HW05 - Circular Motion and Forces
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Course
151
Subject
Physics
Date
Apr 3, 2024
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Pages
26
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HW05 - Circular Motion and Forces
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HW05 - Circular Motion and Forces
Due: 11:59pm on Tuesday, October 18, 2022
To understand how points are awarded, read the Grading Policy for this assignment.
Ladybugs on a Turntable
Two ladybugs are riding on a turntable as it rotates at 15 as shown in .
Part A
What is the period of the turntable?
Express your answer with the appropriate units.
Hint 1. How to approach the problem
15 is the turntable's angular velocity , but not in SI units of . You need to convert using and .
Express your answer in radians per second.
ANSWER:
Hint 2. How to find the period
The period is related to the angular velocity by .
ANSWER:
=
1.57
= 4.0
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Part B
How does the angular velocity of ladybug A compare to that of ladybug B?
ANSWER:
Correct
Both ladybugs make the same number of revolutions per minute. Angular velocity doesn't depend on the
distance from the axis.
Part C
How does the speed of ladybug A compare to that of ladybug B?
ANSWER:
Correct
Tangential velocity is . Objects farther from the axis move faster.
Part D
What is the speed of ladybug A?
Express your answer with the appropriate units.
Hint 1. How to find the speed
Tangential velocity is given by . Speed is the absolute value of velocity. The ladybugs' angular velocity is
that of the turntable, which you have already found.
ANSWER:
Correct
Part E
=
0.16
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What is the angular displacement in degrees of ladybug B during a time interval of 1.4 ?
Express your answer in degrees to two significant figures.
Hint 1. How to find the angular displacement
For uniform circular motion, angular displacement in radians is . That is, the number of multiplied by the number of seconds. Also, don't forget to convert to degrees.
ANSWER:
Correct
Linear and Rotational Quantities Conceptual Question
A merry-go-round is rotating at constant angular speed. Two children are riding the merry-go-round: Ana is riding at point A
and Bobby is riding at point B.
Part A
Which child moves with greater magnitude of linear velocity?
Hint 1. Distinguishing between linear velocity and angular velocity
Ana’s (or Bobby’s) linear velocity is determined by the actual distance traveled (typically in meters) in a given
time interval. The angular velocity is determined by the angle through which (s)he rotates (typically in radians) in
a given time interval.
ANSWER:
=
130
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The linear velocity is often just referred to as the velocity. The tangential velocity and radial velocity are
components of the linear velocity. In this scenario, since the merry-go-round has a constant angular velocity,
the radial velocity is zero, therefore the tangential velocity is equal to the linear velocity.
Part B
Who moves with greater magnitude of angular velocity?
Hint 1. Distinguishing between velocity and angular velocity
Ana’s (or Bobby’s) velocity is determined by the actual distance she (he) travels (typically in meters) in a given
time interval. Her (His) angular velocity is determined by the angle through which she (he) rotates (typically in
radians) in a given time interval.
ANSWER:
Correct
Part C
Who moves with greater magnitude of tangential acceleration?
Hint 1. Distinguishing tangential, centripetal, and angular acceleration
Ana’s tangential and centripetal acceleration are components of her acceleration vector. During circular motion, if
Ana’s speed is changing (meaning the merry-go-round is speeding up or slowing down) she will have a nonzero
tangential acceleration. However, even if the merry-go-round is turning at constant angular speed, she will
experience a centripetal acceleration, because the direction of her velocity vector is changing (you can’t move
along a circular path unless your direction of travel is changing!).
Both tangential and centripetal accelerations have units of , since they are the two-dimensional
components of linear acceleration. Angular acceleration, on the other hand, is a measure of the change in Ana’s
angular velocity. If her rate of rotation is changing, she will have a nonzero angular acceleration. Thus, angular
acceleration has units of .
ANSWER:
Ana has the greater magnitude of linear velocity.
Bobby has the greater magnitude of linear velocity.
Both Ana and Bobby have the same magnitude of linear velocity.
Ana has the greater magnitude of angular velocity.
Bobby has the greater magnitude of angular velocity.
Both Ana and Bobby have the same magnitude of angular velocity.
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Both Ana and Bobby are maintaining a constant speed
, so they both have a tangential acceleration of zero
(thus they are equal)!
Part D
Who has the greater magnitude of centripetal acceleration?
Hint 1. Distinguishing tangential, centripetal, and angular acceleration
Ana’s tangential and centripetal acceleration are components of her acceleration vector. For circular motion, if
Ana’s speed is changing (meaning the merry-go-round is speeding up or slowing down) she will have a nonzero
tangential acceleration. However, even if the merry-go-round is turning at constant angular speed, she will
experience a centripetal acceleration, because the direction of her velocity vector is changing (you can’t move
along a circular path unless your direction of travel is changing!).
