Springs in Series
In this problem you will study two cases of springs connected in series that will enable you to draw a
general conclusion.Two springs in series
Consider two massless springs connected in series. Spring 1 has a spring constant
spring constant
. A constant force of magnitude
, and spring 2 has a
is being applied to the right. When the two springs are
connected in this way, they form a system equivalent to a single spring of spring constant
.
A. What is the effective spring constant of the two-spring system?
Express the effective spring constant in terms of
= (1/k_1+1/k_2)^(-1)
Three springs in series
Now consider three springs set up in series as
and
.
shown.
The spring
constants are
,
, and
, and the force acting to the right again has magnitude
A. Find the spring constant
.
of the three-spring system.
Express your answer in terms of
,
, and
.
= (1/k_1+1/k_2+1/k_3)^(-1)
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Position, Velocity, and Acceleration of an Oscillator
Learning Goal: To learn to find kinematic variables from a graph of position vs. time.
The graph of the position of an oscillating object as a function of time is
shown.
Some of the questions ask you to determine ranges on the graph over which a statement is true. When
answering these questions, choose the most complete answer. For example, if the answer "B to D" were
correct, then "B to C" would technically also be correct--but you will only recieve credit for choosing the
most complete answer.
A. Where on the graph is
?
A to B
A to C
C to D
C to E
B to D
A to B and D to E
B. Where on the graph is
?
A to B
A to C
C to D
C to E
B to D
A to B and D to E
C. Where on the graph is
?
A only
C only
E only
A and C
A and C and E
B and D
D. Where on the graph is the velocity
A to B
A to C
?
C to D
C to E
B to D
A to B and D to E
E. Where on the graph is the velocity
?
A to B
A to C
C to D
C to E
B to D
A to B and D to E
F.
Where on the graph is the velocity
?
A only
B only
C only
D only
E only
A and C
A and C and E
B and D
G. Where on the graph is the acceleration
A to B
A to C
C to D
C to E
B to D
A to B and D to E
?
H. Where on the graph is the acceleration
?
A to B
A to C
C to D
C to E
B to D
A to B and D to E
I.
Where on the graph is the acceleration
?
A only
B only
C only
D only
E only
A and C
A and C and E
B and D
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Mass Hitting a Spring
A block sliding with velocity
along a frictionless floor hits a spring at time
The spring compresses until the block comes to a momentary stop (configuration 2).
Finally, the spring expands, pushing the block back in the direction from which it came.
(configuration 1).
In this problem you will be shown a series of plots related to the motion of the block and spring, and you
will be asked to identify what the plots represent. In each plot, the point labeled "1" refers to configuration
1 (when the block first comes in contact with the spring). The point labeled "2" refers to configuration 2
(when the block comes to rest with the spring compressed).
In the questions that follow, "force" refers to the x component of the force that the spring exerts on the
block and "position" and "velocity" refer to the x components of the position and velocity of the block.
A. Consider graph A.
What might this graph represent?
position vs. time
velocity vs. time
force vs. time
force vs. distance
B. Consider graph B.
What might this graph represent?
position vs. time
velocity vs. time
force vs. time
force vs. distance
C. Consider graph C.
What might this graph represent?
position vs. time
velocity vs. time
force vs. time
force vs. distance
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Extreme Period for a Physical Pendulum
A solid, uniform disk of mass
and radius may be rotated about any axis parallel to the disk axis, at
variable distances from the center of the disk.
A. What is
, the moment of inertia of the disk around its center of mass? You should know this
formula well.
Express your answer in terms of given variables.
= 1/2*M*a^2
B. If you use this disk as a pendulum bob, what is
distance
, the period of the pendulum, if the axis is a
from the center of mass of the disk?
Express the period of the pendulum in terms of given variables.
= 2*pi*sqrt((a^2/2+d^2)/(g*d))
C. The period of the pendulum has an extremum (a local maximum or a local minimum) for some
value of
between zero and infinity. Is it a local maximum or a local minimum?
maximum
minimum
D.
What is
, the minimum period of the pendulum?
Your answer for the minimum period should include given variables.
