This lab is divided into several parts. You must complete the parts in order. 
 Review the theory and methods of L123: Hooke's Law and a Measurement Spring Constant. Also review your original submission of that lab on WebAssign. If the teacher requested that you revise the analysis, you must do that before proceeding with L145, part B.
 Complete WebAssign L145A.
 Wait for the teacher's response to that assignment before continuing with Part B.
To measure the period of a spring in oscillation
to 3 significant figures and compare to theory
 Check your feedback from the instructor on L145A. If rework is required, do that and resubmit your work before continuing with Part B. You'll need an accurate value of the spring constant to continue with Part B.
 The same spring and spring support that you used in L123
 Weight hanger
 Slotted mass of 200 g
 Stopwatch: You may use this program or a handheld stopwatch that reads to 0.01 s. Do not use a sonic ranger or photogate for measuring time in this part of the lab.
 Tape
Subpart 1
Enter your responses to items 15 in the
WebAssign form L145B1. The items below are repeated on the form.

Enter the spring constant
that you submitted in L145A. This should be the revised value if you did the
analysis over. Give the number to 3 significant figures in SI units.
 Place a total of 0.250 kg, including the weight hanger, suspended from the spring. Tape the weight hanger to the spring and the weight to the weight hanger. This will prevent the weight hanger and/or weight from bouncing off the spring to the floor. While this wouldn't damage the weight, it could break the weight hanger. It could also leave a dent in a wooden or lineoleum floor or crack a tile floor. You could also put a cushion on the floor just in case.
Keep
the weight constant throughout this part of the experiment. Set the
spring into oscillation by lifting it upbut not so much as to
completely relax the springand releasing it. A lift of a few centimeters is sufficient. You'll use the stopwatch to measure 5 trials of
the time for 1 complete cycle of the motion. (That's 1 cycle; not 2, or
3, or more.) Before you start taking measurements, read the following important note. Then take your 5 trials and record results to a precision of 0.01 s.
We know this is a crude
measurement technique, but you need this information in order to later
refine your technique appropriately. This is not a contest to see
how close you can get to the same value every time. 

Calculate the mean of your set of 5
measurements.

Calculate the percentage mean deviation of
your set of 5 measurements. Round your result to 1 significant figure. In order to minimize rounding error, carry through all calculations of averages and deviations on your calculator and avoid rounding to the last step. If you know how to use formulas in Excel, this is a good way to do your calculations.

It should be obvious that your
measurements don't meet the goal of the experiment. For a 3 significant
figure measurement, you'll need to get the percentage mean deviation down
to 1%. How many consecutive cycles N must you time in order to achieve
that? Consider that the N cycle method divides the percentage mean
deviation of the one cycle method by N.
Subpart 2
Before continuing with this part, wait for the teacher's emailed response to Subpart 1. Use WebAssign L145B2 to post your responses to the following.

The teacher will confirm the value of N that you will use for your next set of measurements. Enter that value. WebAssign will use it in checking your calculations.

Take a new set of 5 trials of the time for N cycles.

Calculate the mean of your set of 5
measurements.

Calculate the percentage mean deviation of
your set of 5 measurements to 1 significant figure.

Now, using your result from step 3,
calculate the period of the oscillation to
3 significant figures.

Assuming that the period, T, of an oscillating mass on a spring depends only the spring constant, k, and the mass, m, determine the mathematical combination of k and m that gives the right units for period. This result should be correct to within a constant numerical factor,
which will be determined later by experiment. That is, you'll determine an
equation of the form T = Cf(k,m), where f(k,m) is a function of k and m,
and C is a numerical constant to be determined later. Give the function
f(k,m) and show that the units reduce to those expected for period.

Using your measured spring constant, mass and period, and your formula
above, calculate the numerical constant in your formula for period. Give the
result.

Theoretically, what should the constant be? Look
this up if you need to.

Calculate the experimental error between the calculated and
theoretical values of the constant. Take the theoretical value to be
accepted.

