Question

Question 2 The pendulum shown in Figure 2 consists of a concentrated mass m attached to a rod whose mass is small compared to m. The rod's length is L. The equation of motion for this pendulum is Suppose that L 1 m and g 9.81 m/s2. Use MATLAB to solve this equation using symbolic and numerical techniques for, θ(t) for two cases: , θ(0)-0.5 rad and, θ(0)-0.8 rad. In both cases 0(0) 0. Figure 2- A pendulum [3 marks]

Answer 1

For θ(0)= 0.5

*******************Matlab Sxcript**********************

ode_fun=@(t,v) [v(2);-9.81*sin(v(1))]; %create anonymous function such that v1'=v2, v2'=(-g/L)*sin(v(1))
tspan=[0;10]; %find the solution for time t starting at t=0 till time t=10
initial_v=[0.5;0]; %initial condition is [v1(0);v2(0)]=[0.5;0]
[t,v]=ode45(ode_fun,tspan,initial_v); %apply ode45 function on the given system with given initial conditions
y=v(:,1); %the first column of v is v(1)=y whereas the second column is v(2)=y' which we don't need now
plot(t,y) %plot y vs t
title('Plot of solution of theta(y) vs time t') %Add title

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output ...........

Figure 1 File Edit View Insert Iools Desktop Window Help Plot of solution of theta(y) vs time t 0.5 0.4 0.3 0.2 0.1 -0.1 -0.2 -0.3 0.4 0.5 10

For θ(0)= 0.8

*****************Matlab Script**********************

ode_fun=@(t,v) [v(2);-9.81*sin(v(1))]; %create anonymous function such that v1'=v2, v2'=(-g/L)*sin(v(1))
tspan=[0;10]; %find the solution for time t starting at t=0 till time t=10
initial_v=[0.8;0]; %initial condition is [v1(0);v2(0)]=[0.8;0]
[t,v]=ode45(ode_fun,tspan,initial_v); %apply ode45 function on the given system with given initial conditions
y=v(:,1); %the first column of v is v(1)=y whereas the second column is v(2)=y' which we don't need now
plot(t,y) %plot y vs t
title('Plot of solution of theta(y) vs time t') %Add title

*****************************************************

output...................

Figure 1 File Edit View Insert Iools Desktop Window Help 0.8 Plot of solution of theta(y) vs time t 0.6 0.4 0.2 -0.2F 0.4H -0.6 0.8 10