= 2*pi*sqrt((a^2)/(g*a/sqrt(2)))
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Damped Egg on a Spring
A 50.0-g hard-boiled egg moves on the end of a spring with force constant
with an amplitude 0.300 m. A damping force
amplitude of the motion has decreased to 0.100 m.
. It is released
acts on the egg. After it oscillates for 5.00 s, the
A. Calculate the magnitude of the damping coefficient .
Express the magnitude of the damping coefficient numerically in kilograms per second, to
three significant figures.
= .0220 (+/- 1%) kg/s
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Archimedes' Principle
Learning Goal: To understand the applications of Archimedes' principle.
Archimedes' principle is a powerful tool for solving many problems involving equilibrium in fluids. It
states the following:
When a body is completely or partially immersed in a fluid (either a liquid or a gas), the
fluid exerts an upward force on the body equal to the weight of the fluid displaced by the
body.
As a result of the upward Archimedes' force (often called the buoyancy force), some objects may float in a
fluid, and all of them appear to weigh less--this is a well-familiar phenomenon of buoyancy.
Quantitatively, the buoyancy force can be found as
,
where
is the force,
is the density of the fluid, is the magnitude of the acceleration due to
gravity, and is the volume of the displaced fluid.
In this problem, you will be asked several qualitative questions that should help you develop a feel for
Archimedes' principle; you will then solve a quantitative exercise.
An object is placed in a fluid and then released. Assume that the object either floats to the surface (settling
so that the object is partly above and partly below the fluid surface) or sinks to the bottom. (Note that, for
Parts A through D, you should assume that the object has settled in equilibrium.)
A. Consider the following statement:
The magnitude of the buoyancy force is equal to the weight of fluid displaced by
the object.
Under what circumstances is this statement true?
always
only for an object that floats
only for an object that sinks
never
B. Consider the following statement:
The magnitude of the buoyant force is equal to the weight of the amount of fluid
that has the same volume as the object.
Under what circumstances is this statement true?
always
only for an object that floats
only for an object that sinks
never
C. Consider the following statement:
The magnitude of the buoyant force equals the weight of the object.
Under what circumstances is this statement true?
always
only for an object that floats
only for an object that sinks
never
D. Consider the following statement:
The magnitude of the buoyant force is less than the weight of the object.
Under what circumstances is this statement true?
always
only for an object that floats
only for an object that sinks
never
Now apply what you know to some more complicated situations.
A. An object is floating in equilibrium on the surface of a liquid. The object is then removed and
placed in another container, filled with a more dense liquid. What would you observe?
The object would sink all the way to the bottom.
The object would float submerged more deeply than in the first container.
The object would float submerged less deeply than in the first container.
More than one of these outcomes is possible.
B. An object is floating in equilibrium on the surface of a liquid. The object is then removed and
placed in another container, filled with a less dense liquid. What would you observe?
The object would sink all the way to the bottom.
The object would float submerged more deeply than in the first container.
The object would float submerged less deeply than in the first container.
More than one of these outcomes is possible.
C. Two objects, T and B, have identical size and shape and have uniform density. They are carefully
placed in a container filled with a liquid. Both objects float in equilibrium fully submerged in the
liquid without touching the bottom of the container. Object T floats closer to the top surface of the
liquid; object B floats closer to the bottom of the container. Which of the following statements is
true?
Object T has a higher average density than object B.
Object B has a higher average density than object T.
Both objects have the same average density.
Archimedes' principle is helpful in solving quantitative (usually statics) problems as well. If an object is
moving in a liquid, however, the resulting drag forces make the situation much more complicated. A
relatively straightforward example of a statics problem involving a buoyancy force is presented next.
A. A ball of mass
.
and volume
is lowered on a string into a fluid of density
Assume that the object would sink to
the bottom if it were not supported by the string. What is the tension
is fully submerged but not touching the bottom as shown?
in the string when the ball
Express the tension in terms of the given quantities and , the acceleration due to gravity.
= (m_b-rho_f*V)*g
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Crown of Gold?
