Question 3: Simple Pendulum Figure 3 A simple pendulum consists of a mass m attached to a weightless string of length L (see Figure 3 above). The angle that the pendulum makes with the vertical during its swinging motion is given by 8. a) Using small-angle approximation, the equations of motion of the pendulum can be modelled using ODE Egn (3) where g is the acceleration due to gravity. i) State suitable initial conditions for this model. Using the initial conditions you stated in part a), solve the ordinary differential equation above (ODE Egn (3)] to find a particular solution for the motion. ii) What makes the model above unrealistic? UNIVERSITY GREENWICH International College b) We can make the simple pendulum model more realistic by adding a damping term to the ordinary differential equation as shown below Lê + yê + ge =0 ODE Egn (4) where y is the damping coefficient. Note that when y = 0, we go back to the no damping case. i) Write down the auxiliary equation for ODE Egn (4) above and find its roots. i) When considering the case of damping, assume the discriminant of the auxiliary equation is negative. Using the same initial conditions written down in part ai), find a particular solution for the motion of the pendulum in the damping case. c) Graph plotting Using suitable values of constants and a graphing software, plot the solution of your equations from part a) and part b) on the same graph. You must state your reasoning for your choice constants to obtain full marks. The graph should also show your axis clearly labelled, including any units.

Question 3: Simple Pendulum Figure 3 A simple pendulum consists of a mass m attached to a weightless string of length L (see Figure 3 above). The angle that the pendulum makes with the vertical during its swinging motion is given by 8. a) Using small-angle approximation, the equations of motion of the pendulum can be modelled using ODE Egn (3) where g is the acceleration due to gravity. i) State suitable initial conditions for this model. Using the initial conditions you stated in part a), solve the ordinary differential equation above (ODE Egn (3)] to find a particular solution for the motion. ii) What makes the model above unrealistic? UNIVERSITY GREENWICH International College b) We can make the simple pendulum model more realistic by adding a damping term to the ordinary differential equation as shown below Lê + yê + ge =0 ODE Egn (4) where y is the damping coefficient. Note that when y = 0, we go back to the no damping case. i) Write down the auxiliary equation for ODE Egn (4) above and find its roots. i) When considering the case of damping, assume the discriminant of the auxiliary equation is negative. Using the same initial conditions written down in part ai), find a particular solution for the motion of the pendulum in the damping case. c) Graph plotting Using suitable values of constants and a graphing software, plot the solution of your equations from part a) and part b) on the same graph. You must state your reasoning for your choice constants to obtain full marks. The graph should also show your axis clearly labelled, including any units.