Object
To verify that the period of vibration of a body on a spring is independent
of the amplitude, and is given by:
| (1) | |
Theory
| (2) | F = - k s |
The negative sign indicating that the force direction is opposite to
direction of the displacement.
If the vibration amplitude of the body is relatively small, its motion
is Simple Harmonic and its period of oscillation T is given by equation
(1).
Procedure
| 1. With the Lab equipment provided in the counter
tray, clamp a one or two meter stick in a vertical position. Hang
the spring from one of the rod clamps, and then hang the 5 g weight pan
at the bottom of the spring. Carefully record the position yo
of the bottom weight pan.
2. Add 20 g to the weight pan and record the position y1 of the bottom weight pan. 3. Repeat the above point 2 few times and fill in the Table 1 |
Table 1
|
4. With a mass of 50 g on the weight pan, stretch the spring about 2 to 3 centimeters from its equilibrium position. Release the weight and measure and record the time for twenty complete vibrations. In counting the oscillations, count zero at the instant you start the time clock. Repeat this procedure with a vibration amplitude of 6 to 7 centimeters.
| 5. Measure and record the time for twenty complete vibrations
when each of 10, 20, 30, 40 .. 70 g are added to the weight pan.
In this procedure, the time measurement should be done at least 3 or 4
times, preferably with each member of a group doing the timing. Use
as small an amplitude as possible, no more than two centimeters.
6. Weigh the spring, i.e., determine its mass and record this in the data sheet. This may be done at the beginning of the experiment. |
Table 2
mass of spring = __________ [g] |
Data Analysis
Part 1. (graph 1)
1. Determine the spring constant in N/m.
a) graph1 your experimental points
(table 1= raw data): weight [N] vs elongation [m]
b) plot regression line on the
same graph
c) find spring constant from regression
line.
Part 2. (graph2)
2. Plot a function
(1) in the range from 10g to 100 g every 1 g. Use the value of spring
constant from Part 1.
3. Plot a function (3) (Additional Theory)
in the same range and with the same spring constant on the same graph.
4. Plot your experimental data: periodi vs
oscillating massi (table 2 = raw data) on the same
graph.
(graph3)
5. Calculate the experimental errors: 100*[ experimental
valuei - Theory(massi )]/Theory(massi
).
Use the theoretical function (3) from Additional
Theory supplement below.
6. Plot a separate graph 3 of Errori
vs massi
.
The formula for the period of oscillation given above neglects the weight
of the spring. A more exact formula for the period is given by
| (3) |  |
| where Ms is the mass of the spring. If we solve this equation for m, we obtain | |
| (4) |  |
Using MCAD
Find a spring constant k
1. Put the experimental data into two matrices: position
and mass. The units can be changed multiplying the whole matrix
once by appropriate constant. Any constant value can be added (subtracted)
once to the whole matrix and all the matrix elements will include the constant.
Stay
with SI units. Make a graph 1:
position vs mass
2. Calculate the slope of regression line just
typing k:= slope(matrix1, matrix2) where matrix1 is on x axis.
[F = - k s means that k is the slope of
line: F vs. s]
Oscillations:
1. Type a function (equation 1) T1(m)
where m is the oscillating mass. To have the theoretical values
comparable with the experimental ones use
m := 0.01, 0.011 .. 0.1
[m in kg]. Do not use the same letter symbols
for matrices as well as function.
2. Type
a function (equation 3) T2(m) where m is the
oscillating mass.
3. Put your experimental (from procedure 4) data
into two matrices - one for mass, and one for time for 20 full oscillations.
Define the counter [e.g. i] for your experimental data.
i := 0, 1 .. ? (by default matrix elements are numbered
starting from 0)
The units can be changed multiplying the whole matrix once by appropriate
constant. Any constant value can be added once to the whole matrix
and all the elements will include the constant (see 1).
4. Make a graph 2 of your experimental
period vs experimental oscillating mass and calculated function
T1(m) and T2(m) that calculates oscillating period vs oscillating
mass for any (imaginable) masses m.
5. Calculate the error %
[(experimental periodi)-T2(experimental massi)]/T2(experimental
massi) * 100%
6. Make a graph 3: error% vs experimental
mass.
7. Plot a graph 4 of experimental mass vs experimental
period squared:
8. Plot a regression line on the graph 4. To calculate
regression you have to calculate first matrix eg. Tsquarei
:=
Ti 2
and use Tsqare and mass matrices to make regression. [Use slope(Tsquare,mass)
and intercept(Tsquare,mass)]
Discussion
1. Analyze each of the reason for errors:
a) measuring position of pan in procedure 1 and 3
b) timing the oscillations (start and stop)
c) equation (3): Is it correct?
What was the condition when derived? Do you think it should be corrected
in some way?
d) air resistance
Do they introduce systematic error (always positive or always negative
== larger periodi or shorter periodi ) or just random
one?
2. Make suggestions how to improve the experiment
to make errors less significant.
3. If there are significant discrepancies between
the two curves [experimental and theoretical T2(m)] in your graph 2,
give reasons for these discrepancies.
4. Make comment to graph 4 on the slope and intercept
. Explain why your graph should be a straight line.
5. Make CONCLUSION.