Both tangential and centripetal accelerations have units of , since they are the two-dimensional
components of linear acceleration. Angular acceleration, on the other hand, is a measure of the change in Ana’s
angular velocity. If her rate of rotation is changing, she will have a nonzero angular acceleration. Thus, angular
acceleration has units of .
ANSWER:
Correct
Part E
Who moves with greater magnitude of angular acceleration?
Hint 1. Distinguishing tangential, centripetal, and angular acceleration
Ana’s tangential and centripetal acceleration are components of her acceleration vector. For circular motion, if
Ana’s speed is changing (meaning the merry-go-round is speeding up or slowing down) she will have a nonzero
tangential acceleration. However, even if the merry-go-round is turning at constant angular speed, she will
experience a centripetal acceleration, because the direction of her velocity vector is changing (you can’t move
along a circular path unless your direction of travel is changing!).
Both tangential and centripetal accelerations have units of , since they are the two-dimensional
components of linear acceleration. Angular acceleration, on the other hand, is a measure of the change in Ana’s
angular velocity. If her rate of rotation is changing, she will have a nonzero angular acceleration. Thus, angular
acceleration has units of .
Ana has the greater magnitude of tangential acceleration.
Bobby has the greater magnitude of tangential acceleration.
Both Ana and Bobby have the same magnitude of tangential acceleration.
Ana has the greater magnitude of centripetal acceleration.
Bobby has the greater magnitude of centripetal acceleration.
Both Ana and Bobby have the same magnitude of centripetal acceleration.
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ANSWER:
Correct
Both Ana and Bobby are maintaining a constant angular velocity
, so they both have an angular acceleration of
zero (thus they are equal)!
Accelerating along a Racetrack
A road race is taking place along the track shown in the figure . All
of the cars are moving at constant speeds. The car at point F is
traveling along a straight section of the track, whereas all the
other cars are moving along curved segments of the track.
Part A
Let be the velocity of the car at point A. What can you say about the acceleration of the car at that point?
Hint 1. Acceleration along a curved path
Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero
acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the
speed) is constant. Moreover, if the speed is constant, the object's acceleration is always perpendicular to the
velocity vector at each point along the curved path and is directed toward the center of curvature of the path.
ANSWER:
Ana has the greater magnitude of angular acceleration.
Bobby has the greater magnitude of angular acceleration.
Both Ana and Bobby have the same magnitude of angular acceleration.
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Part B
Let be the velocity of the car at point C. What can you say about the acceleration of the car at that point?
Hint 1. Acceleration along a curved path
Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero
acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the
speed) is constant. Moreover, if the speed is constant, the object's acceleration is always perpendicular to the
velocity vector at each point along the curved path and is directed toward the center of curvature of the path.
ANSWER:
Correct
Part C
Let be the velocity of the car at point D. What can you say about the acceleration of the car at that point?
Hint 1. Acceleration along a curved path
Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero
acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the
speed) is constant. Moreover, if the speed is constant, the object's acceleration is always perpendicular to the
velocity vector at each point along the curved path and is directed toward the center of curvature of the path.
ANSWER:
The acceleration is parallel to .
The acceleration is perpendicular to and directed toward the inside of the track.
The acceleration is perpendicular to and directed toward the outside of the track.
The acceleration is neither parallel nor perpendicular to .
The acceleration is zero.
The acceleration is parallel to .
The acceleration is perpendicular to and pointed toward the inside of the track.
The acceleration is perpendicular to and pointed toward the outside of the track.
The acceleration is neither parallel nor perpendicular to .
The acceleration is zero.
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Part D
Let be the velocity of the car at point F. What can you say about the acceleration of the car at that point?
Hint 1. Acceleration along a straight path
The velocity of an object that moves along a straight path is always parallel to the direction of the path, and the
object has a nonzero acceleration only if the magnitude of its velocity changes in time.
ANSWER:
Correct
Part E
Assuming that all cars have equal speeds, which car has the acceleration of the greatest magnitude, and which one has
the acceleration of the least magnitude?
Give your answer as "X,Y" where X is the car with the greatest magnitude of acceleration and Y is the car with
the least magnitude of accelation.
Hint 1. How to approach the problem
Recall that the magnitude of the acceleration of an object that moves at constant speed along a curved path is
inversely proportional to the radius of curvature of the path.
ANSWER:
The acceleration is parallel to .
The acceleration is perpendicular to and pointed toward the inside of the track.
The acceleration is perpendicular to and pointed toward the outside of the track.
