Activity $1$:

Part $1$:

A support beam used in one of the designs for the theme park, is subjected to a free vibration along its length; emanating from two machines situated at opposite ends of the beam. The displacement caused by the vibrations can be modelled by the following equations:

$x_1=3\mathrm{.}5sin(100t+\frac{}{9})$

$x_2=5sin(100t-\frac{3}{5})$

You are required to determine the followings for the vibration of this beam:

i$.$ State the amplitude, phase angle, frequency, and periodic time of each of these waves.

ii$.$ When both machines are switched on$,$ how many seconds does it take for each machine to produce its maximum displacement?

iii. At what time does each vibration first reach a displacement of $-2mm?$

iv$.$ Use the compound angle formulae to expand $x_1$ and $x_2$ into the form.

Asin$100t+-$Bcos$100t,$ where A and $B$ are numbers to be found.

v$.$ Using your answers from part iv$,$ express $x_1+x_2$ in a similar form. Convert this expression into the equivalent form Rsin$(100t+)$

vi$.$ Using appropriate spread sheet software, copy and complete the following table of values:

$\backslash$table$[[t,0\mathrm{.}000,0\mathrm{.}002,0\mathrm{.}004,0\mathrm{.}006,0\mathrm{.}008,0\mathrm{.}010,0\mathrm{.}012,0\mathrm{.}014,0\mathrm{.}016,0\mathrm{.}018,0\mathrm{.}020]]$

$x_1$

$x_2$

vii. Plot the graphs of $x_1$ and $x_2$ on the same axes using any suitable computer package. Extend your table to include $x_1+x_2,x_1-x_2$ and plot this graph on the same axes as the previous two. State the amplitude and frequency of the new wave.

$12$

viii. Using your answers from parts v and vii, what conclusions can be drawn about $x_1+x_2$ and the two methods that were used to obtain this information and compare them together?

X$.$ Find the velocity and acceleration of the position function of a racer with the following function and calculating velocity and acceleration when $t=4s?$

vec$(r)(t)=(2t-3)vec(i)+(t^2+t+1)vec(j)$