In the absence of friction, small free motions of the pendulum are governed by the equation where (t) is the angle of the pendulum inclination at time t and g is the acceleration due to the gravity. (a) Find the frequency of the oscillations of the pendulum. (b) How does the frequency change when the length of the rod doubles? (c) How does the frequency change when the mass of the object doubles? d) On the moon the acceleration due to the gravity is 6 times less than on the earth. How does the frequency change when the pendulum is on the moon?

The ultracentrifuge is an important tool for separating and analyzing proteins. because of the enormous centripetal accelerations, the centrifuge must be carefully balanced, with each sample matched by a sample of identical mass on the opposite side. any difference in the masses of opposing samples creates a net force on the shaft of the rotor, potentially leading to a catastrophic failure of the apparatus. suppose a scientist makes a slight error in sample preparation and one sample has a mass 10 mg larger than the opposing sample. if the samples are 12 cm from the axis of the rotor and the ultracentrifuge spins at 70,000 rpm, what is the magnitude of the net force on the rotor due to the unbalanced samples? ( be thorough on your answer)