The acceleration is neither parallel nor perpendicular to .
The acceleration is zero.
The acceleration is parallel to .
The acceleration is perpendicular to and pointed toward the inside of the track.
The acceleration is perpendicular to and pointed toward the outside of the track.
The acceleration is neither parallel nor perpendicular to .
The acceleration is zero.
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Part F
Assume that the car at point A and the one at point E are traveling along circular paths that have the same radius. If the
car at point A now moves twice as fast as the car at point E, how is the magnitude of its acceleration related to that of
car E.
Hint 1. Find the acceleration of the car at point E
Let be the radius of the two curves along which the cars at points A and E are traveling. What is the magnitude of the acceleration of the car at point E?
Express your answer in terms of the radius of curvature and the speed of car E.
Hint 1. Uniform circular motion
The magnitude of the acceleration of an object that moves with constant speed along a circular path
of radius is given by
.
ANSWER:
Hint 2. Find the acceleration of the car at point A
If , what is the acceleration of the car at point A? Let be the radius of the two curves along which
the cars at points A and E are traveling.
Express your answer in terms of the speed of the car at E and the radius .
Hint 1. Uniform circular motion
The magnitude of the acceleration of an object that moves with constant speed along a circular path of
radius is given by
.
ANSWER:
ANSWER:
D,F
=
=
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Acceleration in an Ultracentrifuge
Part A
Find the required angular speed, , of an ultracentrifuge for the radial acceleration of a point 1.30 from the axis to
equal 3.00×10
5
g
(where is the free-fall acceleration).
Express your answer numerically in revolutions per minute.
Hint 1. Find the tangential speed
Find the tangential speed of a point at radius 1.30 from the axis.
Express your answer numerically in meters per second.
Hint 1. Centripetal acceleration
Recall that the equation for centripetal acceleration is given by
,
where is the velocity of a point at radius that is following a circular path.
Hint 2. Acceleration of gravity
The acceleration due to gravity is equal to 9.8 .
ANSWER:
Hint 2. Converting tangential speed into revolutions
To find the number of revolutions per second from the tangential speed, one just has to divide by the distance
traveled in a single revolution (
).
ANSWER:
The magnitude of the acceleration of the car at point A is twice that of the car at point E.
The magnitude of the acceleration of the car at point A is the same as that of the car at point E.
The magnitude of the acceleration of the car at point A is half that of the car at point E.
The magnitude of the acceleration of the car at point A is four times that of the car at point E.
=
195
=
1.44×10
5
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Problem 4.28 - Enhanced - with Hints and Feedback
Part A
As the earth rotates, what is the speed of a physics student in Miami, Florida, at latitude ? Ignore the revolution of
the earth around the sun. The radius of the earth is 6400 .
Express your answer with the appropriate units.
Hint 1. How to approach the problem
Calculate the radius of the cross section of the earth at the given latitude. Then use the equation for the speed of
a particle undergoing uniform circular motion to determine the speed of the student. Recall that the period of the
earth's motion is 24 hours.
ANSWER:
Correct
Part B
What is the speed of a physics student in Fairbanks, Alaska, at latitude ?
Express your answer with the appropriate units.
Hint 1. How to approach the problem
Calculate the radius of the cross section of the earth at the given latitude. Then use the equation for the speed of
a particle undergoing uniform circular motion to determine the speed of the student. Recall that the period of the
earth's motion is 24 hours.
ANSWER:
Correct
Problem 4.31
=
420 =
200
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Peregrine falcons are known for their maneuvering ability. In a tight circular turn, a falcon can attain a centripetal acceleration
1.5 times the free-fall acceleration.
Part A
What is the radius of the turn if the falcon is flying at 18 ?
Express your answer with the appropriate units.
ANSWER:
Correct
Problem 4.33 - Enhanced - with Hints and Feedback
The radius of the earth's very nearly circular orbit around the sun is .
Part A
Find the magnitude of the earth's velocity. Assume a year of 365 days. Express your answer with the appropriate units.
Hint 1. How to find the earth's velocity
Recall the relationship between velocity, displacement, and time. In this case, displacement is the length of the
earth's orbit around the sun, and time is the period of the earth's motion.
ANSWER:
Correct
Part B
Find the magnitude of the earth's angular velocity.
Express your answer in radians per second.
Hint 1. How to find the earth's angular velocity
Angular velocity is the rate at which a particle's angular position is changing as it moves around a circle. To
calculate it, you only need to know angular displacement and period, not the radius of motion.
ANSWER:
= 22 =
3.0×10
4
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Part C
Find the magnitude of the earth's centripetal acceleration as it travels around the sun. Express your answer with the appropriate units.
