PROBLEM #1: SPRINGS AND EQUILIBRIUM I
You work for a biophysics research group studying the mechanics of DNA during cell
division. One common technique involves securing one end of a DNA molecule to a
surface and, with optical tweezers, measuring the force required to slightly straighten
the DNA molecule from its normal coiled state. The optical tweezers exert a force on a
tiny bead which has been attached to the free end of the DNA molecule. The group is
concerned that the bead’s mass could affect the DNA’s response to the pulling force.
The DNA molecules are expected to behave much like mechanical springs, so you
decide to first use a spring as a model. You decide to calculate, and then measure, how
much a spring stretches when it is subjected to external forces in two situations. In each
situation, a force is exerted by a hanging mass that can be changed. In the first situation
the object will hang directly from the spring. In the second situation the object will
hang from a light string, which runs over a pulley and is attached to a medium-sized
cart on a horizontal track. One end of the spring will be attached to the cart and the
other end to an end-stop fixed to the track.
You want to determine how the stretch of the spring in the two situations will compare.
This will serve as your simple model for how DNA stretching compares with and
without an optical bead.
Instructions: Before lab, read the laboratory in its entirety as well as the required reading in the
textbook. In your lab notebook, respond to the warm up questions and derive a specific prediction
for the outcome of the lab. During lab, compare your warm up responses and prediction in your
group. Then, work through the exploration, measurement, analysis, and conclusion sections in
sequence, keeping a record of your findings in your lab notebook. It is often useful to use Excel to
perform data analysis, rather than doing it by hand.
Read Sternheim & Kane: sections 3.1-3.8, 4.1-4.2, 4.10, 9.1.
EQUIPMENT
You have a spring, mass
set, cart, cart masses,
track, end-stop, pulley,
table clamp, rod, string
and a meterstick.
Cart
If equipment is missing or broken, submit a problem report by sending an email to
labhelp@physics.umn.edu. Include the room number and brief description of the
problem.
SPRINGS AND EQUILIBRIUM I – 1201Lab3Prob1
WARM UP
1. Draw a picture of a spring attached to a hanging object. On your picture, show and
label the extension of the spring.
2. Draw and label all the forces acting on the hanging object.
3. What condition must be satisfied by the forces acting on an object in equilibrium?
Write the equation to express that for the hanging object. How do you know if the
object is in equilibrium?
4. Write an equation that relates the spring’s extension to the force it exerts. Use this
equation and the result of the previous step to write down the relationship between
the extension of the spring and the weight of the hanging object. Sketch a graph of
spring extension vs. weight.
5. Draw a picture of a horizontal spring with one end connected to a cart and the other
to a string which is in turn connected to a hanging object. Identify all the forces
acting on the hanging object; indicate them on the picture as vectors and label them.
Repeat for the forces acting on the cart. What do you assume about the size of the
force exerted by the string on the cart and on the hanging object? How do you
justify this assumption?
6. Write an equation to express the equilibrium condition for the forces acting on (a)
the hanging object and another equation for (b) the cart. Use these equations,
together with the equation that gives the relationship between a spring’s extension
and the force it exerts, to determine the relationship between the extension of the
spring and the weight of the hanging object. Sketch a graph of spring extension vs.
weight.
PREDICTION
Restate the problem in terms of quantities you know or can measure. Beginning with
basic physics principles, show how you get an equation that gives the solution to the
problem. Make sure that you state any approximations or assumptions that you are
making. This equation will be a relationship between the extension of a spring and the
weight of the hanging object for each situation described in the problem (when the
system has reached equilibrium). Illustrate each relationship with a graph of spring
extension vs. weight.
EXPLORATION
Select spring(s) and a series of weights that give a usable range of displacements. The
largest weight should not pull a spring past its elastic limit (about 40 cm for the longer
springs). Beyond that point the spring does not return to its original unstretched length
SPRINGS AND EQUILIBRIUM I – 1201Lab3Prob1
and is permanently damaged. Decide on a procedure to measure the extension of the
spring in a consistent manner, and describe in your lab notebook why it is a reasonable
procedure. Decide on how many measurements using different masses you will need
to test your prediction.
MEASUREMENT
When the spring is vertical, make a series of measurements of the mass of the hanging
object and the extension in the spring in a stationary state. Repeat for the case in which
the spring is horizontal and connected to a cart. To see the effect of the cart’s mass,
make additional measurements with mass added to the cart. As a check, make
measurements with the cart removed so the string that goes over the pulley is
connected directly to one end of the horizontal spring.
Estimate the uncertainty in measuring the extensions. How does friction in the pulley
and in the cart affect your measurement? Is it significant?
ANALYSIS
For each case, make a graph of the extension of the spring against the hanging mass.
Examine the shape of the graphs. If the points in a graph seem to fall on a line, estimate
the slope of that line, with appropriate units, and explain the physical significance of the
slope. Why don’t all of the data points fall exactly on your line?
CONCLUSION
What can you say, based on your measurements, about the relationship between the
extension of the spring and the weight of the hanging object in each case? Compare the
relationships for the two cases. How did the mass of the cart affect the extension of the
spring? Did your prediction agree with your measurements within the uncertainties of
your measurement? If not, why?
Was there a “minimum” force required to stretch your spring beyond its unstretched
length? If so, write a brief justification for how you dealt with that in your
measurements and analysis.
What conclusions can you draw about the corrections you will have to make for the
bead’s mass on the DNA measurement described in the original problem?
SPRINGS AND EQUILIBRIUM I – 1201Lab3Prob1