According to legend, the following challenge led Archimedes to the discovery of his famous principle:
Hieron, king of Syracuse, was suspicious that a new crown that he had received from the royal goldsmith
was not pure gold, as claimed. Archimedes was ordered to determine whether the crown was in fact made
of pure gold, with the condition that only a nondestructive test would be allowed. Rather than answer the
problem in the politically most expedient way (or perhaps extract a bribe from the goldsmith), Archimedes
thought about the problem scientifically. The legend relates that when Archimedes stepped into his bath
and caused it to overflow, he realized that he could answer the challenge by comparing the volume of water
displaced by the crown with the volume of water displaced by an amount of pure gold equal in weight to
the crown. If the crown was made of pure gold, the two volumes would be equal. If some other (less dense)
metal had been substituted for some of the gold, then the crown would displace more water than the pure
gold.
A similar method of answering the challenge, based on the same physical principle, is to compute the ratio
of the actual weight of the crown,
, with the apparent weight of the crown when it is submerged in
water,
. See whether you can follow in Archimedes' footsteps. The figure shows what is meant by
weighing the crown while it is submerged in water.
A. Take the density of the crown to be
to its actual weight?
. What is the ratio of the crown's apparent weight (in water)
Express your answer in terms of the density of the crown
=
B.
and the density of water
.
1-rho_w/rho_c
Imagine that the weight of the crown in water is
, and the true weight is
. Is the crown made of pure gold? The density of water is
density of gold is
Yes
. The
.
No
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Pressure in a Centrifuge
A test tube filled with liquid of uniform density is spun in a centrifuge with angular frequency
.
The test tube lies perpendicular to the axis of rotation of the centrifuge. The pressure in the fluid at a
distance from the axis of rotation is .
A. What is the pressure
rotation?
of the fluid in the test tube at an arbitrary distance from the axis of
= p_0+0.5*rho*(omega)^2*(r^2-(r_0)^2)
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Streamlines and Fluid Flow
Learning Goal: To understand the continuity equation.
Streamlines represent the path of the flow of a fluid. You can imagine that they represent a time-exposure
photograph that shows the paths of small particles carried by the flowing fluid. The
figure
shows streamlines for the flow of an
incompressible fluid in a tapered pipe of circular cross section. The speed of the fluid as it enters the pipe
on the left is . Assume that the cross-sectional areas of the pipe are
its exit on the right.
at its entrance on the left and
at
A. Find , the volume of fluid flowing into the pipe per unit of time. This quantity is also known as
the volumetric flow rate.
Express the volumetric flow rate in terms of any of the quantities given in the problem
introduction.
= v_1 * A_1
B. Because the fluid is assumed to be incompressible and mass is conserved, at a particular moment
in time, the amount of fluid that flows into the pipe must equal the amount of fluid that flows out.
This fact is embodied in the continuity equation. Using the continuity equation, find the velocity
of the fluid flowing out of the right end of the pipe.
Express your answer in terms of any of the quantities given in the problem introduction.
= v_1*A_1/A_2
C. If you are shown a picture of streamlines in a flowing fluid, you can conclude that the __________
of the fluid is greater where the streamlines are closer together.
Enter a one-word answer.
velocity
or
speed
D.
The flow velocity is a function of which property of the streamlines?
density (number per unit area)
length
thickness
straightness
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Venturi Meter with Two Tubes
A pair of vertical, open-ended glass tubes inserted into a horizontal pipe are often used together to measure
flow velocity in the pipe, a configuration called a Venturi meter. Consider such an arrangement with a
horizontal pipe carrying fluid of density . The fluid rises to heights
(see figure). The cross- sectional area of the pipe is
tube 2.
and
in the two open-ended tubes
at the position of tube 1, and
at the position of
A. Find , the gauge pressure at the bottom of tube 1. (Gauge pressure is the pressure in excess of
outside atmospheric pressure.)
Express your answer in terms of quantities given in the problem introduction and , the
magnitude of the acceleration due to gravity.
= rho*g*h_1
B. Find
, the speed of the fluid in the left end of the main pipe.
Express
in terms of
,
,
,
, and .
= (2*g*(h_1 - h_2)/((A_1/A_2)^2 - 1))^(1/2)
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