Hint 1. How to find the earth's centripetal acceleration
After you solve the previous parts, you can use the expression for centripetal acceleration either in terms of
linear velocity and radius or in terms of angular velocity and radius.
ANSWER:
Correct
Problem 4.34
A speck of dust on a spinning DVD has a centripetal acceleration of 19 .
Part A
What is the acceleration of a different speck of dust that is twice as far from the center of the disk?
Express your answer with the appropriate units.
ANSWER:
Correct
Part B
What would be the acceleration of the first speck of dust if the disk's angular velocity was doubled?
Express your answer with the appropriate units.
ANSWER:
=
2.0×10
−7
=
6.0×10
−3
=
38 =
76
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Problem 4.35
Your roommate is working on his bicycle and has the bike upside down. He spins the 56.0 -diameter wheel, and you
notice that a pebble stuck in the tread goes by three times every second.
Part A
What is the pebble's speed?
Express your answer with the appropriate units.
ANSWER:
Correct
Part B
What is the pebble's acceleration?
Express your answer with the appropriate units.
ANSWER:
Correct
Problem 4.38
The figure shows the angular-velocity-versus-time graph for a particle moving in a circle.
5.28 99.5
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Part A
How many revolutions does the object make during the first 4 ?
Express your answer with the appropriate units.
ANSWER:
Correct
Free-Body Diagrams
Learning Goal:
To gain practice drawing free-body diagrams
Whenever you face a problem involving forces, always start with a free-body diagram.
To draw a free-body diagram use the following steps:
1. Isolate the object of interest. It is customary to represent the object of interest as a point in your diagram.
2. Identify all the forces acting on the object and their directions. Do not include forces acting on other objects in
the problem. Also, do not include quantities, such as velocities and accelerations, that are not forces.
3. Draw the vectors for each force acting on your object of interest. When possible, the length of the force vectors
you draw should represent the relative magnitudes of the forces acting on the object.
In most problems, after you have drawn the free-body diagrams, you will explicitly label your coordinate axes and directions.
Always make the object of interest the origin of your coordinate system. Then you will need to divide the forces into x
and y
components, sum the x
and y
forces, and apply Newton's first or second law.
In this problem you will only draw the free-body diagram.
Suppose that you are asked to solve the following problem:
Chadwick is pushing a piano across a level floor (see the figure). The piano can slide across the floor without friction. If
Chadwick applies a horizontal force to the piano, what is the piano's acceleration?
To solve this problem you should start by drawing a free-body diagram.
= 9.5
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Part A
Determine the object of interest for the situation described in the problem introduction. For this situation you should draw
a free-body diagram for
Hint 1. How to approach the problem
You should first think about the question you are trying to answer: What is the acceleration of the piano? The
object of interest in this situation will be the object whose acceleration you are asked to find.
ANSWER:
Correct
Part B
Identify the forces acting on the object of interest. From the list below, select the forces that act on the piano.
Check all that apply.
ANSWER:
the floor.
Chadwick.
the piano.
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Now that you have identified the forces acting on the piano, you should draw the free-body diagram. Draw the length of your
vectors to represent the relative magnitudes of the forces, but you don't need to worry about the exact scale. You won't have
the exact value of all of the forces until you finish solving the problem. To maximize your learning, you should draw the
diagram yourself before looking at the choices in the next part. You are on your honor to do so.
Part C
Select the choice that best matches the free-body diagram you have drawn for the piano.
Hint 1. Determine the directions and relative magnitudes of the forces
Which of the following statements best describes the correct directions and relative magnitudes of the forces
involved?
ANSWER:
ANSWER:
acceleration of the piano
gravitational force acting on the piano (piano's weight)
speed of the piano
gravitational force acting on Chadwick (Chadwick's weight)
force of the floor on the piano (normal force)
force of the piano on the floor
force of Chadwick on the piano
force of the piano pushing on Chadwick
The normal force and weight are both upward and the pushing force is horizontal.
The normal force and weight are both downward and the pushing force is horizontal.
The normal force is upward, the weight is downward, and the pushing force is horizontal. The normal
force has a greater magnitude than the weight.
The normal force is upward, the weight is downward, and the pushing force is horizontal. The normal
force and weight have the same magnitude.
The normal force is upward, the weight is downward, and the pushing force is horizontal. The normal
force has a smaller magnitude than the weight.
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If you were actually going to solve this problem rather than just draw the free-body diagram, you would need
to define the coordinate system. Choose the position of the piano as the origin. In this case it is simplest to let
the y
axis point vertically upward and the x
axis point horizontally to the right, in the direction of the
acceleration.
