Chapter 1: Q5.3-20E (page 1)

In Project C of Chapter 4, it was shown that the simple pendulum equation $θ'' (t) + sinθ (t) = 0$ has periodic solutions when the initial displacement and velocity show that the period of the solution may depend on the initial conditions by using the vectorized Runge–Kutta algorithm with h= 0.02 to approximate the solutions to the simple pendulum problem on
[0, 4] for the initial conditions:
localid="1664100454791" $(a) θ (0) = 0 . 1, θ' (0) = 0 (b) θ (0) = 0 . 5, θ' (0) = 0 (c) θ (0) = 1 . 0, θ' (0) = 0$
[Hint: Approximate the length of time it takes to reach].

Short Answer

Expert verified

(a) The period is 6.28.

(b) The period is 6.4.

Step by step solution

Transform the equation

The equation is $θ'' (t) + sinθ (t) = 0$

The system can be written as:

$x_{1} = θ x_{2} = θ' = x'_{1}$

The transform equation is:

role="math" localid="1664100052779" $x'_{1} = x_{2} x'_{2} = - {sinx}_{1}$

The initial conditions are:

$x_{1} (0) = θ (0) = 0 . 1, 0.5, 1 x_{2} (0) = θ' (0) = 0, 0, 0$

Apply Runge –Kutta method.

For the solution, apply the Runge-Kutta method in Matlab. For h=0.02

h	$θ$ for part a	$θ$ for part b	$θ$ for part c
0	0.1	0.5	1
0.02	0.0999	0.4999	0.99983
0.04	0.09992	0.4996	0.99932
0.06	0.0998	0.49913	0.9984
0.2	0.980	0.492251	0.9863
1	0.054	0.2777	0.6000
2	-0.0415	-0.194	-0.306

Continuing this procedure for value h=4 and getting the result -0.0655,-0.3507,-0.82576, respectively.

From the table and all the results, one gets the first time $- θ (0)$ for part (a) is 3.14.this will be half a period. So, the period of $θ$ is about 6.28.

The first time $- θ (0)$ for part (b) is 3.2.and the period of $θ$ is about 6.4.

The first time $- θ (0)$ for part (c) is 3.34, and the period of $θ$ are about 6.68.

Thus, this is the required result.