4.A satellite is placed into a circular orbit around the planet Jupiter, which has a mass M = 1.90 x 1027 kg. (a) If the radius of the orbit is R (measured from the center of Jupiter), use Newton's laws to derive an
expression for the orbital velocity.
(b) If the satellite increases its orbital radius, how it would change the orbital velocity? Explain.
(c) If the radius of the orbit is R, derive an expression for the orbital period.
(d) The satellite rotation is synchronized with Jupiter's rotation. This requires an equatorial orbit whose
period equals Jupiter's rotation period of 9 hr 51 min = 3.55 x 104 s. Find the required orbital radius.

Industrial processes often require the damping of vibrations. consider a workbench that uses four large ideal springs in place of traditional legs, with each spring supporting one corner of the workbench and exhibiting a force constant of 47500 n/m. this workbench is used when creating products that involve highly unstable chemicals. according to specifications, the acceleration of the manufacturing equipment must be held to less than 22.3 m/s2 for the manufacturing process to proceed safely. the amplitude of vibrations are expected to remain under 0.622 mm throughout. given that the workbench and maufacturing equipment combine to 6.05 kg, find the maximum acceleration expected of the equipment and workbench. (ignore the mass of the springs.)