Chadwick now needs to push the piano up a ramp and into a moving van. The piano slides up the ramp without friction. Is
Chadwick strong enough to push the piano up the ramp alone or must he get help? To solve this problem you should start by
drawing a free-body diagram.
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Part D
Determine the object of interest for this situation. For this situation, you should draw a free-body diagram for
ANSWER:
Correct
Now draw the free-body diagram of the piano in this new situation. Follow the same sequence of steps that you followed for
the first situation. Assume that Chadwick pushes in a direction parallel to the inclined plane. Again, draw your diagram
before you look at the choices below.
Part E
Which diagram accurately represents the free-body diagram for the piano?
ANSWER:
the ramp.
Chadwick.
the piano.
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In working problems like this one that involve an incline, it is most often easiest to select a coordinate system
that is not vertical and horizontal. Instead, choose the x
axis so that it is parallel to the incline and choose the y
axis so that it is perpendicular to the incline.
Tactics Box 5.2 Identifying Forces
Learning Goal:
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To practice Tactics Box 5.2 Identifying Forces.
The first basic step in solving force and motion problems generally involves identifying all of the forces acting on an object.
This tactics box provides a step-by-step method for identifying each force in a problem.
TACTICS BOX 5.2
Identifying forces
1. Identify the object of interest. This is the object whose motion you wish to study.
2. Draw a picture of the situation. Show the object of interest and all other objects—such as ropes, springs, or
surfaces—that touch it.
3. Draw a closed curve around the object. Only the object of interest is inside the curve; everything else is
outside.
4. Locate every point on the boundary of this curve where other objects touch the object of interest. These are the
points where contact forces are exerted on the object.
5. Name and label each contact force acting on the object. There is at least one force at each point of contact;
there may be more than one. When necessary, use subscripts to distinguish forces of the same type.
6. Name and label each long-range force acting on the object. For now, the only long-range force is the
gravitational force.
Apply these steps to the following problem: A crate is pulled up a rough inclined wood board by a tow rope. Identify the
forces on the crate.
Part A
Which of the following objects are of interest?
Check all that apply.
ANSWER:
earth
rope
crate
wood board
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Now that you have identified the object of interest, draw a sketch of the situation and draw a closed curve
around the object, as shown in the figure below.
Part B
Identify the contact
forces exerted on the crate.
Check all that apply.
ANSWER:
Correct
Part C
Identify the long-range forces acting on the crate.
Check all that apply.
ANSWER:
thrust
spring force
tension
normal force
static friction
gravitational force
kinetic friction
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Now that you have identified all the forces acting on the system, your final sketch describing the situation
might look like this:
Problem 5.4 - Enhanced - with Expanded Hints
An arrow has just been shot from a bow and is now traveling horizontally. Air resistance is not negligible.
Part A
Identify the forces acting on the arrow. Check all that apply.
Hint 1. How to approach the problem
To identify the forces acting on the object of interest, follow the steps below.
1. Draw a picture of the situation described. Show the object of interest and all other objects that
touch it.
2. Draw a closed curve, which encloses only the object of interest; everything else should be outside.
3. Locate every point on the boundary of this curve where other objects touch the object of interest.
These are the points where contact forces are exerted on the object.
4. Name and label each contact force acting on the object. There is at least one force at each point of
contact; there may be more than one.
kinetic friction
gravitational force
spring force
normal force
tension
thrust
static friction
drag
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5. Name and label each long-range force acting on the object. For now, the only long-range force is
the gravitational force.
Hint 2. Simplify: agents in contact
Which of the following agents is the arrow in contact with during the flight?
Check all that apply.
ANSWER:
ANSWER:
Correct
VISUALIZE:
Problem 5.1
A chandelier hangs from a chain in the middle of a dining room.
Part A
Identify the forces on the chandelier.
Check all that apply.
ANSWER:
A bow
Air
Ground
Drag Gravity force Force of motion Normal reaction Force of bow string
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Correct
Problem 5.2
A car is parked on a steep hill.
Part A
Identify the forces on the car.
Check all that apply.
ANSWER:
Correct
Conceptual Question 5.1
An elevator suspended by a cable is descending at constant velocity.
Part A
How many force vectors would be shown on a free-body diagram?
ANSWER:
Gravity Static friction Thrust Tension Normal force Drag Friction Traction force Gravity Normal force
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Correct
Part B
Name these forces.
Check all that apply.
ANSWER:
Correct
Score Summary:
Your score on this assignment is 98.2%.
You received 18.65 out of a possible total of 19 points.
0
1
2
3
4
5
Normal force Gravity Thrust Tension Force of motion
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