Consider the frictionless pendulum shown below with mass, m$=3\mathrm{.}3$ kg$,$ length, l $=0\mathrm{.}75$m and it is on the earth$$s surface where ge$=9\mathrm{.}81$ m$/$s$2.$ Problem No$.1$

Consider the frictionless pendulum shown below with mass, $m=3\mathrm{.}3kg,$ length, $I=0\mathrm{.}75m$

and it is on the earth's surface where $g_e=9\mathrm{.}81\frac{m}{s^2}.$

A$)$ Assume small angles and compute the natural frequency in radians per second and

natural period of the pendulum in seconds.

B$)$ If a tangential initial velocity of $0\mathrm{.}25\frac{m}{s}$ is imparted when $=1$ determine the

amplitude of the motion A and the phase angle $.$ Does the assumption of small angles

hold?

C$)$ Plot the angular displacement and velocity response for the first $10$ seconds.

D$)$ Determine the magnitude of the peak restoring moment given the initial conditions in

B$).($hint: the restoring moment can be found using a freebody taken when theta is at its

peak and summing the moments about $O)$

E$)$ Repeat part B$)$ If the pendulum is on the moon's surface where $g_m=1\mathrm{.}625\frac{m}{s^2}$

Problem No $2.$

If a pendulum with length $I=12$ inches is on earth and has an initial position of ${}_0=2$ what is the

maximum angular velocity, ${}_0,$ in rad$/$s that can be imparted on the system to limit the

maximum angle to be less than $10.($note: the equations can be setup and trial and error used

for the solution or you can use the solve command$)$

A$)$ Assume small angles and compute the natural frequency in radians per second and natural period of the pendulum in seconds.

B$)$ If a tangential initial velocity of $0\mathrm{.}25$ m$/$s is imparted when $=1$o determine the amplitude of the motion A and the phase angle$.$ Does the assumption of small angles hold?

C$)$ Plot the angular displacement and velocity response for the first $10$ seconds.

D$)$ Determine the magnitude of the peak restoring moment given the initial conditions in B$).($hint: the restoring moment can be found using a freebody taken when theta is at its peak and summing the moments about O$)$

E$)$ Repeat part B$)$ If the pendulum is on the moon$$s surface where gm$=1\mathrm{.}625$ m$/$s$2$