AM 18 – Homework assignment #2:
pendulum experiment.
The first two parts of the experiment only involve measuring periods of oscillations with a watch.
1. We are interested in proving the following.
(a) The pendulum’s period of oscillation T is independent of the initial
amplitude θ0 of the oscillation, as long as θ0 is reasonably small. To
do this, we shall simply consider one of the two bobs, make it oscillate
starting at different amplitudes and time the period of oscillation by
allowing it to make many oscillations and dividing.
(b) The pendulum’s period of oscillation T is independent of the initial
amplitude and the mass m of the bob. Same as above, except that
we will compare the oscillation periods of two different bobs.
2. We now want to verify that the period of oscillation is proportional to the
square root of the length of the pendulum, i.e.:
√
T ∝ L.
In order to verify the above law, we shall measure the period of oscillation
of the pendulum (using the same bob) at different lengths, and then plot
our data on a graph (possibly using Matlab) and see that the curve we
get resembles the typical one of a square root.
3. We proved in class that the angle θ(t) and the velocity v(t) of a pendulum
follow the (implicit) law:
v 2 + 2k1 (1 − cos θ) = k2 ,
(1)
where k1, k2 are some constants. For small oscillations we may assume
that
θ2
cos θ ' 1 − ,
2
so that (1) becomes:
v 2 + k1 θ2 = k2 ,
or
v2
k1 θ 2
+
= 1.
k2
k2
(2)
An ellipse in the (x, y)-plane has the following equation:
y2
x2
+
=1
a2
b2
where a and b are, respectively, the x and y semiaxes. Therefore equation (2) indicates that θ and v should lie on an ellipse. We shall introduce
appropriate normalizations, so that this ellipse actually becomes a circle.
Here is how we shall proceed. Given a sequence of images, {I1 , . . . , In }, we
display them in Matlab and measure the pendulum’s horizontal position
in pixels {x1 , . . . , xn }. Replacing xi by xi − average(x• ) to make 0 the
mean value, then since the angles are small, this is a good approximation
of {c θ1 , . . . , c θn }, where c as an appropriate constant). Define:
M1 , max {|xi |}
i=1,...,n
xi + xi−1
,
i = 2, . . . , n
2M1
M2 , max {|xi − xi−1 |}
Xi ,
i=2,...,n
Vi ,
xi − xi−1
,
M2
i = 2, . . . , n
so that we get a set of points {(Xi , Vi )}i=2,...,n . We will plot these on
a plane (using Matlab) and verify that these points (roughly) lie on a
circle. In fact, if we consider a large number of points, we will be able to
observe that they slowly spiral towards the origin.
Here is a very simple Matlab program to perform these calculations:
n=6;
x=[x1, x2, x3, x4, x5, x6];
% number of samples
% enter samples here
X =(x(2:n)+x(1:(n-1)))/2;
% average between pairs of
% consecutive measurements
% normalize max swing to be 1
X = X/max(abs(x));
A = x(2:n)-x(1:(n-1));
% differences of consecutive
% measurements, i.e. speed
M2=max(abs(A));
V=A/M2;
% again normalize so max speed is 1
plot(X,V,’x’);
% plots graph
grid on;
title(’Pendulum experiment’);
axis(’square’);
xlabel(’X’); ylabel(’V’);
print -dps pendulum;
4. To see if we have ‘simple harmonic motion’ here, we want to put the
point (Xi , Vi ) in polar cooridnates. The angular variable is given by θi =
atan2(Xi , Vi ). Plot the θi against time, print this out and see how well
the points fit on a straight line.