Experiment #14: Hooke's Law

plantcalicobeansUrban and Civil

Nov 29, 2013 (3 years and 8 months ago)

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Rev 1/01,10/05
,11/06


67

Experiment 14: Hooke’s Law and Simple Harmonic





Motion


Purpose


(1)

To study Hooke’s Law for an elastic spring


(2) To study Simple Harmonic Motion of a mass suspended from an elastic spring


Apparatus


Helical steel spring with supporting stand and
scale, set of slotted weights with hanger,
timer, laboratory balance.


Theory: Hooke’s Law


A spring exerts a force which is given by Hooke’s Law:




1 F
s

=
-

kx

where x is the amount of displacement from the equilibrium position. The negative sign i
n this
equation shows that the spring’s force is
opposite

to x. If the spring is stretched (x is positive)
then the spring pulls back. If the spring is compressed (x is negative) the spring pushes. The
parameter k is the spring constant and is a proper
ty of the spring. It is different for different
springs.


An elastic spring subjected to a stretching force of magnitude F will be stretched from its
equilibrium position by an amount x given by Hooke’s Law until the spring force, which pulls
back when th
e spring is stretched, balances the stretching force.








Experiment 14







68


If you hang your spring on the supporting stand (Figure 1), it will be at its unstretched
length. Hanging a slotted weight of mass m
load

on it will subject it to the force of gravity on the
sl
otted weight F= m
load

g. This will cause the spring to stretch a distance x from its equilibrium
position according to Hooke’s Law until F
s

= m
load

g.



Simple Harmonic Motion (SHM)

If a spring with a weight hanging on it is given an additional displaceme
nt of magnitude
a

(Fig. 2) and released, it will undergo Simple Harmonic Motion. In simple harmonic motion a
spring





will oscillate up and down with an amplitude
a

and a period T. The minus sign in Hooke’s Law
tells us why this happens. When t
he spring is stretched downward, it pulls upwards and then
becomes compressed. When it is compressed, it pushes downward and then becomes stretched,
and so on. The equation for period T is:





2 T = 2

††††
M
敦e

††††††††††††††††††
††††

k


where M
eff

is the “effective mass” = M
load

+ 1/3 M
spring
. This correction follows from a
more accurate theory.




Experiment 14







69

Procedure Part I
-

Testing Hooke’s Law:

1. Measure and record the mass m
hanger

of the hanger, the mass m
spring

of your

spring.


2. Adjust the scale so that the pointer on the hanger is near the upper end of the scale when the
hanger is unloaded. Record its position as S
o

and prepare a table on your data sheet:



m
weights


m
load

= m
weights

+ m
hanger


Scale Rea
ding S


(cm)








The units of the scale are centimeters.


3. Load the hanger by m = 50 grams. Record the scale reading S. Keep increasing the load by
suitable increments (they need not be equal) to obtain 8 data points before the lower

end of the
scale is reached. Record all data.

CAUTION:
Do not stretch the spring excessively
. You may damage it by deforming it
permanently.


4. Reverse the process by unloading the same loads as in (3). Again, record S for each load.



Procedure Par
t II
-

Study of Simple Harmonic Motion
:

1. Prepare a table:


m
weights

m
load

= m
weights

+ m
hanger

N

t
N

T=t
N
/N

m
eff

k




















average



2. Load the hanger with m = 200 g and set the suspended mass into oscillation (as in Fig. 2(b))
with an am
plitude of roughly 1 cm. Measure the time t
N

for N = 100 complete cycles.
Remember that in a complete cycle the mass starts in the original position and goes back to the
original position, i.e. from the top of the cycle down to the bottom and back to the

top of the
cycle.



Experiment 14







70

3. Repeat (2) with an amplitude of about 2 cm and a value of N between 90
-
110.


4. Repeat (1) and (2) with m = 250g.


5. Repeat (1) and (2) with m = 300g.



Lab Report Part I

Theory
: In equation 1 above, x represents the

amount the spring stretched = (S
-

S
o
). You
can calculate the spring constant
k

from your data using equation 1 :




F
s

=
k

(S
-

S
o
)

Since the spring force exactly balances the weight of the mass m
load
g, we can write:



m
load
g =
k

(S
-

S
o
) =
k

S
-

k

S
o

Solving for s we find




S = (g/
k
) m
load

+ S
o

This shows a linear relationship between m and s with the slope = g/
k

and the y intercept of S
o
.
For a plot of S vs m


s
lope =

S/

m
load

= g/
k


k

= g / slope


1. Plot S vs m
load

for all your data points (some points may be on top of each other). Label all
axes and write the units.





Calculate the slope =

s/

m (don’t forget

the units). Then calculate
k

using the equation

k

= g / slope. What are the units of k? (Remember g = 980 cm/sec
2
)


Experiment 14







71

Question #1

What is the purpose of measuring S both when loading and unloading. What is your
conclusion of this effect, based on your dat
a?


Question #2

From looking at your graph, can you claim that you verified Hooke’s Law? Explain
your reasoning.


Lab Report Part II

Theory
:

You can calculate
k

for all runs from this data using the equation for the period T:




k
m
2
eff
T



Squaring both sides and solving for
k
. we find:









2 Find the average value of
k
for Part II using Equation 3. Remember to write all units.


Question #3

Did you verify that the period of SHM is independent of ampl
itude? Explain.



Conclusion

3, Compare your results from Part I to your average results from Part II by calculating the
discrepancy as follows:






k
(Part I)


k

(Part II)



100%




.5*
k

(Part I) +
k

(Part II)



What are some reasons for the d
iscrepancy?

2
2
T
m
4
π
k

3
eff