Describe 2 possible and significant
sources of experimental error. Be specific in describing how the source
could contribute to error. As always, phrases like human error and rounding
error are not acceptable.
To determine the
equation of motion of a mass oscillating on a spring
Complete and submit the previous parts of L145.
 The same spring and spring support that you used in the previous parts of the lab
 Weight hanger
 Slotted mass of 200 g
 Computer with available USB port
 LabQuest Mini interface and USB cable
 Vernier Motion Detector 2 and cable
Enter your responses in the
WebAssign L145C form. The items below are repeated on the form.
 Enter the following items for reference.
 spring constant in SI units
 mean period that you determined in Part B2
 Place a total mass of 0.250
kg on the spring. This includes the weight hanger. Connect the motion detector and position it on the floor under the weight hanger as you did in L123. Start the Logger Pro software and then do the following.
 Go to Experiment Menu, Data Collection, Collection tab, and change the sampling rate to 30 samples/sec. Also, set the experiment length to 3 seconds. Click OK.
 Delete the velocity vs. time graph so that you only have position vs. time. Click Page > Auto Arrange to maximize the graph size.
 Set the spring into oscillation as you did in L145B. Click Collect to start data collection. You should obtain a smooth, sinusoidal graph.
You may need to click the autoscaling icon at the top. There may be stray points toward the end of the time interval if the spring starts to sway from side to side.You may also find that the amplitude decreases significantly with time. If your graph shows evidence of either of these, just collect new data until you get a goodlooking graph.
 Save
your file and upload it to WebAssign under item 2 of the L145C assessment form.
Next you'll determine the equation of the function
that you just saved. That means you'll need to determine the equilibrium
position, the amplitude, the period, and the phase shift. You should have already practiced these things.
One method you may not use at this point is a curve fit. That would sidestep the
knowledge that this lab expects you to demonstrate.
 Describe the methods that you
will use to determine the indicated characteristics below.
 equilibrium
position
 amplitude
 period (Use
the position vs. time graph rather than the value that you
determined in Part B2 of the lab.)
 phase shift
 Now
carry out your methods. Be sure to have your calculator in radians mode. Give final values below
to 3 significant figures.
 equilibrium
position
 amplitude
 period
 phase shift
(Assume a cosine function for the waveform.)
 Write the equation of the position vs.
time function. Use the same values that you gave above.
 Perform the following checks on your equation.
Substitute the time coordinate for one of the data points into your equation to see if you get the correct position.
Enter the following:
 time
coordinate that you selected
 position
coordinate calculated from your equation (remember to have your
calculator in radians mode)
 position
coordinate read from your graph
 percentage
difference between items b and c (give absolute value)
 percentage
difference between the period measured in Part L145B2 and that measured
from your graph (give absolute value)
To examine the
position, velocity, and acceleration of the spring as a function of
time
Use the same equipment as in L145C. In addition, you'll need the 100 g slotted mass.
Collect your data as follows:
 Suspend a total of 350 g of mass from the spring, including the weight hanger.
 You'll need position, velocity, and acceleration vs. time graphs to be displayed in Logger Pro. Insert any graphs that you need, and make sure that the axis labels are appropriate. Autoarrange the page so that the graphs don't overlap.
 Go to Experiment,
Data Collection, Collection tab, and set the experiment length to 2
seconds and the sampling rate to 30 samples/sec. Press OK.
 Set the mass
into oscillation and collect data.
 You should get a
smooth curve for the position graph, but the velocity and
acceleration graphs may show some scatter in the points. (This has
to do with the method used to calculate velocity and acceleration
from the position data.) Select View, then Autoscale if the curves
are not showing on any of the graphs. Then collect data as before.
 Obtain a curve fit to each of the graphs as follows: Select Analyze, then Automatic Curve Fit,
then Sine.
 Save your file and
upload it to item 1 of the WebAssign L145D form. You'll use the file in the instructions to follow.
For the following, use your Logger Pro file with the fits to the position, velocity, and acceleration vs. time graphs. Enter your answers on the L145D WebAssign
form. The questions are repeated below for reference.
Note: While the textbook makes a point of distinguishing between
the terms angular velocity and angular frequency, they are
both defined by ω = 2π/T. We will use the term angular velocity in the following.
Enter all values to 3 significant figures unless indicated
otherwise.
 Angular velocity and period
 What fit coefficient represents the angular
velocity in all of the fits?
 Give the value of the angular velocity obtained
from the fit coefficient for the position vs. time fit.
 Use the angular velocity from the previous item
