Aparticle of mass m moves in the 2-d potential v=4wy+1 1/(2* +34) with no and 21 positive. (a) find the stable equilibria for the particle. (b) give the lagrangian appropriate for small oscillations about one of the equilibrium positions. (c) give the normal frequencies and modes of vibration corresponding to the equilibrium position in (b).

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)

The experiment done in lab is repeated, using a ball that has unknown mass m. you plot your data in the form of f 2 versus m/l, with f in rev/s, m in kg, and l in m. your data falls close to a straight line that has slope 3.19 m/(kg · s2). use g = 9.80 m/s2 and calculate the mass m of the ball.

You are doing a science experiment with a fahrenheit thermometer. your data must be in degrees celsius. if you measure a temperature of 125°f, what is this temperature in degrees celsius