; and the angle ?

Express your answer in terms of some or all of the variables , , , and .


The acceleration in the direction is defined as follows:

radians/s

lb lb/ft Fr F

ar a

a

a 0

k r0

0 k r0

= r + 2a r

The expression for is given in the problem statement and .


What is , the first time derivative of the r coordinate, in terms of some or all of the following variables: the angular velocity of the bar, ; the

spring constant, ; the constant ; and the angle ?


ANSWER:

Hint 3. Find the value of

What is the value of , the second time derivative of the coordinate?

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

ANSWER:

ANSWER:


What is , the acceleration component in the r direction, in terms of some or all of the following variables: the angular velocity of the bar, ; the spring

constant, ; the constant ; and the angle ?



r = 0

r

r 0

k r0

0 k r0

= r sin()r00

= 0.000 radians/s2

= a 2 sin()r002

ar

ar 0

k r0

0 k r0

The acceleration in the r direction is defined as follows:

The expression for is given in the problem statement and .


What is , the second time derivative of the coordinate, in terms of some or all of the following variables: the angular velocity of the bar, ; the

spring constant, ; the constant ; and the angle ?


ANSWER:

ANSWER:

ANSWER:

All attempts used; correct answer displayed

In cylindrical coordinates, the r component is positive when directed away from the origin and negative when directed toward the origin. The negative value

for , therefore, means that the component of the force in the r direction is toward point O.

Part C

What are and , the magnitudes of the tangential force, , and the normal force, , acting on the roller by the path when = 124 ?

Express your answers numerically in pounds to three significant figures separated by a comma.

= rar r 2

r = 0

r

r 0

k r0

0 k r0

= r cos()r002

= ar 2 (cos() 1)r002

, = 1.36,0.721 Fr F lb

Fr

Ft N Ft N


Draw the free-body diagram of the roller. The tangential force is always tangential to the path of a particle. The normal force exerted by the path is

always perpendicular to the path's tangent. The spring is aligned with the slotted bar and only exerts a force in the direction. is the angle between

the extended radial line and the path's tangent; it is found using the following equation:

where is the particle's path and is the path's derivative with respect to . After finding , the components of the tangential force and the normal

force in the and directions can be related to and .

Hint 2. Draw the free-body diagram of the roller

Complete the free-body diagram of the roller by adding the missing forces that act on it.

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

vectors will not be graded.

ANSWER:

Hint 3. Find

What is , the angle between the tangential force and the extended radial line?

Express your answer numerically in degrees to four significant figures.


The equation for is as follows:

What is , the path's derivative with respect to , in terms of and ?

Express your answer in terms of and .

ANSWER:

ur

tan = rdr/d

r dr/d

ur u Fr F

dr/d

tan = rdr/d

dr/d r0

r0

ANSWER:

ANSWER:

All attempts used; correct answer displayed

Problem 13.89

Rod rotates counterclockwise with a constant angular velocity of = 5 . The double collar is pin-connected together such that one collar slides over

the rotating rod and the other slides over the horizontal curved rod, of which the shape is described by the equation = 1.5(2 ) .

= dr/d sin()r0

= 72.05 degrees

, = 0.876,0.365 Ft N lb

OA rad/s Br cos ft

Part A

If both collars weigh 0.85 , determine the normal force which the curved rod exerts on one collar at the instant = 110 . Neglect friction.


ANSWER:

Correct

Problem 13.96

The particle has a mass of 0.5 and is confined to move along the smooth horizontal slot due to the rotation of the arm .

Part A

Determine the force of the rod on the particle and the normal force of the slot on the particle when = 30 . The rod is rotating with a constant angular velocity

= 2 . Assume the particle contacts only one side of the slot at any instant.

Express your answers using three significant figures separated by a comma.

lb

= 2.86 N lb

kg OA

rad/s

ANSWER:

Correct

Score Summary:

Your score on this assignment is 99.7%.

You received 99.7 out of a possible total of 100 points.

, = 1.78,5.79 F N N

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