to calculate the period of the mass on the spring.
 Use the graph (not the fit) to determine the
period of the mass on the spring. Describe the method that you use to achieve an accurate result. (Consider how you increased the accuracy of the measurement of period in L145B2.)
 Now give the value of the period that you
determined from the graph (not the fit).
 Calculate the experimental error between the two
values of period given above. Use the value obtained from the fit as the
accepted value.
 Comparision of angular velocity values
 There's another way to determine the angular velocity. Calculate the angular
velocity using the spring constant and the mass.
 Calculate the experimental error between
the values of angular velocity in 1b and 2a. Use the
former value as accepted.
 Position characteristics
 Which two coefficients do
you need from the fit to the
position graph in order to determine the amplitude and
equilibrium position of the
motion?
 Give the values of the fit coefficients
that represent the amplitude and equilibrium position.
 Velocity characteristics
 Which direction is
positive, up or down? (The answer isn't an arbitrary
choice, because it's determined by the position of the sonic ranger and
how the Logger Pro software records position.)
 When the mass has its
greatest positive displacement, what value do you expect for the
velocity?
 When the mass is passing
the equilibrium position going down, do you expect the velocity to be
positive or negative?
 Does your velocity graph agree with your predictions?
 Acceleration characteristics
 When the mass has its
greatest positive displacement, is the acceleration positive or negative?
 If the mass were
oscillating horizontally so that the only horizontal force on the mass
was the spring force, why would one expect that when the mass had its
greatest positive displacement, the acceleration would have its greatest
negative value? Use Newton's 2nd Law and the defining relationship
of simple harmonic motion in your answer. See section 132 if necessary.
(We will see later that the verticallyoscillating system can, in fact,
be treated the same as a horizontallyoscillating system.)
 Assuming now that the
verticallyoscillating system can be treated the same as a
horizontallyoscillating system, what value would you expect the
acceleration to be as the mass passed the equilibrium position going
down?
 Does your acceleration graph agree with your prediction?
 Tension characteristics: While you don't have data for the tension force of the spring on the mass as a function of time, you can make sensible predictions below.
 What direction is the tension at all
times? Why is this expected?
 What value do you expect for the tension
when the mass is in equilibrium?
 More about acceleration
 What one coefficient from your
fit to the Acceleration vs.Time graph can be used to determine the maximum and minimum
values of the acceleration?
 Give the value of the coefficient.
 Given your result from
part b and a law of physics, determine the maximum and minimum values of the
net force.
Draw force diagrams for these three situations: i) mass displaced above equilibrium position, ii) mass at the equilibrium position, iii) mass displaced below the equilibrium position. Show the force vectors with approximately correct relative magnitudes. Refer to your diagrams as you answer the following.
 More about tension
 The magnitude of the
tension force is _____ (less than, equal to, more than) the weight
when the object is displaced above the equilibrium position.
 The magnitude of the
tension force is _____ (less than, equal to, more than) the weight
when the object is displaced below the equilibrium position.
 The net force on the object is _____
(up, down) when the object is displaced above
the equilibrium position.
 The net force on the object is _____
(up, down) when the object is displaced below
the equilibrium position.
 Net force
 Assuming +x is up,
write the one net force equation that applies to the object in all
positions while oscillating vertically. Use the symbols T, m, and g.
 Using the net force equation and the maximum and minimum values of the net force from 7c, calculate the maximum and minimum values of the tension.
This part is primarily theory. There's no data to take, although you'll use a result from L145D. Write your responses on paper and submit your scanned file to BrainHoney.
To show that a net force and energy analysis for a verticallyoscillating spring yields the same results as for a horizontallyoscillating spring
Open and run this animation to familiarize yourself with the situation.
The diagram below shows 4 sidebyside
snapshots (numbered 14) of the system. Here are the situations represented
in each snapshot:
Snapshot 1: The spring is
motionless with no added mass.
Snapshot 2: Total mass m has been added to the spring. The system is
in equilibrium. The equilibrium position is assumed to be 0.
Snapshot 3: The mass is moving either up or down, and has a positive
displacement y from the origin. The direction of +y is defined to
be up.
Snapshot 4: The mass is moving either up or down, and has a negative
displacement y from the origin.
Also note the following:
Now we'll do a net force problem for Snapshot 2. Refer to
the force diagram associated with Snapshot 2 above.
Net force analysis for Snapshot 2 

