A system is rotated and solved. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

The region R is bounded by the graphs of x-2y = 3 and x=y^2. Set up (but not evaluate) the integral that gives the volume of the solid obtained by rotating R around the line x=-1.

For what initial velocity and direction of the puck will it (the puck) appear motionless when viewed from above (ie the motionless reference frame).
See attached file for full problem description.
Note for clarification. The initial position refers to the puck is: (x = -.5R, y = 0)

A cylinder of mass M and radius R is rotated in a uniform V grove (which looks like a rectangular block with an 45-45-90 degree triangular prism cut from the center- that the cylinder rotates in) The cylinder has a constant angular velocity w and a coefficient of friction u with each surface. What torque must be applied to the c

Rotating Space Stations. One problem for human living in outer space is that they are apparently weightless. One way around this problem is to design a space station that spins about its center at a constant rate. This creates "artificial gravity" at the outside of the station.
A) If the diameter of the space station is 8

Assume the life of a rotating item follows a Weibull distribution with the shape parameter (or slope) = 2 and the scale parameter = 10,000 hours.
a. Determine the probability that the rotating item lasts at least 8000 hours.
b. Determine the mean time until failure of the item.
c. If 10 rotating items are used in a larger com

A string is wrapped several times around the rim of a small hoop with radius r and mass m . The free end of the string is held in place and the hoop is released from rest. Calculate the angular speed of the rotating hoop after it has descended a distance, h. (See attached file for diagram and figures)

A wheel with mass less spokes has mass 1 kg and radius 10 cm and is mounted on one end of a mass less axle as figure.
The axle rests on a pivot at a point 16 cm from the mounting point and 10 cm from the wheel.
At the other end, a mass of 0.8 kg is attached.
The wheel spins at an angular frequency of 10 rad/s.
What is the