Springs in Series

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 , and spring 2 has a spring constant . A constant force of magnitude 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 and .

= / (1/k_1+1/k_2)^(-1)

Three springs in series
Now consider three springs set up in series as 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 (configuration 1). 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.

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 , the period of the pendulum, if the axis is a distance 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 . It is released with an amplitude 0.300 m. A damping force acts on the egg. After it oscillates for 5.00 s, the amplitude of the motion has decreased to 0.100 m.

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 in the string when the ball is fully submerged but not touching the bottom as shown?
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 . What is the ratio of the crown's apparent weight (in water) to its actual weight?

Express your answer in terms of the density of the crown and the density of water .

= / 1-rho_w/rho_c

B.  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 . The density of gold is .

Yes 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 of the fluid in the test tube at an arbitrary distance from the axis of rotation?