The net force equation is written. 
There is no acceleration, so the net force is 0.
Hooke's Law is used to substitute ky_{1} for T.
Note
that the value of ky_{1} is positive as expected with +y pointing up. 
y_{1} is solved for. This gives us an expression for y_{1} that will
be useful later. 
Next we'll do a net force problem for Snapshot 3. Once
again, refer to the force diagram above associated with Snapshot 3. The tension force is now less than the weight, and the acceleration is down.
We'll make an exception to the usual practice of selecting the positive
direction to be that of the acceleration. The reason for this exception is
to keep the direction of positive displacement the same for all snapshots.
Otherwise, sign inconsistencies would result.
Net force analysis for Snapshot 3 

The net force equation is written. 
Since y_{1} and y are both positive in
this case, the quantity (y_{1}  y) < y_{1}. This
results in a smaller tension force than for Snapshot 2. 
The result obtained for y_{1} from the
analysis of Snapshot 2 is substituted. 
Since k and y are positive, the net force is
negative. This is consistent with a downward (negative)
acceleration. 
The last result shows that F_{net} is proportional
to the displacement from the equilibrium position. There is no dependence on g. Hence, we can analyze
this system for forces as if it were a horizontallyoscillating spring. Note that in this
situation the equilibrium position is not the unstretched position of the
spring. Rather, the equilibrium position is the position shown in Snapshot
2.

Following the guide of the
net force analysis for Snapshot 3, do the net force analysis for
Snapshot 4. Start with the relevant force diagram. Consider the following questions in order to avoid sign mistakes.
You don't have to write the answers to the questions, but you do have to show a net force analysis similar to what was done above.

Does y represent a positive or negative
number for this analysis?

Is (y_{1}  y) less than or greater than y_{1}?

Does F_{net} point up (positive)
or down (negative)?

Using the default data in the applet and results of the force analyses above, calculate the following. Show your work.

spring constant

maximum and minimum values of the tension force

maximum and minimum values of the acceleration of the mass
Goal: For the
verticallyoscillating spring, derive an equation for the velocity of the
mass as it passes through y = 0 in terms of the amplitude of the motion, the
spring constant, and the mass on the spring.
Introduction:
We'll take the system for the conservation of energy analysis to be the
spring, mass, and Earth. We'll continue to use the same coordinate
system as for the net force analysis. These are the states:
initial: the mass as it's passing through y = y_{i} = 0 on the way down
final: the lowest position of the
mass, y_{f} = A. The negative sign is necessary, because
the amplitude A is defined to be positive.
Since we've defined the system
in such a way that there are no external forces, the conservation of energy
equation that applies is 0 = ΔE_{sys}.
 ΔE_{sys} can be expanded to 6 individual energy terms (3 initial and 3 final).
Enter the formulas for these 6 terms in a table using only the following
symbols: k, m, v_{i}, v_{f}, y_{1}, y_{i}, y_{f} . Don't substitute 0's at this point. Take careful note of signs when you
write your terms. Gravitational potential energy has a positive value
for y > 0 and a negative value for y < 0. Elastic potential energy must
always be positive, and the value depends on the total extension of the
spring below y_{1}. Give the sign for each nonzero energy
term or write 0 in the Sign column if the energy term evaluates to 0.
Energy term 
Formula 
Sign 
K_{i} 


K_{f} 


U_{gi} 


U_{gf} 


U_{ei} 


U_{ef} 



Now write out an expression for ΔE_{sys}, substituting for terms that equal 0 and
for y_{f} = A. Simplify your expression as much as possible. The result from the Snapshot 2 analysis will be
helpful. Any terms
involving g must cancel. If they don't, you've made a mistake. Your
final result should be an equation for v_{i} in terms of A, k,
and m only. Write your complete solution in stepwise form vertically
down your paper. Organization and clarity are essential.

How does your result in the previous step compare to
that for the velocity of a horizontallyoscillating springmass
combination as the mass passes through the equilibrium position?
What do you learn from this?

Use the equation for v_{i}from
step 4 to
calculate the velocity using the default values of mass and amplitude from the applet and the spring constant that you found previously. Give your answer to 3 significant figures.

Now open your Logger Pro file from L145D. Do the following:

Using your equation from step 4, calculate v_{i} for your spring.

Using the velocity vs. time fit in your Logger Pro file, given the value of the coefficient that represents v_{i}.

Calculate the experimental error in v_{i} using the value in 7b as the accepted value.
